Island Networks: Communication, Kinship, and Classification Structures in Oceania (Structural Analysis in the Social Sciences (No. 11))

In their previous book, Exchange in Oceania, anthropologist Per Hage and mathematician Frank Harary demonstrated that models from graph theory, a branch of pure mathematics, provide the essential basis for analyzing the great variety of exchange systems in Micronesian, Melanesian, and Polynesian societies. In this new book the authors extend these models and apply them to the analysis of communication, kinship, and classification structures in the island societies of Oceania, presenting the relevant topics from graph theory in a form accessible to the nonmathematical reader. The research problems include the formation of island empires, the social basis of dialect groups, the emergence of trade and political centers, the evolution and devolution of social stratification, the transformations of marriage and descent systems, the historical development of kinship terminologies, and the reconstruction of protosocieties. Island Networks is at once a unique and important contribution to Oceania studies, anthropology, and social network analysis in general.

Structural analysis in the social sciences Island networks

Structural analysis in the social sciences Mark Granovetter, editor Other books in the series: Ronald L. Breiger, ed., Social Mobility and Social Structure

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Mark S. Mizruchi and Michael Schwartz, eds., Intercorporate Relations: The Structural Analysis of Business Philippa Pattison, Algebraic Models for Social Networks Barry Wellman and S. D. Berkowitz, eds., Social Structures: A Network Approach Stanley Wasserman and Katherine Faust, Social Network Analysis: Methods and Applications Philippe Bourgois, In Search of Respect: Selling Crack in El Barrio Gary Herrigel, Industrial Constructions: The Sources of German Industrial Power

The series Structural Analysis in the Social Sciences presents

approaches that explain social behavior and institutions by reference to relations among such concrete entities as persons and organizations. This contrasts with at least four other popular strategies: (a) reductionist attempts to explain by a focus on individuals alone; (b) explanations stressing the causal primacy of such abstract concepts as ideas, values, mental harmonies, and cognitive maps (thus, "structuralism" on the Continent should be distinguished from structural analysis in the present sense); (c) technological and material determinism; (d) explanations using "variables" as the main analytic concepts (as in the "structural equation" models that dominated much of the sociology of the 1970s), where structure is that which connects variables rather than actual social entities. The social network approach is an important example of the strategy of structural analysis; the series also draws on social science theory and research that is not framed explicitly in network terms but stresses the importance of relations rather than the atomization of reductionism or the determinism of ideas, technology, or material conditions. Though the structural perspective has become extremely popular and influential in all the social sciences, it does not have a coherent identity, and no series yet pulls together such work under a single rubric. By bringing the achievements of structurally oriented scholars to a wider public, the Structural Analysis series hopes to encourage the use of this very fruitful approach.

Island networks Communication, kinship, and classification structures in Oceania Per Hage University of Utah

Frank Harary New Mexico State University and the University of Michigan

CAMBRIDGE UNIVERSITY PRESS

CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, Sao Paulo Cambridge University Press The Edinburgh Building, Cambridge CB2 2RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521552325 © Cambridge University Press 1996 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 1996 This digitally printed first paperback version 2006 A catalogue record for this publication is available from the British Library Library of Congress Cataloguing in Publication data Hage, Per, 1935Island networks : communication, kinship, and classification structures in Oceania / Per Hage, Frank Harary. p. cm. - (Structural analysis in the social sciences) Includes bibliographical references. ISBN0-521-55232-X 1. Ethnology — Oceania — Mathematical models. 2. Structural anthropology - Oceania. 3. Graph theory. 4. Kinship - Oceania Mathematical models. 5. Social networks - Oceania - Mathematical models. 6. Oceania - Social life and customs - Mathematical models. I. Harary, Frank. II. Title. III. Series. GN663.H37 1996 306'.099-dc20 95-31639 CIP ISBN-13 978-0-521-55232-5 hardback ISBN-10 0-521-55232-X hardback ISBN-13 978-0-521-03321-3 paperback ISBN-10 0-521 -03321 -7 paperback

To Claude Levi-Strauss

Islands have always gripped man's imagination. Ernest Sabatier, Astride the Equator The legitimacy of the comparative method does not rest on massive and superficial resemblances. Analysis has to take place on a level deep enough to allow us to discern, at the base of all social life, the simple features that combine into rudimentary systems, which may eventually become the stuff of more complex and more completely integrated systems with entirely new characteristics. Claude Levi-Strauss, The View from Afar

Contents

List of figures, tables, and maps Preface Acknowledgments 1 Island networks and graphs Graph theoretic models Geographical, linguistic, and anthropological terms

page ix xv xix 1 3 17

2 Trees E
22 22 30 35 43 45

3 The minimum spanning tree problem Dialect groups and marriage isolates in the Tuamotus The evolution of the Lakemban matanitu The Renfrew-Sterud method of close-proximity analysis On deconstructing a network

51 52 66 75 89

4 Search trees: I Independent discoveries of the conical clan Social stratification in Polynesia A structural model of the conical clan Prestige-good systems

90 92 101 107 116

5 Search trees: II The Marshallese conical clan The devolution of social organization in Nuclear Micronesia

125 126 142

viii

Contents Search trees and the organization of genealogical knowledge

162

6 Centrality Southern Lau, Fiji: "A natural trade area" Power centers in the Greater Lauan trade network Political and mythological centers in Ralik and Ratak Expeditions in Torres Strait On the position of Delos in the Archaic Aegean network Self-centered networks

165 166 174 178 180 194 201

7 Dominating sets Local domination in the Caroline Islands Alliance structures in the western Tuamotus Pottery monopolies in Melanesian trade networks

204 205 207 212

8 Digraphs Basic definitions Murdock's maze: The bilateral hypothesis of Proto-Malayo-Polynesian social organization Sibling classification and culture history in Island Oceania

218 219

231

9 Conclusion

262

References Index

269 289

222

Figures, tables, and maps

Figures 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.10 1.11 1.12 1.13 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 2.11 2.12 2.13

Alternative representations of a graph page 4 Equivalent representations of a rooted tree as (a) a graph 5 and (b) nested sets Two binary trees 5 A native model of social organization in Polynesia (from 6 E. and P. Beaglehole 1938) An electrical network N, its underlying graph G, and a 7 spanning tree T Boruvka's illustration of a minimum spanning tree 8 algorithm (from Graham and Hell 1985) Depth-first and breadth-first search trees 10 A breadth-first search of a graph 12 A graph to illustrate centrality concepts 13 One solution to the Five Queens Problem 14 Digraphs that are (a) weakly and (b) strongly connected 15 Asemilattice 16 The Austronesian family tree (from Blust 1990) 19 The 11 graphs with four nodes 23 A graph to illustrate adjacency 23 A graph, a subgraph, and a spanning subgraph 24 Two labeled graphs 24 A graph to illustrate walks, trails, paths, and cycles 25 Four of the 11 graphs with four nodes 26 A graph to illustrate cutnodes and bridges 26 Two isomorphic graphs _ 26 A graph G and its complement G 27 Three bigraphs 27 A rooted graph and a doubly rooted graph 27 The product of two graphs 28 Planar, plane, and nonplanar graphs 28 IX

x 2.14 2.15 2.16 2.17 2.18 2.19 2.20 2.21 2.22 2.23 2.24 2.25 2.26 2.27 2.28 2.29 2.30 2.31 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11

Figures, tables, and maps The digraphs with three nodes and three arcs The 11 trees with seven nodes The rooted trees with four nodes Yapese communication structures (tha6): (a) the outerisland tribute system; (b) Gacpar political networks Hocart's (1929) model of "perpetual dichotomy" in Fijian social organization Social organization in the Lau Islands (from Hocart 1929) Eyde's (1983) model of "recursive dualism" in the Admiralty Islands J. J. Fox's (1989) model of "recursive complementarity" in eastern Indonesian exchange Three rooted plane trees Three full binary trees The binary trees with four nodes The twin binary trees with seven nodes An out-tree and its dual in-tree An in-tree model of 'ati affiliations in the Tuamotus (from Ottino 1972) A Southeast Solomons-Vanuatu-New Caledonia Lapita network (from Hunt 1988) A graph and all its spanning trees A graph G, a spanning tree T, and the set of independent cycles obtained from T Illustrations of the alpha index of planar graphs (from Haggett 1967) A minimum spanning tree (MST) of a network N Generating an MST using Kruskal's algorithm The MST of the Tuamotus network, clustered to show dialect groups Generating an MST using Boruvka's algorithm Modeling the evolution of the Lakemba matanitu with Boruvka's MST algorithm Illustration of the Renfrew-Sterud method of double-link close-proximity analysis applied to a series of Aurignacian burins Construction of the close-proximity graph in Fig. 3.6 using Kruskal's algorithm Generating an MST using Prim's algorithm Illustration of a matrix method for using Prim's MST algorithm (from Wilson and Watkins 1990) A network with two MSTs Renfrew and Sterud's (1969) close-proximity structure

29 29 30 32 36 36 38 39 40 41 42 42 43 45 46 47 48 49 56 57 59 73 74 76 77 79 80 81 82

Figures, tables, and maps

3.12 3.13 3.14 3.15 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10 4.11 4.12 5.1 5.2 5.3 5.4 5.5 5.6 6.1 6.2 6.3 6.4 6.5

for the Early Cycladic cemeteries, using presence-absence similarity coefficients Two different MSTs generated by the Cycladic cemetery matrix The universal subtree of the MST of the Cycladic cemetery matrix The graph in Fig. 3.13 relabeled with island names An MST of linked Lapita pottery motifs, based on data in Green (1978, 1991) Sahlins's (1958) model of the conical clan (ramage) in Polynesia White's (1959) model of the conical clan The graph Ky drawn as a rooted labeled plane graph The labeled trees with four nodes A breadth-first search of T2 Goody's (1966) implicit BFST model of the "inclusive system of agnatic hereditary succession" A depth-first search of T2 The details of a depth-first search A left-to-right DFST model of the conical clan in Polynesia, where rank is defined by primogeniture A right-to-left DFST model of the conical clan in Kachin gumsa society, where rank is defined by ultimogeniture A graph of the protohistoric long-distance exchange network of the Tongan maritime empire (from Kirch 1988a) A prestige-goods cycle linking Tonga, Fiji, and Samoa Mason's (1954) coding of rank in the Bikini clan A product graph of intra- and interclass marriage and status of the children in Marshallese society A graph of the Ralik-Ratak voyaging network Wife-giving and tribute relations between noble and royal Marshallese lineages A schematic map showing languages in Micronesia (from Bender 1971) The "ideal scheme for reciting genealogies" among the Tory Islanders (from R. Fox 1978) Illustrations of degree, closeness, and betweenness centrality in a graph The graph of the southern Lau trade network A graph of the Greater Lauan trade network A graph and the eccentricities of nodes A graph in which the center and the median are disjoint

xi

85 86 86 88 102 109 111 112 112 113 113 114 114 115 118 123 128 133 138 141 144 164 169 172 176 179 179

xii 6.6 6.1 6.8 6.9 6.10 6.11 6.12 6.13 6.14 6.15 6.16 7.1 7.2 7.3 7.4 7.5 7.6 8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8 8.9

Figures, tables, and maps A rooted graph of the western Torres Strait-Papua New Guinea canoe-purchasing trade routes implicit in Harris (1979) A rooted graph of the western and central Torres StraitPapua New Guinea canoe-purchasing trade routes, based on Haddon (1904, 1935) A doubly rooted graph of the western, central, and eastern Torres Strait-Papua New Guinea canoe-purchasing trade routes, based on Haddon (1904, 1935) A graph of the Torres Strait-Papuan coast-Cape York trade network A graph and its adjacency, reachability, and distance matrices A network N and its cost matrix C The Archaic Ionian city-state network, based on J. L. Davis (1982) The four automorphisms of the graph K4 - e Three node-symmetric graphs A self-centered graph that is not node-symmetric A nonregular self-median graph (from Sabidussi 1966) A graph to illustrate dominating sets A graph of the western Carolines voyaging network (from Hage and Harary 1991) A graph of the overnight-voyaging network in the Vahitu, Tapuhoe-Tauro, and Parata districts of the western Tuamotus The kula ring (after Irwin 1983) The square G2 of a graph G The graph of the Mailu network (from Irwin 197r4) A digraph to illustrate the classification of nodes A digraph to illustrate walks Digraphs to illustrate connectedness categories Illustrations of converse digraphs A BFST of Murdock's (1949) maze with Normal Hawaiian as the root A BFST of Murdock's (1949) maze with Normal Iroquois as the root Nerlove and Romney's (1967) ideal types of sibling terminologies Polynesian sibling terminologies identified by Firth (1970) A graph of Epling, Kirk, and Boyd's (1973) "uppersemilattice of [Polynesian] sibling terminologies showing the two reconstructed evolutionary chains"

186 187 190 192 195 196 199 202 202 203 203 205 206 211 214 215 216 219 220 221 222 229 230 235 237 239

Figures, tables, and maps 8.10 8.11 8.12 8.13 8.14 8.15

8.16 8.17 8.18 8.19 8.20 8.21 8.22

A digraph of Clark's (1975) model of the evolution of Polynesian sibling terminology A graph of Marshall's (1984) "chain of structural patterns of sibling classification in Island Oceania" Marshall's (1984) "developmental core of structural patterns of sibling classification in Island Oceania" Marshall's (1984) graph of "relationships among structural patterns of sibling classification in Island Oceania" Marshall's (1984) digraph of "probable genetic relationships among the eleven structural patterns of sibling classification in Island Oceania" Marshall's (1984) digraphs of "probable developmental sequences of major structural patterns of sibling classification in Polynesia, Micronesia, and Melanesia" Two depictions of the same rooted tree A directed path (a) Lattice, (b) semilattice, (c) oriented graph A semilattice of the evolution of Proto-Oceanic (POC) sibling terminologies, based on Milke (1938) Marshall's (1984) model of the evolution of Nuclear Micronesian (NM) sibling terminologies A lattice model of the evolution of Nuclear Micronesian (NM) sibling terminologies Lexical relationships in the evolution of Trukese sibling terminology

xiii 242 243 244 245 246 247

251 252 253 254 257 258 259

Tables 3.1 3.2 6.1 6.2 6.3 6.4

Population of the Tuamotus in 1951 Presence-absence similarity coefficient matrix of Early Cycladic cemeteries, based on Renfrew and Sterud (1969) and Renfrew (1972) Relative centrality of islands in the southern Lau trade network Relative centrality of islands in the Greater Lauan network Estimated pre-European (ca. 1840) populations and population densities of the western Torres Strait islands The position of the western Torres Strait islands in the canoe trade with Papua New Guinea

63 84 173 177 184 188

xiv 6.5 6.6 8.1 8.2

Figures, tables, and maps Relative centrality of locations in the Greater Torres Strait-Papuan coast-Cape York trade network Relative centrality of city-states in the Archaic Ionian network A digraph (list of arcs) of Murdock's (1949) evolutionary model of social organization Murdock's (1949) classification of Malayo-Polynesian societies

193 200 227 228

Maps 1.1 3.1 3.2 6.1

The distribution of the Austronesian and Oceanic languages (from Bellwood 1978) The Tuamotu Archipelago (from Emory 1934) The Fiji Islands (from Thompson 1940) Torres Strait (adapted from Beckett 1987)

18 54 61 182

Preface

This book is the third work in a comprehensive program of research on applications of graph theory to anthropology. Graph theory is an explosively developing branch of pure mathematics with increasingly important applications to many fields, including architecture, biology, chemistry, computer science, cognitive science, economics, geography, and operations research. It is our belief that anthropology belongs with this company of subjects. Our aims are (1) to solve certain theoretical and methodological problems in anthropology by using the concepts, theorems, and techniques of graph theory; (2) to provide a common framework for structural analysis by demonstrating the applicability of graph theory to a wide spectrum of social and cultural phenomena; (3) to promote connections between various areas of anthropology and between anthropology and other disciplines in which graph theoretic modeling has proven useful; (4) to preserve continuity with the historical tradition of structural analysis in anthropology; and (5) to make graph theoretic models accessible to all structurally minded anthropologists and other social scientists. In our first book, Structural Models in Anthropology (Hage and Harary 1983), we presented graph theory as a family of models for the analysis of social, symbolic, and cognitive relations. We used graphs, digraphs, and networks, together with their associated matrices, to study such diverse topics as mediation and power in exchange systems, reachability in social networks, efficiency in cognitive schemata, and productivity in subsistence modes. We exploited duality laws for graphs and the interaction between graphs and groups to analyze transformations and permutations in myths and symbolic systems. Much of the inspiration for that book, as for all of our research, came from Claude LeviStrauss's (1949, 1962) theories, which focus on the logical, combinatorial, and isomorphic properties of kinship and classification systems, prefiguring the application of finite mathematics to anthropology. In our second book, Exchange in Oceania (Hage and Harary 1991), we extended the graph theoretic analysis in Structural Models in AnXV

xvi

Preface

thropology to provide an essential basis for the description, quantification, simulation, enumeration, and notation of the great variety of exchange systems found in Polynesian, Micronesian, and Melanesian societies. We used bipartite graphs and hamiltonian digraphs and networks to elucidate the cyclic structure of marriage and ceremonial exchange systems, and markov chains to simulate network flows. We introduced the concept of sex duality in graphs to study systematic variation in kinship structures and used binary operations on graphs, and group theory, to reveal the underlying structure of anatomical and physiological beliefs associated with different types of exchange structures. The theory of relations, which is coextensive with graph theory, provided a means for analyzing the higher-order logical structures implicit in an array of exchange and communication networks. The present volume introduces a set of graph theoretic models for the study of communication, kinship, and classification networks in Island Oceania. The research problems concern the formation of overseas empires, the social basis of dialect divisions, the emergence of trade and political centers, the evolution and devolution of social stratification, the transformation of marriage and descent systems, the replication of ideological systems, the historical development of kinship terminologies, and the reconstruction of protosocieties. The graph theoretic models essential for the study of these problems are six in number: (1) Trees, including rooted trees and binary trees and the spanning tree of a graph, describe the basic anatomy of kinship and communication networks and taxonomic structures. (2) Minimum spanning tree algorithms provide methods for analyzing clustering and classification in numerical networks and provide models for simulating processes of network growth. (3) Search trees serve as models of stratified descent groups and are the basis of the shortest-path algorithm used in the exploration of evolutionary mazes. (4) Centrality concepts define the different senses of advantageous location in voyaging and trade networks. (5) Dominating sets describe distributional aspects of economic and political power. (6) Digraphs, including semilattices, articulate the underlying structure of evolutionary theories of social organization based on genetic reconstructions as well as formal typologies. Our study broadens the scope of Oceanic anthropology in several important respects: (1) We increase the range of theoretical problems that can be formulated and solved as network problems. The definition of central location in a trade network is obviously a network problem, but so is the derivation of a kinship terminology from a prototype. (2) We provide a common framework for network analysis in anthropology, and in certain types of research in linguistics and archaeology, by showing that all three fields can advance in parallel through the application

Preface

xvii

of common graph theoretic models. For example, minimum spanning tree algorithms are equally useful for modeling the evolution of political networks, describing the breakup of archipelagoes into language groups and marriage isolates, and constructing pottery-design networks. (3) In contrast to most research on kinship in Oceania after the time of W. H. R. Rivers (1914a), we give as much weight to marriage alliance as we do to descent groups. Thus in our comparative analysis of social organization in Micronesia we discuss changes from elementary to complex and semicomplex marriage systems as well as shifts from matrilineal to double and patrilineal descent. (4) We exploit and promote connections between Oceanic and Indonesian anthropology, two fields that will inevitably become part of the larger field of Austronesian anthropology. An Indonesianist perspective is implicit in our treatment of Nuclear Micronesia as a "field of ethnological study" and in our support for the idea that much of the Austronesian world represents an eastward extension of Levi-Strauss's (1949) Sino-Tibetan axis of generalized exchange. (5) Finally, we restore part of the intellectual tradition of Oceanic anthropology by integrating into our account the important but often overlooked contributions of earlier scholars such as Paul Kirchhoff, Edward Winslow Gifford, Wilhelm Milke, and Leonard Mason. Kirchhoffs (1955) discovery of the conical clan, Gifford's (1929) analysis of descent and marriage alliance in Tonga, Milke's (1938) reconstruction of Proto-Oceanic sibling terms, and Mason's (1947, 1954) analysis of stratification in the Marshall Islands in Micronesia provide an essential basis for comparative studies of social organization and social networks in Oceania. In a classic paper, Barnes (1972:5) distinguished between analytical as opposed to metaphorical uses of the network concept. He observed that "a few simple notions taken from graph theory have proved useful in the analysis of social networks, but at present the supply of mathematical tools available far outstrips the supply of social data to which the tools might be applied." Although this statement is still true, we will show that the applicability of graph theory to real-world networks is far greater than commonly imagined. The applications in this book are highly varied, and the interested reader will no doubt discover analogues to every research problem we present. The entire work is therefore intended as a general contribution to network analysis in anthropology. With each book we have expanded the range of empirical structures amenable to graph theoretic analysis, as well as the repertoire of graph theoretic and network models for studying them. Thus in Structural Models in Anthropology we described abstract trees in a section of the chapter on undirected graphs and used them to analyze mnemonic

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structures. In Exchange in Oceania we briefly introduced minimum spanning trees in the chapter on matrices, noting their potential application to simulating the evolution of exchange networks. In this book we devote four chapters to trees, minimum spanning trees, and search trees, giving applications to classification systems, kinship networks, the evolution and devolution of social and linguistic networks, and the structure of stratified descent groups. Clearly, it is possible to write an entire volume just on interesting and useful applications of trees to anthropology. There is a parallel here with the second author's book, Graph Theory•, published in 1969, which became in 1978-9 the fifth most cited reference in the research literature of mathematics. Virtually every section of every chapter of that book has become a special field of research and is now the subject of a separate book. Phonetic note: The spelling of proper names and indigenous terms sometimes varies according to author and publisher (e.g., Tui Tonga vs. Tu'i Tonga, Lakeba vs. Lakemba, ldti vs. *ati etc.).

Acknowledgments

We wish to express our warm thanks to a number of individuals for their contributions and assistance. John Barnes, Ward H. Goodenough, and William Alkire in anthropology, Robert Blust in linguistics, and Fred Buckley in mathematics made particularly helpful and encouraging comments on various chapters. We also thank N. J. Allen, Jeremy Beckett, Peter Bellwood, Terry Hunt, Dell Hymes, David Jenkins, David Klein, David Lawrence, Mona Letourneau, Leonard Mason, and A. Kimball Romney. Bojka Milicic provided invaluable research assistance and contributed to discussions of many topics in the book. Brent James gave generously of his time and did the computer work. We are especially grateful to Ursula Hanly, who again transformed a handwritten manuscript into an impeccable typescript and drew the graceful and beautiful graph diagrams. The first author thanks Andrea Morguloff Hage for generous and enthusiastic support. For permission to reproduce the following figures, maps, and table, we thank: The Bishop Museum Press, Honolulu, for Fig. 1.4, from E. and P. Beaglehole, Ethnology of Pukapuka (1938); Figs. 2.18-19, from A. M. Hocart, Lau Islands, Fiji (1929); Map 3.1, from K. P. Emory, Tuamotuan Stone Structures (1934), and Map 3.2, from L. Thompson, Southern Lau, Fiji: An ethnography (1940); the Institute of Electrical and Electronics Engineers, Inc., for Fig. 1.6, from R. L. Graham and P. Hell, "On the History of the Minimum Spanning Tree Problem," Annals of the History of Computing (1985); the School of Oriental and African Studies, University of London, for Fig. 1.13, from R. Blust, "Three Recurrent Changes in Oceanic Languages," in J. H. C. S. Davidson (ed.), Pacific Island Languages (1990); P. Bellwood for Map 1.1, from Man's Conquest of the Pacific (Oxford University Press, 1979); the Musee de l'Homme for Fig. 2.20, from D. B. Eyde, "Recursive Dualism in the Admiralty Islands," Journal de la Societe des Oceanistes, 39 (1983); the University of Michigan Press for Fig. 2.21, from J. J. Fox, "Category and Complement: Binary ideologies and the organization of dualism in eastern Indonesia," in D. Maybury-Lewis and U. Almagor XIX

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(eds.), The Attraction of Opposites (1989); Cujas for Fig. 2.27, from P. Ottino, Rangiroa: Parente etendue, residence et terres dans un atoll Polynesien (1972); the Burke Museum for Fig. 2.28, from T. L. Hunt, "Graph Theoretic Models for Lapita Exchange," in P. V. Kirch and T. L. Hunt (eds.), Archaeology of the Lapita Cultural Complex: A critical review (1988), and Fig. 4.11, from P. V. Kirch, Niautoputapu: The prehistory of a Polynesian chiefdom (1988); Methuen and Co. for Fig. 2.31, from P. Haggett, "Network Models in Geography," in R. J. Chorley and P. Haggett (eds.), Models in Geography (1967); the Society for American Archaeology for Figs. 3.6-7 and 3.12, from C. Renfrew and G. Sterud, "Close-Proximity Analysis: A rapid method for the ordering of archaeological materials," American Antiquity, 34 (1969); John Wiley and Sons for Fig. 3.10, from R. J. Wilson and J. J. Watkins, Graphs: An introductory approach (1990); the University of Washington Press for Fig. 4.1, from M. D. Sahlins, Social Stratification in Polynesia (1958); McGraw Hill for Fig. 4.2, from L. A. White, The Evolution of Culture (1959); Cambridge University Press for Fig. 4.6, from J. Goody, Succession to High Office (1966), Fig. 5.6, from R. Fox, The Tory Islanders (1978); and Map 6.1, from J. Beckett, Torres Strait Islanders (1987); L. Mason for Fig. 5.1, from "Relocation of the Bikini Marshallese: A study in group migration," Ph.D. dissertation, Yale University (1954); Mouton de Gruyter for Fig. 5.5, from B. W. Bender, "Micronesian Languages," in T. A. Sebeok (ed.), Current Trends in Linguistics (1971); Academic Press for Table 6.3, from D. R. Harris, "Foragers and Farmers in the Western Torres Strait Islands: An historical analysis of economic, demographic, and spatial differentiation," in P. C. Burnham and R. F. Ellen (eds.), Social and Ecological Systems (1979); Mankind for Fig. 7.6, from G. J. Irwin, "The Emergence of a Central Place in Coastal Papuan Prehistory: A theoretical approach," Mankind, 9 (1974); and the University of Chicago Press for Figs. 8.13-15, from M. Marshall, "Structural Patterns of Sibling Classification in Island Oceania: Implications for culture history," Current Anthropology, 25 (1984).

Island networks and graphs

In the course of transforming verbal propositions into images many things are made explicit that were previously implicit and hidden. Herbert A. Simon, Models of My Life

Oceanists have increasingly come to recognize the limitations of the laboratory analogy that treats island societies as isolated experiments in adaptive radiation. Reconstructions of regional exchange systems (Hage and Harary 1991), archaeological evidence of sustained inter-island contacts (Kirch 1988a), firsthand accounts of traditional voyaging techniques (Lewis 1972), and the evident contradiction between neoevolutionist assumptions and the facts of Oceanic ethnography and prehistory (Friedman 1981) conduce to a network perspective that views island societies as elements of communication systems. Most islands in the Pacific are, in fact, distributed in groups, and most island societies are, or once were, connected to other island societies - as colonists, trade partners, tributaries, allies, wife-givers, and in various other ways. In acknowledging the importance of these connections many researchers, including anthropologists, archaeologists, and linguists, are now using or recommending the application of network concepts to answer a range of fundamental questions concerning 1. the settlement of island groups (Levison, Ward, and Webb 1973; Ward, Webb, and Levison 1976; Green 1979; Kirch 1988a; Irwin 1992); 2. the location of trade centers (Irwin 1974, 1978, 1983; Kirch 1988b; Hunt 1988); 3. the development of social stratification and social complexity (Reid 1977; Friedman 1981; Kirch 1984a; Lilley 1985; Graves and Hunt 1990); 4. the differentiation of cultural complexes (Green 1978); 1

2

Island networks 5. the diversification of dialects and languages (Pawley and Green 1984; Marck 1986); 6. the distribution of physical and cultural traits (Terrell 1986); 7. the selection of subsistence practices (Harris 1979); 8. the evolution of kinship structures (Epling, Kirk, and Boyd 1973; Marshall 1984).

It is clear that network analysis can lead to the solution of many longstanding problems and open up entirely new lines of inquiry in Oceania studies, provided that the handful of technical terms and structural metaphors now in use are replaced by a full and consistent set of mathematical concepts that are all part of some underlying model. Without such a model, many research problems cannot be correctly formulated or even imagined. Among all mathematical models, those from graph theory are most naturally suited for network analysis (Harary, Norman, and Cartwright 1965; Harary 1969; Hage 1979a; Hage and Harary 1983, 1991; Buckley and Harary 1990). The models, called graphs, are topological systems whose abstract properties define the significant structural features of all networks. For island networks these features include their distances, reachability, and connectedness and hence their centers, sources, cycles, orders, partitions, and spanning substructures. Graphs have the intuitive appeal of iconicity and the computational advantage of matrix methods. Graph theory contains theorems and algorithms that permit deductive approaches and precise and efficient solutions to numerous structural problems. Our purpose is to provide a set of graph theoretic models for studying the structure and formation of island networks and certain of their kinship armatures. We show how island networks are internally connected by a variety of social, cultural, and linguistic relations, and how they are partitioned into subgroups and organized into hierarchies. We introduce elementary techniques for simulating processes of network growth, where historical data are lacking, and we clarify and extend the application of graph theoretic models to the study of culture history. We provide a large repertory of concepts for analyzing the effects of network location on the economic and political status of island communities, and we show how complex networks can be simply represented to reveal essential connections - how to see the trees obscured by the forests. Our presentation is organized in a logical, graph theoretic manner. As in our previous books, Structural Models in Anthropology and Exchange in Oceania, the account is self-contained and readily accessible to the nonmathematical reader. We now give informal illustrations of our models together with a preview of the research presented later in this book.

Island networks and graphs

3

Graph theoretic models We use six basic graph theoretic models in the analysis of island networks and associated kinship structures: trees, minimum spanning trees, search trees, centrality concepts, dominating sets, and digraphs. The models, and hence our illustrations, are suggested by applications of graph theory to network analysis in several different fields: computer science, operations research, recreational mathematics, social networks, and transportation geography. Graphs

A graph G consists of a finite set V of nodes together with a set E of edges where each edge joins two nodes. This is illustrated by the diagram in Fig. 1.1a. For purposes of algebraic manipulation a graph can also be represented by an adjacency matrix, denoted A(G), in which each node has a row and a column and in which the entries in the cells are either 1 or 0 to show the presence or absence of an edge joining a pair of nodes (see Fig. 1.1b). A third way to represent a graph, useful for certain algorithmic procedures and in computer implementations, is to list its edges, as shown in Fig. 1.1c. Typically, a real-world problem involving structure is modeled by a graph and then solved through the application of concepts, theorems, and algorithms from graph theory. Trees Trees are the simplest of all graph theoretic models. In Chapter 2 we present a theorem that gives several structurally equivalent properties of a graph, each serving to characterize a tree. For present purposes, a tree is simple because it has no cycles. Particularly useful in network analysis are a rooted tree (which has a special node called its root) and a spanning tree of a connected graph. Rooted trees serve as models of (1) networks in which one island is distinguished from all others, (2) kinship groups defined by reference to an ego or an ancestor (Goodenough's 1970 "ego-focused" and "ancestor-focused" kinship groups), (3) hierarchical classification systems headed by a unique beginner. Fig. 1.2a shows a rooted tree, with the root indicated by an encircled node. In a collection of nested sets, either any two sets are disjoint or one set is a subset of the other. Fig. 1.2b shows an equivalent representation of a rooted tree as nested sets. Some anthropologists, including several whose work is discussed in Chapter

Island networks

A(G) =

G:

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(a) List of edges: 1,2 1,3 1,4 2,3 3,4 (c) Figure 1.1. Alternative representations of a graph.

8, implicitly use nested sets to model kinship structures, but we will always use graphs, which have the advantages of clarity and intuitiveness. Binary trees, as illustrated in Fig. 1.3, with the top node taken as the root, constitute an important special class of rooted trees and are often used to model classification systems. In Chapter 2 we use twin binary trees to give a single characterization of an apparently widespread and primordial type of Austronesian classification system variously referred to as "recursive dualism" (Eyde 1983), "recursive complementarity" (J. J. Fox 1989), and "perpetual dichotomy" (Hocart 1952), among other designations. In modeling kinship structures as rooted trees we note that EvansPritchard (1940), who was renowned for his diagrammatic virtuosity, used several different models, all of which are equivalent to a rooted tree, to represent the segmentary lineage.1 He used conventional kinship diagrams to describe its branching structure and nested sets to elucidate its alliance structure. He even drew pictures of trees, a method that he 1 As Geertz (1988:44) has observed, "The outstanding characteristic of E-P's [EvansPritchard's] approach to ethnographic exposition and the main source of his persuasive power is his enormous capacity to construct visualizable representations of cultural phenomena - anthropological transparencies."

Island networks and graphs

(a)

(b) Figure 1.2. Equivalent representations of a rooted tree as (a) a graph and (b) nested sets.

Figure 1.3. Two binary trees. felt would "commend itself to Nuer, who sometimes speak of a lineage as kar, a branch." We will not use pictures here, but the native drawing in Fig. 1.4 (where the root is as usual) of moiety organization, from Pukapuka Atoll in East Polynesia, suggests that the rooted tree model would also commend itself to Oceanic thought. A spanning tree of a graph G contains all the nodes of G. This concept was discovered by the pioneering physicist Gustav Kirchhoff in the

Island networks UILA

TAWOLA Yalo Yelu Manu

Kenakena

PUNGA-PUNGAMOMOTO

Figure 1.4. A native model of social organization in Polynesia: "Branching of maternal sub-lineages (keinanga), from informant's drawing. Main trunk of tree represents total population; main branches four main groupings of maternal lineages (wua); remaining small branches individual sub-lineages" (from E. and P. Beaglehole 1938). course of solving a network problem in electrical engineering. The circumstances of his discovery provide an example of graph theoretic modeling that can be interpreted generically. Kirchhoff (1847) developed the theory of trees in 1847 in order to solve the system of simultaneous linear equations that give the current in each branch (edge) and around each circuit (cycle) of an electrical network. The resulting graph is often called the "topology of the network." As a physicist he abstracted and deliberately oversimplified an electrical network, with its resistances, condensers, inductances, and so forth, and replaced it with its corresponding combinatorial structure consisting only of nodes and edges without any indication of the type of element represented by individual edges. Thus, in effect, Kirchhoff replaced each network with its underlying graph and showed that it is not necessary to consider every cycle in the graph of a network separately in order to solve the system of equations. Instead, he pointed out, by a simple but powerful construction which has since become standard procedure, that the independent cycles of a graph, determined by any one of its spanning trees, will suffice. A contrived electrical network N, its underlying graph G, and a spanning tree Tare shown in Fig. 1.5. In graph theory, the independent cycles of a graph determine its cycle

Island networks and graphs

Figure 1.5. An electrical network (N), its underlying graph (G), and a spanning tree (T).

rank. In geography, cycle rank provides a standard measure of the connectedness of a transportation network known as the alpha index of a graph (Garrison and Marble 1962). In Chapter 2 we give this index a firm mathematical foundation by supplying the necessary theorem on cycle rank in graphs and propose it as a tool for studying the relation between network connectedness and linguistic and cultural differentiation. In Chapter 4 we give a political interpretation of the alpha index as the potential for elite control of an exchange network. Minimum spanning trees

When we assign numerical values, or "weights" or "costs," to the edges of a graph G, we obtain a network N, also called a weighted graph, W. A minimum spanning tree, denoted MST, is a most economical spanning tree of a network. An intuitively appealing illustration of an MST, and one which immediately suggests anthropological applications, is that of Boruvka (1926a, b). Boruvka discovered MSTs when he was asked to give a mathematical solution to the problem of finding the most economical electrical network for a region. He described his method using the example in Fig. 1.6. The points (nodes) are in the plane, and the distances between them are euclidean. Boruvka writes,

8

Island networks

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Figure 1.6. Boruvka's illustration of a minimum spanning tree algorithm (from Graham and Hell 1985).

I will join each of the given points with the point nearest to it. Thus, for example, point 1 with point 2, point 2 with point 3, point 3 with point 4 (point 4 with point 3), point 5 with point 2, point 6 with point 5, point 7 with point 6, point 8 with point 9 (point 9 with point 8), etc. I will obtain a sequence of polygonal strokes [i.e., fragments] 1, 2 , . . . , 13 [Fig. 1.6b]. I will join each of these in the shortest possible way with the stroke nearest to it. Thus for example, stroke 1 with stroke 2 (stroke 2 with stroke 1), stroke 3 with stroke 4 (stroke 4 with stroke 3), etc. I will obtain a sequence of polygonal strokes 1, 2 , . . . , 4 [Fig. 1.6c]. I will join each of these in the shortest possible way with the stroke nearest to it. Thus stroke 1 with stroke 3, stroke 2 with stroke 3 (stroke 3 with stroke 1), stroke 4 with stroke 1.1 will finally obtain a single polygonal stroke [Fig. 1.6d], which solves the given problem (Boruvka 1926a, translated by and quoted in Graham and Hell 1985:51-2).

Island networks and graphs

9

Boruvka's method is based on an algorithm, that is, a finite, systematic (step-by-step) procedure for solving a problem.2 Boruvka's procedure is one of three standard algorithms for solving the minimum spanning tree problem (MSTP), as it is called in applied combinatorics. As informally stated by Graham and Hell (1985:44), they are: 1 (two nearest fragments). Add a shortest edge which joins different fragments.

ALGORITHM

2 (nearest neighbor). (A vertex [node] v is arbitrarily chosen.) Add a shortest edge which joins the fragment containing v to another fragment.

ALGORITHM

3 (all nearest fragments). For every fragment add the shortest edge which joins it to another fragment.

ALGORITHM

MST algorithms provide models for simulating processes of network growth and for analyzing clustering in geographically and culturally defined networks. In Chapter 3 we use Algorithm 3 to simulate the evolution of overseas chiefdoms in the Lau Islands, Fiji. Then Algorithm 1 and the concept of clustering in a minimum spanning tree describe the partitioning of the Tuamotu archipelago into dialect groups and marriage isolates. We regard this clustering as one way of interpreting the Pawley and Green (1984) "network-breaking model" of diversification in the Austronesian languages. In an archaeological application, we show that the complicated method of "close-proximity analysis" developed by Renfrew and Sterud (1969) in Mediterranean archaeology and applied by Green (1978) to the analysis of pottery-design motifs in the Lapita network in Oceania, is an independent discovery of an MST algorithm. As such it can be much more simply stated and efficiently computed using Algorithm 2. All three algorithms are "greedy" in the sense that they add shortest edges of a network first. At the conclusion of Chapter 3 we note two additional algorithms that proceed dually by removing longest edges first. These could serve as models of network devolution. Search trees Search trees are rooted trees in which all the nodes are ordered in a specified way. They are basic tools of computer science, used in sorting colThe term derives from the name of a ninth-century Arabian mathematician, Abu Ja' far Mohammed Ibn Musa al-Khowarizm, who wrote an important text entitled Kitab al jabr wy al-muqabala. For a fascinating discussion of algorithmic methods and artificial intelligence, see Penrose (1989).

10

Island networks 1

Figure 1.7. Depth-first and breadth-first search trees. lections of items into their natural orders (Roberts 1984).3 They should seem congenial to anthropologists, since they are conventionally described in the language of kinship, with nodes referred to as "fathers and sons" and "ancestors and descendants." Thus a computer file search resembles a genealogical search. In a breadth-first search tree (BFST) the nodes are ordered horizontally, whereas in a depth-first search tree (DFST) they are ordered vertically, as illustrated in Fig. 1.7. Each of these search trees has a directional dual, obtained by reversing the ordering of labeling. It is also possible to combine breadth-first and depth-first searches of a rooted tree. Search trees provide ideal models of hierarchically structured kinship groups. In Chapters 4 and 5 we use DFSTs to model rank in the conical clan, a stratified type of descent group that is to Oceanists what the segmentary lineage is, or once was, to Africanists. The conical clan constitutes the basic structural design of many Polynesian societies (Sahlins 1958; Goodenough 1959), of Ancestral Polynesian Society (Kirch 1984a; Kirch and Green 1987),4 and, we conjecture, of Proto-Nuclear Micronesian and Proto-Oceanic society as well. In spite of its theoretical importance to neoevolutionists, social anthropologists, and Oceanists, the conical clan has never been accurately defined, and as a result it has been independently discovered, but only partially characterized, on numerous occasions. Our analysis is structural, historical, and comparative. In depicting the conical clan as a DFST, we provide a precise, clear, intuitively appealing model capable of including all of its variants as de3 Search trees are also basic models in artificial intelligence, where they are used to represent decision-making processes. 4 Kirch and Green use the terms "protolanguage," "ancestral culture," and "parental population" to distinguish between linguistic, cultural, and biological reconstructions.

Island networks and graphs

11

fined by alternative rules of descent and succession. In comparison with cumbersome verbal definitions, this is indeed a case in which one picture is worth many words. In reviewing the intellectual genealogy of the conical clan, we seek to illuminate the forms it takes in Oceania. The most significant contributions to the analysis of the conical clan are those of Paul Kirchhoff (1955), who recognized its evolutionary implications; Gifford (1929) and Leach (1954), who articulated its integration with matrilateral cross-cousin marriage; Bott (1982), who revealed it as the infrastructure of the Tongan Empire; and Friedman (1981), who integrated the asymmetries of descent and alliance in a regional systems analysis of social stratification in Oceania. The generically important point of Friedman's model - that stratification is tied to the elite control of exchange networks through kinship and marriage links - is the basis of our analysis of social organization in Nuclear Micronesia. In Chapter 5 we describe the matrilineal variant of the conical clan that Mason (1954) independently discovered in the Marshall Islands in eastern Micronesia and that he modeled using a notational scheme resembling a DFST. We compare the Marshallese to the Tongan variant, describing a similar rule of marriage and an analogous function of descent and alliance in the formation of inter-island empires. We infer, on the basis of linguistic evidence (Pawley 1982), that these two variants are genetically related, having a common origin in Proto-Oceanic society. We then treat the Marshallese variant as the ancestor of all the differently permuted forms of social organization in Nuclear Micronesia. We show that these forms, which include primogenitural and age-based descent hierarchies as well as elementary, complex, and semicomplex marriage systems, are determined by a combination of ecological, demographic, and network variables. Search trees have many potential uses in network analysis.5 For example, Barnes (1979) implicitly recommends the use of depth-first searches of genealogical trees to obtain exhaustive descriptions of ancestry (with right and left searches, respectively, for patrilineal and matrilineal descent). At the conclusion of Chapter 5, we describe BFSTs and DFSTs as alternative mental models in the organization of genealogical knowledge. Any graph, whether or not it is a tree, can be searched in order to discover certain of its structural properties, such as its connectedness, planarity, and isomorphism to other graphs (Tarjan 1972). A breadth-first search of a graph is illustrated in Fig. 1.8. The edges marked by arrows showing the order of the search form a BFST. This procedure is the basis of the shortest-path algorithm that we use in Chapter 8 to evaluate 5 In linguistics, DFSTs serve as mental models of sentence parsing (Pinker 1994).

12

Island networks 2

3 7 Figure 1.8. A breadth-first search of a graph. Murdock's (1949) evolutionary model of Malayo-Polynesian social organization. Centrality concepts Centrality models are based on the concept of distance in graphs (Buckley and Harary 1990). They are well known in the fields of geography (Haggett 1967), social networks (Freeman 1979; Freeman, Borgatti, and White 1991), and operations research (Hakimi 1964). A typical problem in operations research is that of choosing a site for a facility on the basis of a specific criterion (Slater 1981; Buckley 1987). In some cases the problem is to choose a site that minimizes response time to any other location. In other cases it is to minimize the total travel time between a site and all other locations. When modeled as a graph, the first problem is solved by finding the set of nodes whose maximum distance to any other node is least. This is the center of the graph. The second problem is solved by finding the set of nodes whose total distance to all other nodes is least. This is called the median of the graph. In the graph of Fig. 1.9, nodes d and e constitute the center, and nodes e and /"the median. In social network analysis, three different models are used to study the effects of central location on group processes (L. C. Freeman 1979). Closeness (median) centrality, as already defined, is an index of "communication efficiency." Betweenness centrality refers to the frequency of occurrence of a node on the geodesies (shortest paths) joining all pairs of nodes and is an index of the "potential for control of communication." In Fig. 1.9, node /"is the betweenness center of the graph. Degree centrality refers to the number of edges incident with a node and is an index of "communication activity." In Fig. 1.9, node /"is also the degree center of the graph. In Chapter 6 we use six different models to describe the effects of cen-

Island networks and graphs

13

o

c

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k

Figure 1.9. A graph to illustrate centrality concepts. tral location in island networks. Degree, median, and betweenness centrality account for the emergence of trade and political centers in the Lau Islands, Fiji. Our analysis provides a corrective to the ecological interpretations of Thompson (1940) and Sahlins (1962) and develops a network explanation for the success of Lakemba over competing power centers in the Greater Lauan trade network. The classic, unmarked concept of centrality predicts the location of political capitals and mythological centers in the Marshall Islands in Micronesia. We introduce the concept of betweenness in a rooted graph to evaluate Harris's (1979) network explanation for the selection of horticulture as opposed to foraging in the economies of the western Torres Strait islands. We then show the relation between location, trading success, and stratification in a larger network comprising all the islands of Torres Strait as reconstructed from the detailed information in Haddon (1890, 1904, 1935). In order to accommodate situations in which physical distances must be taken into account, we describe a matrix method for finding the median of a network N. Our illustration is the Archaic Aegean network, as described by Davis (1982). Here, as in many Melanesian networks, structural and environmental variables jointly determined the location of a trade center. We conclude by defining the theoretically limiting case of networks whose graphs are self-centered. Dominating sets Solutions to many network problems can be found by decomposing a graph into subgraphs - for example, its independent edges, disjoint cycles, or spanning trees. In certain types of communication problems, a graph may be decomposed into its dominating sets. A dominating set S

14

Island networks

o o o o o Figure 1.10. One solution to the Five Queens Problem.

of a graph G has every node of G either in it or adjacent to it. A classic example of the concept of a dominating set is the Five Queens Problem in chess: How can five Queens be positioned so that they occupy or command every square on the board without threatening each other? One solution is shown in Fig. 1.10. There may also be six, seven, or eight Queens in a minimal dominating set (so that the removal of any one Queen leaves a nondominating set). Hence five Queens constitute a minimum dominating set. If we model this problem as a graph, the squares of the board become nodes, two of which are adjacent if a Queen can move from one square to the other. The solution consists of finding a dominating set with five nodes. This is known as the Queens Graph. Of course there are also 64node graphs, determined by the admissible moves of the other chess pieces: Bishop, King, Knight, and Rook. These and other related graphs suggest many enjoyable two-person games (Harary, to appear). Clearly the number of Kings required to dominate the chessboard is nine, and the number for Bishops and Rooks is eight. The concept of a dominating set can be generalized to include independent dominating sets and «-step dominating sets, among many other variations (Haynes, Hedetniemi and Slater, to appear). In Chapter 7 we use these concepts to analyze distributional aspects of power relations in the western Caroline Islands, political alliances in the northwestern Tuamotu Islands, and pottery monopolies in Melanesia. Digraphs

A directed graph, or digraph, has directed edges, or arcs, to represent the ordered pairs in a relation (Harary et al. 1965). Digraphs have two

Island networks and graphs

(a)

(b)

Figure 1.11. Digraphs that are (a) weakly and (b) strongly connected. important properties that distinguish them from graphs. Unlike graphs, which represent symmetric relations, digraphs may be symmetric, asymmetric, or neither. Whereas graphs are either connected or disconnected, digraphs may be connected in various ways. A digraph is strongly connected if any two nodes are mutually reachable and weakly connected if its underlying graph is connected (see Fig. 1.11). In Chapter 8 we examine the digraphic basis of two controversial evolutionary models of social organization in Oceania: Murdock's (1949) reconstruction of Proto-Malayo-Polynesian society as "Hawaiian" in type, and Marshall's (1984) derivation of Island Oceanic sibling terminologies from a "Melanesian" prototype. Murdock's model is, implicitly, a strongly connected digraph whose nodes represent types of social organization defined by variations in descent, residence, and kinship terminology and whose arcs represent permissible one-step transitions between them. In this digraph, or, to use Murdock's metaphor, "evolutionary maze," the prototype of all the different types of social organization found in societies belonging to the same linguistic family is the one from which they can all be derived in the least total distance. In the Malayo-Polynesian family it is claimed that the prototype is Normal Hawaiian. A strict application of Murdock's method, however, shows that the prototype is "Iroquois," or "Nankanse." This eliminates the "bilateral hypothesis" as an alternative to Blust's (1980) linguistic reconstruction of early Austronesian society as one based on unilineal descent and matrilateral cross-cousin marriage and to our conjecture that Proto-Nuclear Micronesian society was similarly structured and socially stratified. Marshall's model is, implicitly, a weakly connected digraph whose nodes represent the different types of sibling terminologies found in the Austronesian and non-Austronesian languages of Island Oceania and whose arcs represent one-step transitions between them. The transitions take the form of successive deletions or additions of the binary features of sex, seniority, and parity, as described for Polynesian terminologies

16

Island networks

Figure 1.12. A semilattice.

by Epling, Kirk, and Boyd (1973). The prototype of all Island Oceanic sibling terminologies is that terminology whose total distance to all other terminologies is least. An examination of the digraph of Marshall's model, however, shows little support for this claim. In view of the serious linguistic objections to Marshall's typological model (Bender 1984; Blust 1984; Clark 1984), we propose that it be replaced, for the Oceanic languages, by the genetic model contained in Milke's (1938) reconstruction of Proto-Oceanic sibling terms. The digraph implicit in Milke's model is an upper-semilattice: a partially ordered set in which every pair of nodes has a least upper bound, that is, a node which can reach each of them by a directed path, as illustrated in Fig. 1.12. A semilattice is a very general model that can take the form of a directed path (evolutionarily interpreted, a unilinear structure), an oriented branching tree (a multilinear structure), a lattice (a diverging and converging structure), or a digraph like the one in Fig. 1.12 (a diverging and partially converging structure).6 We will see that the evolutionary sequence of sibling terminologies in the Oceanic languages has the structure of a semilattice. This semilattice contains as subdigraphs the Polynesian sequence proposed by Clark (1975) and the Nuclear Micronesian sequence proposed by ourselves, both of which are consistent with the hypothesis that Proto-Oceanic society was stratified. Implicit uses of semilattices as evolutionary models of kinship structures include LeviStrauss's (1969) "classification of the principal types of kinship systems, evolved from dual organization" (which is isomorphic to Fig. 1.12) and Needham's (1984) "scheme of transformations" of symmetric and asymmetric prescriptive marriage systems in eastern Indonesia.

Island networks and graphs

17

Geographical, linguistic, and anthropological terms In order to expedite our presentation in the following chapters, it will be helpful to define in advance a number of geographical and linguistic terms. (See Map 1.1.) For reasons of convenience and theoretical interest we use the conventional triadic division of Oceania into Melanesia, Micronesia, and Polynesia. We also refer to a fourth area, Indonesia, in the discussion of certain classification and kinship systems. Island Oceania includes Micronesia, Polynesia, and Melanesia exclusive of New Guinea. For archaeological and linguistic purposes Oceania can also be divided into Near Oceania, consisting of New Guinea and the Bismarck and Solomon Islands as far east as San Cristobal, and Remote Oceania, consisting of all the remaining islands (Pawley and Green 1975). New Guinea and the islands in Near Oceania form a chain of intervisible land masses connected to the Southeast Asian mainland by the Indonesian archipelago. The islands and archipelagoes of Remote Oceania are separated by a gap of 350 kilometers from those of Near Oceania (the distance from San Cristobal to the Santa Cruz group) and often by large gaps from each other. The distance from the New Hebrides to Fiji is 900 kilometers and from the Marquesas Islands to Hawaii about 3,000 kilometers. There are two groups of languages in Oceania: Austronesian (AN) and non-Austronesian (NAN), or Papuan. In the older literature, such as Murdock (1949), the AN languages are referred to as "Malayo-Polynesian" (MP), but in current usage MP is a first-order node in the AN family tree. Some authors prefer "non-Austronesian" to "Papuan" to indicate that the historical relationships of these languages are not well understood. There are some 750 NAN languages, which are found mainly in New Guinea (Foley 1986). A small number are found in islands to the west, including Timor and North Halmahera in Indonesia and in islands to the east in the Bismarcks (New Britain and New Ireland), the Solomons, and the Santa Cruz group. AN languages are interspersed with NAN languages on the northern coast of New Guinea. There are between 1,000 and 1,200 AN languages, stretching from Madagascar to Easter Island (Grimes 1992; Tryon 1993). Proto-Austronesian (PAN) was probably spoken 5,000 to 6,000 years B.P. (Pawley and Ross 1993). The AN family tree, as reconstructed by Blust (1990), is shown in Fig. 1.13. "Formosan" (F) is a convenient cover term that includes at least 6 primary branches of AN, represented by the 14 remaining Formosan aboriginal languages (R. Blust, personal communica-

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Island networks and graphs AN

19

MP

WMP

CEMP

CMP

EMP

SHWNG

OC

Figure 1.13. The Austronesian family tree (from Blust 1990). (Abbreviations: AN: Austronesian; F: Formosan; MP: Malayo-Polynesian; WMP: Western Malayo-Polynesian; CEMP: Central-Eastern MalayoPolynesian; CMP: Central Malayo-Polynesian; EMP: Eastern MalayoPolynesian; SHWNG: South Halmahera-West New Guinea; OC: Oceanic.) tion). Malayo-Polynesian (MP) includes all the AN languages outside of Taiwan. Western Malayo-Polynesian (WMP) comprises the "languages of the Philippines and western Indonesia, including Palauan and Chamorro of western Micronesia, the Chamic languages of mainland South East Asia, and Malagasy" (1990:8). Central Eastern MalayoPolynesian (CEMP) includes all other MP languages. Central MalayoPolynesian (CMP) includes the "AN languages of the Lesser Sundas and of the southern and central Moluccas" (1990:8-9). Eastern MalayoPolynesian (EMP) includes the languages of South Halmahera-West New Guinea (SHWNG), and Oceanic (OC). The latter includes some 450 languages spoken in Melanesia, Polynesia, and Nuclear Micronesia.7 According to the "Oceanic hypothesis," which is accepted by most linguists, Proto-Oceanic (POC), called "Proto-Melanesian" (PMN) by Dempwolf (1934-38) and Milke (1938), was ancestral to all the AN languages of Melanesia, Polynesia, and Nuclear Micronesia: Pawley and Ross (1993) accept EMP, SHWNG, and OC as subgroups of MP but regard WMP, CMP, and CEMP as problematic. We will be primarily concerned with AN, MP, and OC.

20

Island networks The decisive Austronesian (AN) settlement of Melanesia was by the descendants of a single ancestral language community, which entered northwest Melanesia from eastern Indonesia. All the so-called "Melanesian" (MN) languages spoken east of 136° E fall into a subgroup, i.e., they represent genetic continuations of one Austronesian (AN) interstage language. "ProtoMelanesian" was Dempwolf's name for this interstage but it is now generally called Proto-Oceanic and the subgroup, Oceanic. Proto-Oceanic was also ancestral to the Polynesian and Nuclear Micronesian languages . . . While other AN languages may have entered Melanesia at various times, none beside PMN left any descendants in Melanesia east of 136° E, in Polynesia, or in Micronesia east of Palau and the Mariana Ids (Pawley 1981:273).

The following divisions of OC are relevant for our purposes. On the basis of linguistic and archaeological evidence Pawley and Green (1975) locate the home of POC in Near Oceania, "specifically in the region of the north coast of New Guinea and the Bismarck Archipelago." By 2000 B.C. POC split into New Guinea Oceanic and Eastern Oceanic (EO). The EO group includes "all the languages of the Southeast Solomons, Fiji and Polynesia, and Rotuman. The Nuclear Micronesian languages can be added to this list, together with all the better known languages of the Central and Northern New Hebrides and the Banks Islands" (1975:25). Nuclear Micronesian (NM) includes the languages of the Gilbert, Marshall, and Caroline Islands and probably Nauruan, but not Palauan or Chamorro, which are AN but not OC. Yapese is AN, but its status is otherwise uncertain. Pawley and Green (175:51) mention that the "NM group exhibits greater internal diversity than the Polynesian group, thus suggesting a time depth of settlement exceeding that of Polynesia." There are, however, no dates as of yet to support this conjecture. Proto-Fijian (PF) and Proto-Polynesian (PPN) are descendants of Proto-Central Pacific, which was located in Fiji around 1500 B.C. PPN was located in the Tonga-Fiji-Samoa region and by 200 B.C. split into ProtoTongic (the ancestor of Tongan and Niuean) and Proto-Nuclear Polynesian (PNPN) (the ancestor of all the other Polynesian languages). PNPN split into Proto-Samoic-Outlier (18 languages) and Proto-Eastern Polynesian (PEPN) by 300 A.D. PEPN split into Easter Island and Proto-Central-Eastern Polynesian by 400 to 700 A.D. The latter split into ProtoMarquesic and Tuamotuan, Tahitian, Maori, and Rarotongan by 800 A.D. Finally, Proto-Marquesic split into Southeast and Northwest Marquesan and Hawaiian.

Island networks and graphs

21

In several chapters of this book we refer to different types of marriage systems as distinguished in the well-known theories of Levi-Strauss (1949, 1966, 1969). Elementary structures are those that have a positive rule prescribing the choice of spouse. This results in fixed or determinate alliances between wife-exchanging groups. Elementary structures include restricted and generalized exchange. Restricted exchange is a symmetric relation joining pairs of groups (lineages or clans) to each other as wife-givers (WGs) and wife-takers (WTs). It is expressed in bilateral cross-cousin marriage. In the case of first cousins, marriage is with the mother's brother's daughter (MBD) who is also the father's sister's daughter (FZD).8 Generalized exchange is an asymmetric relation in which WGs are distinguished from WTs. It is expressed in a rule of matrilateral cross-cousin marriage, that is, marriage with the MBD. The relation is hypergamous if women marry up, and hypogamous if they marry down. The simplest model of generalized exchange is a cyclic triple in which group A is WG to group B, B is WG to group C, and C is WG to A. Generalized exchange is sometimes referred to as "asymmetric marriage alliance." Complex structures have only a negative rule prohibiting close marriages. Alliances are therefore not repeated in successive generations but are indeterminate or probabilistic. Semicomplex structures, sometimes identified with Crow and Omaha systems (LeviStrauss 1966; Heritier 1981), stand in between elementary and complex structures: they have marriage rules like the former, but the rules are all negative as in the latter.

We use the conventional kin type notation: F = father, M = mother, B = brother, Z = sister, D = daughter, S = son, H = husband, W = wife, with MB = mother's brother, FZ = father's sister, MBD = mother's brother's daughter, and so forth. M.s. = man speaking, and w.s. = woman speaking. In the earlier kinship literature, kin types were denoted by the first two letters of each term: Br = brother, Si = sister, and so on.

Trees

Tis education forms the common mind, Just as the twig is bent, the tree's inclined. Alexander Pope, Moral Essays Trees are the most elementary and versatile of all graphs and hence of all network models. We will now use rooted trees to describe the structure of communication in a Micronesian prestige-good system, binary trees to elucidate the structure of an wr-Austronesian classification system, in-trees to represent the flexibility of cognatic kinship networks found in many atoll societies, and spanning trees to measure the connectedness of exchange networks. Trees and all graph theoretic models are based on the following definitions.

Basic definitions Graphs Before defining a graph, we show in Fig. 2.1 the 11 different graphs with four nodes. Later we will see that 1. 2. 3. 4. 5. 6. 7. 8.

Every graph of order 4 is isomorphic to one of these. Graphs (a) through (e) are disconnected. Graphs (f) through (k) are connected. Graph (k) is complete. Graph (h) is a cycle. Graph (f) is a path. Graph (g) is a star. Graphs (f) and (g) are trees.

A graph G consists of a finite nonempty set, V = V(G) of p nodes, together with a prescribed set E of q unordered pairs of distinct nodes of 22

23

Trees O

O

o

r>——Q

o

o

o

6..

(a)

(b)

(c)

Q

(g)

O

(h)

Or

O

(0

(e)

(d)

Q

(i)

0)

(k)

Figure 2.1. The 11 graphs with four nodes.

Figure 2.2. A graph to illustrate adjacency.

V. We also write G = (V, E). We say G has order p and s/ze g. Each pair e = {«, t/} of nodes in E is an edge1 of G, and e is said to ;om u and u We also write e = uv and say that u and ^ are adjacent nodes. Adjacent nodes are said to be neighbors. Node u and edge e are incident with each other, as are v and e. If two distinct (different) edges are incident with a common node, then they are adjacent edges. A graph with p nodes and q edges is called a (p, q) graph. The (1,0) graph, consisting of just one node, is called trivial, mainly in order to exclude it by specifying that a graph be nontriviaL A graph is usually represented by a diagram that is referred to as the graph. In the graph G of Fig. 2.2, the nodes u and v are adjacent, but u and w are not; edges a and b art adjacent, but a and c art not. Although 1 Synonyms of "node and edge" are "vertex and edge," "node and branch," and "point and line."

24

Island networks

Figure 2.3. A graph, a subgraph, and a spanning subgraph.

6

5

Figure 2.4. Two labeled graphs. the edges b and c intersect in the diagram, their intersection is not a node of the graph. A subgraph of G is a graph having all of its nodes and edges in G. It is a spanning subgraph if it contains all of the nodes of G. In Fig. 2.3, G± and G2 are subgraphs of G; G2 is a spanning subgraph of G, but Gr is not. A labeling of a graph G is an assignment of labels 1 to p, or sometimes vx to fp, to its nodes. The degree of a node vt in a graph G, denoted
(2.1)

In words, the sum of the degrees of the nodes of a graph is twice the number of edges. This equation (2.1) follows from the fact that each edge contributes 2 to this sum, 1 for each of its two nodes. A walk of a graph G is an alternating sequence of nodes and edges v09 eu vu . . . , vn_ l5 em vn, beginning and ending with nodes, in which each edge et is incident with the two nodes vt _ x and v{ immediately preceding and following it. Both nodes and edges may occur more than once in a walk. The length of a walk is the number of occurrences of edges in it.

Trees

25

2

6

7

Figure 2.5. A graph to illustrate walks, trails, paths, and cycles. This applies to each type of walk defined in this section. This walk connects VQ and vm and may also be denoted v0, v^ v2, . . . , vn (or, more briefly, simply by 0, 1, 2 , . . . , » , when there is no confusion); it is sometimes called a vo-vn walk. It is closed if v0 = vm and is open otherwise. It is a trail if all the edges are distinct and a path if all the nodes (and thus necessarily all the edges) are distinct. If n ^ 3 and the walk has distinct edges and also distinct nodes, except for its endnodes v0 = vm it is a cycle. By definition, this cycle has length n; it is sometimes called an n-cyde. In the labeled graph G of Fig. 2.5, the sequence of nodes 1, 2, 3, 2, 6 is a walk that is not a trail, and 1, 2, 3, 4, 5, 3 is a trail that is not a path; 1, 2, 3, 4, 5 is a path, and 2, 3, 5, 6, 2 is a cycle. Finally, 1, 2, 3, 4, 5, 6, 7 is called a spanning path, one that contains all the nodes. By definition, the length of a path is the number of edges in it. The distance between two nodes vt and v^ denoted d(vh Vj) or J//5 is the length of any shortest path, or geodesic, g, that joins them. In the graph G of Fig. 2.5, d13 = 2 and d45 = 1, as nodes 4 and 5 are adjacent. The length of a cycle is the number of edges in it. In Fig. 2.5 there is one 3-cycle, 3, 4, 5, 3; one 4-cycle, 2, 3, 5, 6, 2; and one 5-cycle, 2, 3, 4, 5, 6, 2. We denote by Pn the graph that is a path with n nodes (and hence has length n - 1) and by Cn the cycle with n nodes (and length n). A graph is complete if every pair of nodes are adjacent, and it is connected if there is a path between every pair of nodes. A component of a graph is a maximal connected subgraph. If a graph has only one component, it is connected; otherwise, it is disconnected. In Fig. 2.6 the first graph is disconnected and consists of two components, and the other three are connected. The last graph is complete. The complete graph with p nodes is denoted Xp, so that the last graph in Fig. 2.6 is K4. It is easy to see that the number of edges in Kp is p(p - l)/2. For example, K4 has (4 • 3)/2 = 6 edges. A cutnode of a graph is one whose removal (together with its incident edges) increases the number of components, and a bridge is such an

26

Island networks

-o

6

Figure 2.6. Four of the 11 graphs with four nodes.

W

Figure 2.7. A graph to illustrate cutnodes and bridges.

G,:

J

2'

Figure 2.8. Two isomorphic graphs.

edge. Thus if v is a cutnode of a connected graph G, then G - v is disconnected. An endnode of a graph is one whose degree is 1. In Fig. 2.7, v is a cutnode but w/ is not; edge x is a bridge, but y is not; z is an endnode, but w is not. An isomorphism between two graphs G1 = (Vl3 £x) and G2 = (V2, E2) is a one-to-one correspondence between the node sets Vt and V2 that preserves adjacency. This correspondence is written Gx = G2, or sometimes G1 = G2. The two graphs in Fig. 2.8, although they appear to be different, are isomorphic. Fig. 2.1 shows all 11 nonisomorphic graphs with four nodes. The complement of a graph G is that graph G having the same set of nodes as G but in which two nodes are adjacent if and only if they are not adjacent in G, as illustrated in Fig. 2.9. A bipartite graph (or bigraph) G is a graph whose node set V can be

27

Trees

G:

G:

Figure 2.9. A Graph, G, and its complement, G.

K 3,3

Figure 2.10. Three bigraphs.

Figure 2.11. A rooted graph and a doubly rooted graph.

partitioned into two subsets, Vj and V2, such that every edge of G joins (a node of) V1 with (one in) V2. If G contains every edge joining Vj and V2, then G is a complete bigraph. If Vx and V2 have m and w nodes, we write G = iCmw = K(m,n). The graph K l w is called a star. The second graph in Fig. 2.10 is X3?3, and the third is X 1 4 . The first is the path P 6 . A rooted graph G is one in which a specified node, called the root, is singled out. Fig. 2.11 shows a rooted graph and a doubly rooted graph, one in which two nodes are distinguished from all others.

28

Island networks o U

2

V

2

O-

,:

w2 —o

6

Figure 2.12. The product of two graphs.

Figure 2.13. Planar, plane, and nonplanar graphs.

To define the (cartesian) product of two graphs, G1 x G2, consider any two nodes u = (uu u2) and v = (vl9 v2) in V = Vx x V2. Then w and v are adjacent in G] x G2 whenever \ux - V\ and u2 adj v2] or [«2 = i/2 and ux adj i/a]. The product of Gx = P2 and G2 = P3 is shown in Fig. 2.12. A graph G is embedded in a surface, S, when it is drawn on S so that no two edges intersect. A graph is planar if it can be embedded in a plane; a plane graph has already been embedded in a plane. The graph G1 in Fig. 2.13 is planar, although as drawn it is not plane (it has three pairs of edge intersections); the graph G2 is both planar and plane and is

Trees

29

Figure 2.14. The digraphs with three nodes and three arcs.

Figure 2.15. The 11 trees with seven nodes.

isomorphic to Gx as both of them are forms of the complete bipartite graph K2j; the graph G3, which is the bigraph K3j3, is not planar.2 Finally, a directed graph or digraph D consists of a finite nonempty set V of nodes, together with a collection A of ordered pairs of distinct nodes in V. We also write D = (V, A). The elements of A are called arcs, or directed edges. A symmetric pair of arcs joins two nodes, u and v, one in each direction, that is, arcs (u, v) and (v, u). An oriented graph is a digraph having no symmetric pair of arcs. In Fig. 2.14 all four digraphs with three nodes and three arcs are shown; the last two are oriented graphs. Trees We can now give a precise definition of a tree. A graph is acyclic if it has no cycles. A tree is a connected, acyclic graph. Any graph without cycles, whether or not it is connected, is a forest; thus each component of a forest is a tree. The 11 different (nonisomorphic) trees of order 7 are shown in Fig. 2.15. The first tree in the first row is the path P7, and the 2 Kuratowski's (1930) theorem states that a graph is planar if and only if it contains no subgraph homeomorphic to Ki3 or K5. See the discussion in Hage and Harary (1991).

30

Island networks

Figure 2.16. The rooted trees with four nodes.

last tree in the second row is the star K16. In a tree, every node is either a cutnode or an endnode, and every edge is a bridge. A rooted tree is obtained from a tree by distinguishing one of its nodes and calling that node the root. Fig. 2.16 shows all the rooted trees with four nodes. There are numerous structurally equivalent ways of defining a tree, as stated in the following result from Harary (1969). THEOREM 2 A. For a graph G with p nodes and q edges, each of the following equivalent properties can define a tree.

1. G is connected and acyclic. 2. G is connected, and p - q + 1. 3. G is acyclic, and if any two nonadjacent nodes of G are joined by an edge e, then G + e has exactly one cycle. 4. Every two nodes of G are joined by a unique path. 5. G is connected but loses this property if any edge e is deleted.

A Micronesian prestige-good system The Yapese Empire, in the Caroline Islands in western Micronesia, was one of the largest "systems of areal integration"3 in Oceania. Although it is usually described from the standpoint of the outer-island tributaries, it can also be described, from the standpoint of Yap, as a "prestige-good 3 This term derives from Schwartz's (1963) prophetic paper on regional networks in Oceania.

Trees

31

system" (Friedman 1981). In general terms, a prestige-good system is one in which an elite, through its control of a long-distance exchange network, is able to monopolize goods necessary for social reproduction. In the Yapese system this control was achieved by ensuring that all legitimate forms of communication within Yap and between Yap and the outer islands took the form of a rooted tree - in Yapese conception, the form of a tha\ Yap is a compact cluster of four high, volcanic, islands with varied and abundant resources. Yapese social organization was complex, based on double descent, ranked social classes, and a dual division into upper and lower, landowning and land-using castes. In a kinship metaphor that was used throughout Micronesia to designate social and political asymmetries, the upper and lower castes were said to be related as "father" and "child." In the Yapese tribute system, 14 low, coral, islands and atolls stretching some 1,200 kilometers eastward were joined to the Gagil district of Yap by three types of gift relation (Lessa 1950). In the pitigil tamol relation, the low islanders paid tribute to the highest-ranking matrilineal chief of Gagil. In the sawei relation, individual low island matrilineages paid "rent" to individual Gagil patrilineages that owned (at least nominally) individual districts of the low islands. In a symbolic replication of intercaste relations, the low island tenants stood as "children" to their Yapese landlords and "fathers." In the mepel relation, low island matrilineages made religious offerings to Yapese ancestral ghosts. Tribute gifts consisted of woven fiber cloth, mats, coconutfiber rope, and shell valuables. The sawei gift was reciprocated by "optional" Yapese gifts of food, including taro, yams, sweet potatoes, and bananas, craft goods, and raw materials such as basalt, timber, and turmeric, all of which were scarce or unavailable on the low islands. The sanctions for the tribute payments were supernatural: Yapese magicians had the power to promote the fertility of women and crops and to cause or ward off storms, epidemics, and pestilence. There is no record of Yapese conquest of the low islands. A little-noted but significant aspect of the sawei relation concerned marriage between Yapese men and outer-island women. Lessa mentions that "A Yapese [man] . . . may marry any Ulithi woman, since there is no barrier to marrying into his sawei sib, even though such persons are his 'children.'" Many Ulithi women have migrated to Yap, where they have married and settled down. Their children have been adopted into the lineages of their sawei, and, largely because of the depletion of some of these lineages in recent years, they have attained positions of headship. On account of the caste restric-

32

Island networks Ulithi

Yap (Gagil)

Fais

Namonuito Faraulep Lamotrek Pulap

Sorol Ifaluk

Elato

Satawal

Eauripik

Puluwat Pulusuk

(a)

10 14

17

(b)

Figure 2.17. Yapese communication structures (tha'): (a) the outer-island tribute system; (b) Gacpar political networks (villages are numbered). tions which apply to males from other islands, the number of men who have migrated to Yap is far less. And, while they may marry lower caste women, Ulithians ordinarily avoid marriage to any Yapese women whatever (Lessa 1950:45). In the payment of tribute, an order demanding tribute originated at Yap and was then transmitted through a specified sequence of islands, beginning with Ulithi in the west and ending with Pulap, Pulusuk, and

Trees

33

Namonuito in the east, with tribute gifts flowing back in the converse direction. This structure, known as the "Yapese chain of authority" (Lessa 1950), is shown as a rooted tree in Fig. 2.17a. By condition 4 of Theorem 2.1, if G is a tree, then every two nodes are joined by a unique path. This means that orders for tribute could only be communicated in one way. This is precisely the property of the chain of authority that the islanders emphasize: "the 'chain of authority' is scrupulously observed, so that a small island will not comply with orders from Yap unless they have been sent in accordance with the proper pattern" (Lessa 1966:38). The tree structure meant that there could never be conflicting orders and hence questions about their legitimacy. We note that all the islands in this structure were connected by unique paths of communication, but this does not imply, as often assumed (Lessa 1950; Alkire 1965; Bellwood 1978), that all the islands were uniquely ranked. There is no empirical basis for labeling all the nodes in this tree in a complete order. (See Chapter 4 on search trees.) In Yapese conception, the structure in Fig. 2.17a is an application of the tha% literally, "a series of things tied together with string."4 The concept implies that political communication must flow through properly designated channels. When used in Yapese politics, the concept of tha( designates a long line of communication that ties together the various geographical and political units of Yap. Any legitimate request or message must follow the channels of communication, or tha\ This is a very serious matter to the Yapese and if word is passed improperly, regardless of its importance, it may be disregarded. On the other hand, a properly communicated message has the force and power of the highest chiefs and to disregard it brings serious consequences. The hierarchy of communication is one of the keys to the power of the paramount chiefs (Lingenfelter 1975:131). Fig. 2.17b shows two principal tha% depicted as rooted trees, leading out of the chiefly village of Gacpar in the Gagil district of Tamil to villages on the adjoining islands of Map and Rumung. (The root is placed below in accordance with geography.) The tha' provides the basic framework for the collection of tribute, and it organizes many forms of social interaction, including payments for breaches of custom, which move up the tree in a manner similar to the flow of tribute from the outer islands to Gagil: 4 A similar metaphor is used to designate tributary relations in the Lau Islands, Fiji. See Chapter 3, on the nggali relation.

34

Island networks Breaches of custom regarding marriage or even murder may be carried to the high chiefs on the tba\ In such a case, the accused pleads with his chief for support and intervention. The chief then takes a piece of shell money and presents it with his plea to the next higher chief on the tha\ who passes it on to the highest chiefs, with each chief adding a piece of shell money if he approves, until it returns to the victim's family. To refuse such a plea passed along the tha' of all the high chiefs of Yap would be utter folly. It is invariably accepted and pardon given (Lingenfelter 1975:132).

The Yapese tribute system is usually regarded as something of an enigma, since the atoll dwellers obtained substantial material goods in return for apparently modest gifts of rope, mats, and shells (Lessa 1966; Bellwood 1978). For the Yapese, however, these gifts were highly valued, exotic goods that figured prominently in their internal exchange system. Because of their scarcity on Yap, they were of "much greater value to the chiefs in Gacpar than regular Yapese tribute" (Lingenfelter 1975:153). Spondylus shells from the outer islands together with stone money quarried in the Palau Islands 400 kilometers to the southwest of Yap were among the highest-ranked valuables, macaf, given in ceremonial exchanges (Alkire 1980). A Yapese marriage, for example, required complementary exchanges of shell money for stone money between the sides of the groom and bride (Labby 1976). Through the monopolization of outer-island tribute, paramount chiefs in Gacpar village, in the Gagil district of Yap, were able to substantially increase their power: Using the outer island tribute, the high chiefs of Gacpar were very effective in expanding their alliances through distribution of that wealth. The Banpagael alliance of which Gacpar is chief was without question the most powerful, with the most extensive network of allies. The deciding factor in the accretion of power appears to be the distribution of trade goods from the outer islands and the reciprocal obligations incurred by that distribution. Even in present day research informants were quick to mention that one tha' coming to their particular estate had a part in the trade distribution from Gacpar. Furthermore informants in Tamil tell the story of a chief who prayed earnestly that foreigners would come to Yap and bring a new trade to Tamil and Rull [districts]. It seems significant that early traders were received readily in Tamil and Rull, but with hostility in Gagil. The chiefs of Rull and Tamil welcomed new

Trees

35

trade goods that might swing the balance of power in their favor again (Lingenfelter 1975:152-3). Although little is known about the history of Yapese society, the Gagil district, at least, had some of the elements of a West Polynesian prestige-good system (Friedman 1981). These include: (1) an elite monopolization of imported goods essential for social reproduction; (2) a flow of tribute and women toward a center (although not a corresponding flow of men toward the periphery); (3) an asymmetry in the kinship system, the pairing of Yapese landowning patrilineages with outer-island land-using matrilineages; (4) an asymmetric political dualism, expressed symbolically in the opposition between parent and child, superior and inferior, center and periphery.

"Recursive dualism" in Austronesian classification systems In certain parts of Fiji, island networks have a symbolic structure that combines the properties of duality, repetition, and hierarchy. The same type of structure has been independently discovered in a number of other Austronesian societies, where it is described in terms of its mediational logic, generative capacity, and historical primordiality. This structure lacks a formal definition and an agreed-upon name, but its underlying graph is a twin binary tree. According to Hocart (1929, 1952), the most distinctive feature of Lauan society in eastern Fiji is the repeated division of all social groups, including islands, districts, villages, and clans, into two sides - a "noble" (turanga) side, and a "land," or "border" (vanua) side. In Lauan conception, superiority in rank implies a difference in manners: "nobles are gentle, they just speak, but the border is aggressive and does. The border are the chief's right arm, as we should say, the hand of the club as the Fijians put it. They champion the chief - whether in war, or against insolence, or in preparing feasts" (Hocart 1952:29). The distinction between the two sides is based on "a relativity which is typically Fijian: a group which is noble to another may be border to a third. Thus Natewa, noble in relation to Kama, is land in relation to Thakaundrove" (1952:30). Hocart's generic model of this structure, which he characterized as a system of perpetual dichotomy, is reproduced in Fig. 2.18. The division of the Lau Islands (see Map 3.2) is shown in part in Fig. 2.19. Lakemba and Naiau are noble islands in relation to all other is-

36

Island networks Tribe

Nobles

Land

I

I

Nobles

Land

I

I

Nobles

Land

Figure 2.18. Hocart's (1929) model of "perpetual dichotomy" in Fijian social organization. Lau

Lakemba-Naiau

I

I

Navuaira

Natokalau

I Tumbou

Lau

I Navuaira

Nasangalau

Nukunuku

i Tai

Tumbou

I Katumbalevu

I Valelailai

Thekena

Tumbou

I Navuanirewa

Kalumbalevu

Figure 2.19. Social organization in the Lau Islands (from Hocart 1929).

lands in Lau and are themselves subdivided into noble and land groups on the left and right nodes respectively. Although it is not shown in Fig. 2.19, some islands in Lau (e.g., Kambara and Mothe) are border to Lakemba and Naiau but noble in relation to other islands. In a structural analysis of Moalan society, also in Fiji, Sahlins (1976) emphasizes the "symbolic productivity" of this dualism as expressed in its social, spatial, and temporal correspondences and in its mediational or "reciprocal logic":

Trees

37

Similarly for the Moalans, their island and each of its villages are essentially made up of two "kinds" of people: the Land People (kai vanua) and the Chiefs {turaga). The Land People are also known as the "owners" (taukei) an expression synonymous with first occupants or original settlers. The Chiefs came later, by sea, to assume the rule over a numerous host that had filled the inland regions - so the Land People are also the "Thousands" (Udolu) or "Animal People" (Yavusa Manumanu) . . . A difference of social groups corresponds to the distinction of land and sea on the geographic plane, itself an instance of a general spatial differentiation of interior and peripheral, correlated with oppositions of indigenous and foreign, earlier and later, even animal and cultural; the same groups again are inferior and superior politically, ritual and secular functionally . . . Local legends of the coming of the Chiefs as well as many customary practices reveal a definite structure of reciprocities. In its most general terms the reciprocal logic is that each "kind" mediates the nature of the other, is necessary for the realization and regulation of the other, so that each group necessarily contains the other (Sahlins 1976:24-5). Referring to Hocart's diagram in Fig. 2.18, Sahlins defines this structure as "a four-part code operated by the replication of a master dichotomy" (1976:25). He also notes Bourdieu's analysis of a similar structure in Kabyle symbolism: Bourdieu (1971) analyzes a Kabyle structure of this type under the general diagrammatic formula a:b::bl:b2. He notes of its generative capacity: "doubtless one of the simplest and most powerful [structures] that may be employed by a mythico-ritual system since it cannot oppose without simultaneously uniting (and inversely), while all the time being capable of integrating in a unique order an infinite number of data, by the simple application of the same principle of division indefinitely repeated" (1971, p. 749) (Sahlins 1976:25). Eyde (1983) has discovered the Sahlins-Bourdieu structure, which he calls recursive dualism^ in the kinship and spatial categories of the Admiralty Islanders in Melanesia. "In this organization, opposed male and female categories are themselves further subdivided into opposed male and female categories. Such subdivision is infinitely repeatable as indicated in [Fig. 2.20]" (1983:3). The Manus house, for example, is partitioned into male and female spaces, each of which has a male and fe-

38

Island networks UNITY

Female

Male

Male

Male

Female

Male

Female

Male

Female

Male

Female

Female

Male

Female

Figure 2.20. Eyde's (1983) model of "recursive dualism" in the Admiralty Islands. male side analogous to the quadripartite noble-commoner divisions of a Fijian house. In an essay on organizational principles in Micronesian cultures, Alkire and Fujimura identify recursive dualism in the evolution of exchange relations in the outer islands of the Yap state: The Micronesian world view, like that of many Austronesian speakers, emphasizes dualistic oppositions, quadripartite divisions and mid-points as loci of control and mediation . . . In the outer islands an attempt is made to define all exchanges as involving two parties. Dualistically balanced districts and lagoons are examples of this. When more than two units occur, they frequently derive from earlier dualistic divisions that have been further subdivided into quadripartite units. David Eyde (1983) has named a similar tendency in Melanesia recursive dualism (Alkire and Fujimura 1990:77). In the view of Alkire and Fujimura, these classificatory tendencies derive from a common Austronesian substratum: "The differences frequently emphasized between Polynesia and Melanesia (Sahlins 1963) probably do not derive from fundamental differences in principles of organization but rather in the way such 'Austronesian' principles are manifested, emphasized and combined in the respective culture areas" (1990:75-7). J. J. Fox (1989) has identified a symbolic process in eastern Indonesian cultures that he calls recursive complementarity. By this he means that a pair of complementary categories (in Needham's [1978] terms,

Trees

39 Exchange Goods Bridewealth

Dowry/Counter gift

Male goods

Female goods

M

F

Horses

M

Stallions

valuables F

M

Mares

Chains

Pendants M

F

I

I

Decorated pendants

Undecorated pendants

Figure 2.21. J. J. Fox's (1989) model of "recursive complementarity" in eastern Indonesian exchange.

"primary factors") are the "operators" that organize classification in diverse contexts - social, ceremonial, botanical, and so forth - and at "many levels of signification." Fig. 2.21 shows how the categories male-female organize the classification of exchange goods associated with marriage. Fox emphasizes the relative, and, to borrow an expression from Dumont (1980), the "encompassing" nature of this principle: By this principle of recursive complementarity, nothing is exclusively of one category; anything that is categorized according to one component of a complementary pair can potentially contain elements of its complement. A great deal of the symbolic elaboration of dualistic structures in eastern Indonesia involves playing with this principle of recursive complementarity: male contains female, female contains male; inside contains outside, and outside, the inside; black, white, and white, black. Similarly, wife-givers are also wife-takers, and a group that is classified as elder to one group may be younger to another (Fox 1989:46).

40

Island networks

(a)

(b)

(c)

Figure 2.22. Three rooted plane trees.

Boon (1990), in his study of Balinese symbolism, calls this principle relational contrast as opposed to substantive contrast. He emphasizes its magical aspect: "Following the views of Marcel Mauss (1967) a categorical distinction reapplied to itself generates manalike power. Moreover, the male valence of a female side, or the female valence of a male side may exhibit special charisma or enchantedness, heightened affinities attached to the gift, magic or love" (1990:125). In characterizing the formal properties of recursive complementarity, Fox maintains that "this principle should not be confused with hierarchy, since it is not wholly systematic and it rarely achieves great taxonomic depth" (1989:47). We can explicate this confusing statement and give a precise definition of all the structural principles discovered by Hocart, Sahlins, Bourdieu, Eyde, Fox, and Boon by using the model of a "twin binary tree." We require some preliminary definitions. In the mathematical literature, rooted trees have the root below and grow up, as shown in Fig. 2.16; in computer science, however, Knuth (1968) introduced the custom of placing the root above (uncircled) and having it grow down. We will follow the latter custom. A plane tree is a drawing of a tree in the plane: that is, it is a plane graph that is a tree. A rooted plane tree is a plane tree with a root node. It is depicted by placing the root at the top of the diagram. Fig. 2.22 shows three rooted plane trees, which (as we shall soon see) depict (a) a full binary tree, (b) a binary tree that is not full, and (c) a ternary tree. In describing the structure of a rooted tree, it is helpful to use the kinship metaphors of computer science. The root of a tree, T, is the ancestor of all the other nodes. In general, u is an ancestor of v (and v is a descendant of u) if v is reachable from u (in the "arborescence" of T).

Trees

41

T

2

Figure 2.23. Three full binary trees.

When u is adjacent to z/, one says that u is the father of v and v is a son of u. Some authors say, "parent-child," but none yet write "motherdaughter."5 The height of a rooted tree is the maximum distance from the root to some other node. Of course, this other node must be an endnode. The full binary tree Th of height h is the rooted plane tree such that 1. Every edge has slope either +1 or -1 in the plane. 2. There are 2h endnodes. 3. Every endnode has distance h from the root. The full binary trees Tl9 T2, T3 are shown in Fig. 2.23. A binary tree is a subtree of a full binary tree with the same root. A binary tree of height h is a binary tree contained in Th that has at least one endnode of Th. Fig. 2.24 shows all the binary trees with four nodes.6 A twin binary tree is one in which every father has two sons. In his otherwise splendid book Applied Combinatorics, Roberts (1984) defines a binary tree as one in which each father has two sons - that is, as a twin binary tree. Referring to Fig. 2.24, then, there would be no such binary tree with four nodes, nor with any even number of nodes. Fig. 2.25 shows all the twin binary trees with seven nodes. A ternary tree, also known as 3-ary, and in general an m-ary tree, are defined analogously. The structures identified by Hocart, Sahlins, Bourdieu, Eyde, Fox, and Boon are all implicitly twin binary trees in which the same pair of labels are applied to the two sons of every father. The four-part code de5 Linguists say, "mother-daughter." 6 It is not a coincidence that Fig. 2.24 shows 14 binary trees and that 14 is bn = (2n/n)/{n + 1). Rather it is a theorem that this Catalan number formula for bn is the number of binary trees with n nodes.

42

Island networks

Figure 2.24. The binary trees with four nodes.

Figure 2.25. The twin binary trees with seven nodes.

scribed by Sahlins is the full binary tree T2. Eyde's illustration of recursive dualism is T3, but his general model is Th. Fox's characterization of recursive complementarity as "not hierarchical because not wholly systematic" simply refers to twin binary trees that are not full. Thus the structure he displays in Fig. 2.21 is a hierarchy; specifically, it is a twin binary tree of height 4. The tree defined by Bourdieu's formula is a twin binary tree of height 2. Hocart's generic model of perpetual dichotomy in Fig. 2.18 is a full binary tree, but the example in Fig. 2.19 is a twin binary tree of height 6. In regard to the work of Fox and Boon, we should mention that other Indonesianists - for example, Barraud (1979), Forth (1985), and Howell (1985) - have adopted Dumont's (1980) terms hierarchical opposition^ or the encompassing of the contrary, to designate what Fox and

Trees

(a)

43

(b)

Figure 2.26. An out-tree and its dual in-tree. Boon call recursive complementarity and relational contrast. Dumont's concept of hierarchical opposition is structurally identical to and was inspired by Hocart's concept of perpetual dichotomy. Thus the opposition between noble and land groups in Fiji corresponds to that between pure and impure castes in India. As shown in Hage, Harary, and Milicic (1995), Dumont's nested-set diagram of this structure is very misleading. The "logical scandal" it purports to represent disappears completely when hierarchical opposition is modeled as a twin binary tree (or, equivalently, by a correct nested-set diagram).

Cognatic kinship networks A rooted tree can be represented digraphically as an unrooted oriented tree with every arc directed away from the root, as illustrated in Fig. 2.26a. This is called an out-tree, or in French an arboresence. The dual of an out-tree is an in-tree, in which all the edges are directed toward a single node, as illustrated in Fig. 2.26b, where the in-tree is the converse of the out-tree. In a twin binary in-tree, every child has two parents. Anthropologically interpreted, this is the parent digraph. It is particularly useful for describing personal kinship networks based on cognatic descent. In a society with cognatic descent groups, an individual can claim membership in as many different descent groups as he has ancestors who are members. This flexibility gives cognatic groups an advantage in small island societies faced with the twin problems of scarce land and fluctuations in the size of landowning kinship groups. Goodenough (1955) in fact has proposed that cognatic descent groups were a feature of Proto-Malayo-Polynesian (PMP) society. The type case is the Gilbertese ooi in Micronesia:

44

Island networks With the ooi type of group a person has membership in as many ooi as there are distinct landowning ancestors of which he is a lineal descendant. While he can expect little from those ooi which have become numerically large, he can expect a lot of land from those which have few surviving members. The overlapping memberships inevitable with unrestricted [cognatic] descent groups make them an excellent vehicle for keeping landholdings equitably distributed throughout the community (Goodenough 1955:80).

Another case, which has been studied in much greater detail and which illustrates the interaction between the rules of land tenure and the constraints of exogamy, comes from the Tuamotu atolls in East Polynesia. In the Tuamotus, cognatic landholding descent groups, 'ati, are variably distributed over a number of different atolls. Effective affiliation with a particular 'ati or 'ati segment is determined by residence, but an individual is always free to change his or her affiliation by moving to another district or island. In his monograph on Rangiroa Atoll in the northern Tuamotus, Ottino (1972) illustrates the spread of an individual's 6ati connections with the twin binary in-tree in Fig. 2.27 (arrows added). The root of this in-tree represents an individual, Mama Teipo, who resides in the Tipuka district of Rangiroa. The paths converging on the root show her connections through her parents and ancestors (males in capitals, females in lower case) to various other cati on Rangiroa and on other islands, including 'Ana'a to the southeast and Borabora in the Society Islands. This is only a segment of Mama Teipo's personal kinship network, as connections could be traced through still more remote ancestors to 'ati on numerous other islands in the Tuamotus. In the Tuamotus, where atoll populations typically number a few hundred or less, as opposed to the Gilberts, where they number a few thousand or more, the reason for remembering all these 'ati connections is not simply to gain access to land but rather to gain access to land in the inter-island search for a spouse. Without these inter-island cognatic connections, many inhabitants of the Tuamotus would be faced with the impossible choice of remaining at home, where they have rights to land but no chance of finding a spouse, or setting out for another island, where they could find a spouse but as "strangers" would have no access to land. In Ottino's view, exogamy was more important than politics and economics in accounting for "la mobilite polynesienne." In Chapter 3 we will see that every atoll in the Tuamotus belongs to a voyaging and marriage network.

Trees

45

PAPATA ('ati Fariua 'ati Ta'aroa)

Taupe ('ati Hoara)

TAPORA

Teua

TEHAU (ne a Teu 'ati Fariua)

Maua te Kaunuku ('§ti Fariua 'ati Taia)

TEVARIA ('ati Mota'i 'ati Marere)

Te hinano atoll de 'Ana'a

Teipo ('ati Mota'i 'ati Marere)

PUHENUA ('ati Fariua)

TEHAUA FA'ARERE ('ati Fariua 'ati Tetua)

TAHANIA AUTAI de Borabora

Tahuri a Tapuni

Tearevahine ('ati Fariua 'ati Tetua)

Tevahinenuihau ('ati Fairua 'ati Tetua)

MAURUA (*&ti Fariua Mota'i et Marere)

Mama Teipo

Figure 2.27. An in-tree model of 'ati affiliations in the Tuamotus (from Ottino 1972).

Cycle rank and network connectedness The third property of a tree in Theorem 2.1 is linked to the concept of the "cycle rank" of a graph. Cycle rank provides a basis for measuring the connectedness of a network. 7 In a recent archaeological study of the Lapita cultural complex, described in Chapter 3, Hunt (1988) gives an informal characterization of the "connectedness" of a number of exchange networks in Melanesia and West Polynesia. Referring to the Lapita exchange network in the Southeast Solomons-Vanuatu-New Caledonia region in Fig. 2.28, he observes that such networks "of linear configuration . . . are generally poorly connected. Such an observation may help to explain the great The concept of cycle rank has a variety of applications. It was first discovered by Kirchhoff (1847) in the study of electrical networks, as described in Chapter 1. In computer science, McCabe proposed the cycle rank of a digraph as what is now known as the "McCabe measure of software complexity."

46

Island networks SOLOMON ISLANDS

SOUTHEAST SOLOMONS

N

13

12 TIKOPIA

VANUATU

300

km

NEW CALEDONIA

Figure 2.28. A Southeast Solomons-Vanuatu-New Caledonia Lapita network (from Hunt 1988). human diversity known for the region, particularly evident in the chainlike structure of linguistic variation known for both New Caledonia (Grace 1973) and Vanuatu (Clark 1985)" (Hunt 1988:140). The terms "linear configuration" and "chain-like structure" suggest a tree in graph theory. A tree is in fact poorly connected, in the sense that every pair of nodes are joined by a unique path or, equivalently, that the removal of any edge will disconnect it, as stated in conditions 4 and 5 of

Trees

47

(b)

Figure 2.29. A graph and all its spanning trees.

Theorem 2.1. Graphs such as the one in Fig. 2.28 are not trees, although they are "tree-like." Since graphs vary in their resemblance to trees, we require a measure of their tree-likeness, or connectedness. In an early social science application of graph theory, the geographers Garrison and Marble (1962) developed just such a measure, which they called the "alpha index" of a graph. The alpha index measures the connectedness of a transportation network, that is, the degree to which towns and cities are joined by alternative paths, but it can be applied to many other types of communication networks. In order to explicate this measure, we need to define the concept of a "spanning tree." A spanning tree T of a connected graph G is a spanning subgraph of G that is a tree. The edges of T are called its branches. To illustrate, the graph in Fig. 2.29a, which is K4 - e, has eight different spanning trees,8 as shown in Fig. 2.29b. 8 A formula for the exact number of spanning trees of a given graph was derived by Kirchhoff (1847).

48

Island networks

T:

G:

Figure 2.30. A graph, G, a spanning tree, T3 and the set of independent cycles obtained from T. Consider the graph G and the spanning tree T in Fig. 2.30. In a connected graph G, a chord of a spanning tree T is an edge of G that is not in T. In Fig. 2.30 the edges 13, 24, 25, 35 are the chords of T. Clearly, the subgraph of G consisting of T and any chord of T has exactly one cycle; see Theorem 2.1, statement 3. Moreover the set Z(T) of cycles obtained in this way (one from each chord) is independent, since each contains an edge not in any of the others. The bottom half of Fig. 2.30 shows the set of independent cycles obtained from T. We define m(G)9 the cycle rank of G, to be the number of independent cycles of G. Kirchhoff observed that this invariant is given by the following result. THEOREM 2.2. The cycle rank of a connected (p, q) graph G is equal to the number of chords of any spanning tree ofG, that is,

m(G) = q-p + 1. COROLLARY

2.2

2.3 (a). If G is a (p, q) graph with k components, then

m(G) = q-p + k.

2.3

The cycle rank of a tree is obviously 0. The cycle rank of the graph in Fig. 2.30 is

Trees

a =0

a =66.6%

49

a = 33.3%

a = 100%

Figure 2.31. Illustrations of the alpha index of planar graphs (from Haggett 1967).

m(G) = 9 - 6 + 1 = 4. The set Z(T) of independent cycles, so obtained from a tree T, is called the system of fundamental cycles, determined by T, while the cycle rank, m(G), is also known as the cyclomatic number, or the "first Betti number," of a graph. (The "zero'th Betti number" of a graph is the number of components.) Garrison and Marble (1962) interpret the cyclomatic number of a graph as a crude measure of network redundancy: by itself it does not provide a readily intelligible measure of structure, since its upper bound is determined by the number of nodes in a graph. They therefore define an alpha index, which is based on the ratio of the observed number of independent cycles to the maximum number of independent cycles in a planar graph. The maximum number is 2p - 5. Multiplying a by 100 gives it a range of 0 to 100, which can be interpreted as "percent redundancy." Hence the alpha index of G is defined by

50

Island networks

Illustrations of this index are shown in the four 10-node graphs in Fig. 2.31 from Haggett (1967). The first graph is a tree with a = 0 percent, while the fourth graph is maximally planar, having a = 100 percent. In developing Hunt's intuition concerning the structure of exchange networks in Melanesia, we suppose that the magnitude of alpha is inversely related to the degree of linguistic (and cultural) differentiation in a regional network. The greater the connectedness or circuitry of a network, the greater the probability of linguistic or cultural innovations spreading from community to community along alternative communication paths. The alpha index of the Southeast Solomons-Vanuatu-New Caledonia Lapita graph, which is planar although not plane, is relatively low, with a = 44 percent. There are, of course, exchange and communication networks whose graphs are nonplanar, for example the Mailu network in Fig. 7.6. In that case (Haggett and Chorley 1969), the alpha index is / q-p+k \

The minimum spanning tree problem

A savage's mind is anything but defective. He is just a normal social animal whose chief interest in life is his relation to his neighbors. A. M. Hocart, "Psychology and Ethnology" The minimum spanning tree problem (MSTP) is a well-known topic in combinatorial optimization. We illustrate it with an instance of how to determine the monthly telephone charge to a generic large corporation G with offices in many cities, vu . . . , vp. All the distances d(vh vj) are known and are distinct. The corporation G does not wish to pay the phone company an amount proportional to the sum ^d(vh vj) of all the distances, since not all pairwise connections are needed for each office to be able to communicate with every other office. What is needed is a tree on these p nodes having minimum total distance. In their definitive article on the history of the MSTP, Graham and Hell (1985) give three reasons for its popularity and importance. (1) It has efficient solutions that make it applicable to the analysis of large graphs, including graphs with thousands of nodes. (2) It has direct applications to the design of all kinds of networks, including communication, computer, transportation, wiring, flow, and other networks. (3) It has applications to numerous other problems, including network reliability, speech recognition, classification, and clustering. Graham and Hell also note that the greedy method common to MST algorithms is a source of many further applications. For similar and analogous reasons, the MSTP has potential applications to anthropology. (1) It offers a means for analyzing large realworld networks, including those that are known to be connected but whose internal structure is not known in detail. (2) It provides the basis for a nonlinear, branching alternative to standard techniques of seriation. (3) Under the heading of design problems, MST algorithms provide models for simulating processes of network growth. In solving an 51

52

Island networks

MSTP, any of three well-known algorithms can be used. For theoretical and practical reasons, we will use all three. We will use Kruskal's (1956) algorithm and the concept of clustering in an MST to analyze linguistic and matrimonial subgrouping in the Tuamotu Islands in Polynesia, and Boruvka's (1926a, b) algorithm to simulate the evolution of overseas chiefdoms in the Lau Islands, Fiji. In an archaeological application, we will show that the method of "close-proximity analysis," developed in Mediterranean studies and applied in Oceania (Renfrew and Sterud 1969; Green 1978), is an independent discovery of an MST algorithm. As such, it can be replaced by Prim's (1957) algorithm, which has an explicit mathematical foundation and a more efficient means of computation.

Dialect groups and marriage isolates in the Tuamotus Linguists have discovered that diversity in many Austronesian (AN) regions is the result of a weakening of ties between sister dialects rather than the isolation of daughter languages. Pawley and Green (1984) have therefore proposed that an adequate account of AN subgrouping will require a "network-breaking model" to complement the traditional "radiation model." The radiation model, familiar from Polynesian linguistics, "posits an initial period of unified development undergone by a localized, homogeneous language community, followed by a period of geographic expansion, leading to the creation of dispersed, isolated daughter communities which develop independently from the time of dispersal" (Pawley and Green 1984:138). The network-breaking model takes account of recent evidence for the rapid expansion of AN over a vast region. It assumes the capacity of early AN speakers to "maintain a fairly unified speech tradition across a network of local communities dispersed across an archipelago - a unity that may last for many centuries, even millennia, before there is a decisive divergence of local dialects" (1984:138-9). In this case the protolanguage breaks up, not through geographical expansion but through a "gradual weakening of ties [in] the network of sister dialects," leading eventually to distinct language boundaries. Applications of the network-breaking model are indicated for Proto-Fijian, Proto-Central and North Vanuatu, Proto-Southeast Solomon, and ProtoTrukic and the application of both models for different stages of ProtoCentral Pacific and Proto-Remote Oceanic. Geographical barriers in the form of remoteness or topography will

The minimum spanning tree problem

53

obviously lead to the breakup of a language network, but there are social factors as well. Even a network of communities distributed over a homogeneous surface, having an adequate communication technology, will break up into dialect groups and eventually separate languages through the spontaneous formation of marriage isolates. Three questions arise concerning these isolates. (1) How are they formed? (2) How large are they, that is, what are their upper and lower bounds? (3) What is their internal structure? Some preliminary answers are provided by data from the Tuamotu Islands in East Polynesia. The Tuamotu Islands, known to early European navigators as the Dangerous Islands or the Labyrinth Islands, consist of a single raised coral island, Makatea, and 75 atolls, all of which trend southeast to northwest between 135° and 145° W. and 14° and 23° S. The islands lie in the eastern trade wind belt and are subject to occasional severe storms in the fall and winter months. The environment is relatively uniform, with rainfall averaging 45 to 60 inches per year (O. W. Freeman 1951). Crops are limited to pandanus, supplemented in some islands by coconut, arrowroot, and taro. Marine resources include fish, turtles, and shellfish, especially the tridacna clam. Half of the islands, found mainly in the southeastern, more scattered part of the archipelago, are uninhabited or only temporarily inhabited by seminomadic populations. With the notable exception of 'Ana'a, which numbered 2,000 or more inhabitants in the eighteenth and nineteenth centuries (Emory and Ottino 1967), the populations of most islands are small, ranging from less than 100 to a few hundred. The total population is about 6,700. In former times periodic severe food shortages and raiding were constant features of island life. Map 3.1 (from Emory 1934) shows the layout of the archipelago, its relation to Tahiti and the Marquesas Islands, and the names of many of the more prominent atolls. Tuamotuans were renowned shipwrights and seafarers (Haddon and Hornell 1975). For example, Takumeans (nearest neighbors of Raroians in Map 3.1), sailed 400 kilometers to 'Ana'a, an island favored by its relative wealth and proximity to Tahiti (Lucett 1851; Dening 1963), while 'Ana'ans undertook military expeditions as far east as Vahitahi (Emory and Ottino 1967). Western Tuamotuans regularly sailed 290 to 370 kilometers to Tahiti, where they exchanged mats, pearl shells, and red feathers for basalt adzes and other high island goods. Most voyaging within the archipelago, however, was for purposes of warfare, migration, and marriage rather than trade (Emory 1975). Because of the islands' relative isolation, inaccessibility, and lack of exploitable resources, Tuamotuans retained many features of ancient East Polynesian culture, including types of artifacts and religious ideas (Emory 1975). Tuamotuan social organization was based on landhold-

54

Island networks

Marquesas Is.

Tepoto

Makatea o

O

oTakaroa

'©Napuka

^Rairoa U Takapc)tO ^aODAPataki

o Pukapuka

15° s. lat.

Fagatau „ . , . Q ^ Fakahina u au ura a x ) - . Q Kati Hikueni ^ Amanu Marokau \ 9\Hao Akiaki ^ Pukarua ^Vahitahi Reao

o

C7/"\ Tahiti

135*

o

Vairaatea ® ^Nukutavake ° Pinaki

a

20* s. lat.

* 0

100

•

o Tureia

200 miles

S. Marutea

§>

AUSTRAL IS.

0

p

MANGAREVA

Raivavae

rimoe

Map 3.1. The Tuamotu Archipelago (from Emory 1934). ing cognatic descent groups known in the western and central islands as 'dti (Ottino 1967).1 'Ati were distributed over a number of islands, while individual segments of an 'dti, sometimes called S6pu, were associated with specific estates and island districts. Effective 'dti or opu membership was activated by residence. Within an 'dti a senior patrilineal line of descent, an 'dti ariki, defined chiefly rank and succession. The status of an 'dti was determined by its depth, the number of its branches, its historical associations, and the number of illustrious names in its genealogies. 'Ati or 'dti segments were associated with maraes, religious structures consisting of rectangular stone-enclosed courts with altars in which ancestral deities were invoked or worshiped. Marae ceremonies included the sacrifice of turtles, sacred fish, and enemies slain in warfare, the celebration of rites of passage - birth, succession, and death 1 In Pukarua, in the southern part of the Tuamotus, this kinship group was known as a pupu, a generic term for a group or class of things (Hatanaka 1971).

The minimum spanning tree problem

55

and the offering of prayers to end famine and secure blessings for a voyage (Emory 1947). All the islands of the Tuamotus were connected by multiple, overlapping bonds of kinship and marriage, as indicated by the extensive interisland sharing of marae names (Emory 1934) and as implied by the fact of small populations having a broad bilateral extension of the incest taboo. Unfortunately, little is known about specific connections between particular islands, making it impossible to construct a conventional network model of the archipelago. It is known, however, that the islands were divided into a number of named dialect groups (Stimson 1964). These groups, which are regarded by Tuamotuans as "ancient cultural and linguistic divisions" (Danielsson 1956), can be identified as clusters in the network's "minimum spanning tree." A network, N (or weighted graph, W) is a graph with a numerical value, f(e), assigned to each edge e. In particular, every graph can be regarded as a network in which all the edges have value 1. A minimum spanning tree (MST) is a spanning tree of N with minimum edge-value sum. Fig. 3.1 shows a network N and an MST. The two best-known MST algorithms are by Kruskal (1956) and Prim (1957), although a third algorithm, discovered by Boruvka (1926a, b), has historical priority. All three algorithms are "greedy" in the sense that they bite off smallest pieces of the network, that is, they add edges of the smallest weight first. Kruskal's algorithm is conceptually the simplest. It operates by adding a new edge of minimum value that does not create a cycle. ALGORITHM

3.1. KruskaVs algorithm for constructing an MST.

Given An undirected network N, with distinct positive integer values (equivalently, positive real numbers) f(e) on each edge e of N. Wanted A spanning tree T of N with minimum edge-value sum. We will specify such a tree by building its edge set ET. Step 1. Label the edges of N by eu elr>. .., eq, such that whenever / < /, we have f(et) < f(ej). Call this sequence of edges a. Step 2. Place ex in ET. Step 3. On arriving in d at a generic edge eh place e{ in ET if and only if e{ together with the edges of N already in ET do not contain a cycle. If e{ is not placed in £ T , go to step 4. Step 4. If l£Tl = p - 1, stop. Otherwise repeat step 3. THEOREM 3.1. Given a connected network N in which all the edge-values are distinct, KruskaVs algorithm will terminate with N's unique minimum spanning tree T.

Kruskal's algorithm is illustrated in Fig. 3.2. In step 1 we order the

56

Island networks b

N:

a

MST:

f

1

1

c

a

e

Figure 3.1. A minimum spanning tree (MST) of a network N. edges of the network from lowest to highest value, from 1 to 12, to obtain the sequence cr. In step 2 we add to ET the edge be, which corresponds to the edge el9 in a (Fig. 3.2a). In step 3 we add the edge de (Fig. 3.2b). Repeating step 3 two more times, we add the edges a/and fg9 neither of which creates a cycle (Fig. 3.2c and 3.2d). We cannot add the next edge, ag, because this would create a cycle (afga) so we proceed to step 4. The number of edges in ET is p - 4, and ET does not yet have p 1 = 6 edges, so we return to step 3. The next edge examined is dg, which does not create a cycle, so we add it to ET (Fig. 3.2e). We cannot add the next two edges, ef and eg, because they would create the cycles (defgd) and (degd) respectively. We then add the edge eg Fig. 3.2f). Turning to step 4, we see that ET now has six edges, so we stop. We have constructed the unique MST of this network N. As stated, Kruskal's algorithm builds a unique MST when all values of N are distinct. When they are not distinct, one of several MSTs may be obtained. We note that the graphs in Figs. 3.1 and 3.2 show all the edges of N with their values e,. In our first two - geographical - applications of an MST all the edges of a complete network Np are implicitly present, with their values given by their physical - map - distances. Thus we proceed by connecting the dots. In constructing an MST of the Tuamotus, we will restrict the nodes to the set of permanently inhabited islands. On the basis of information in Emory (1939, 1975), Lucett (1851), Friederici (1911), and Robineau (1977), it appears that 38 of the 76 islands were uninhabited or only temporarily inhabited, leaving 38 islands as the node set of the Tuamotus network.2 The MST of this network is shown in Fig. 3.3. Since the 2 This set of islands corresponds to the set in the official French Census in 1951 (Tessier 1953) with two exceptions: the census includes Tureia but excludes Aratika. It apparently merges two closely adjoining island pairs: Tepoto North-Napuka is listed simply as "Napuka," and Takume-Raroia as "Raroia." Our analysis would not be changed in any significant way by including Tureia and excluding Aratika.

The minimum spanning tree problem 1

b

57

c

N:

b

,

c

b

,

(a)

(b)

(d)

(e)

c

b

i

C

(f)

Figure 3.2. Generating an MST using Kruskal's algorithm. The illustrated network N has q - 12 and /"(e,) = / for / = 1, . . . , 12.

distances between all pairs of islands are different, the edges et all have distinct values, making this the unique MST. In his Dictionary of Tuamotuan Dialects, Stimson (1964) classifies the Tuamotuan language as a "complex of interrelated dialects" descended in the AN family tree from Proto-Eastern Polynesian (PEPN).3 He identifies nine major culturally recognized dialect groups: Mihiroa, Vahitu, Tapuhoe, Tapuhoe-Tauaro, Marangi, Parata, Napuka, Reao, and Fangatau, each consisting of two or more islands. 3 Pawley and Green (1975) classify Tuamotuan together with Maori, Rarotongan, Marquesan, and Tahitian as descendants of Proto-Central-Eastern Polynesian, which derives from PEPN.

58

Island networks

By definition, an MST connects all the nodes of a network N. It therefore does not define clusters, but it does contain information useful for clustering.4 Thus an edge with a large value (in this case physical distance), relative to other edges incident with a node, can be "cut" to form two clusters. Proceeding on this basis, if we cut the edges joining Pukarua and Tatakoto, Hao and Vairaatea, Napuka and Fangatau, Takume and Fangatau, Katiu and Faaite, Niau and Fakarava, Arutua and Ahe, we get eight clusters that represent eight of the nine dialect groups of the Tuamotus. The ninth cluster and dialect group, Parata, is obtained by cutting a second (longest) edge incident with Faaite. These clusters are shown as encircled subtrees of the MST in Fig. 3.3. Besides these major clusters in the network, there are minor clusters representing subdialects (incipient dialects): (Amanu, Hao) and (Vahitahi, Nukutavake, Vairaatea) are subdialects of Tapuhoe and Marangi respectively. There are also superclusters, representing superdialects (former dialects): Tapuhoe included Tapuhoe and Tapuhoe-Tauaro, and Marangi once included Reao. Vahitu formerly included Mihiroa, "before the speech of these islands had become modified by contact with Tahiti" (Stimson 1964:22). All of these partitions are consistent with the rule of clustering in the network's MST. Danielsson (1956) identifies virtually the same divisions as Stimson. The only apparent difference in his map is the inclusion of Aratika in Vahitu, which would violate the rule of clustering. He also classifies Fangatau and Napuka as a single division, Tupitimoake, which would combine these two island clusters into a single cluster. The partitioning of the Tuamotu Islands into dialect groups, then, is structurally determined, although not uniquely so, since some different edge cuts could have been made, resulting in a somewhat different clustering. The four largest dialect groups are Mihiroa, Vahitu, Tapuhoe, and Marangi. The first three are named after eponymous tribal ancestors who discovered and colonized these areas. The fourth group is named after the southeast trade wind, an allusion to its route of entry into the Tuamotus (from Mangareva?). Most of the atolls were probably settled from Tahiti or other islands in the Societies. Mihiroa, in the northwest, is the most Tahitianized of all dialects. It is distinguished by the replacement of the velar stop /k/ and the velar nasal /rj/ by a glottal stop. Tapuhoe, in the center, is the most common dialect and has two main subdivisions, Tapuhoe and Tapuhoe-Tauaro, as indicated in Fig. 3.3. For alternative approaches to clustering, based on multidimensional scaling, see the volumes edited by Shepard, Romney, and Nerlove (1972) and Romney, Shepard, and Nerlove(1972).

VAHITU NAPUKA

MIHIROA

REAO

MARANGI Figure 3.3. The MST of the Tuamotus network, clustered to show dialect groups. (Island locations are based on the official French Survey Map, 1947-52.)

60

Island networks

Each of the five remaining dialect groups consists of a single pair of islands. Parata includes 'Ana'a and its colony Hereheretue. The name refers to a ferocious species of man-eating shark evocative of the qualities of the 'Ana'an warriors who conquered the western Tuamotus in the eighteenth century. (See Chapter 7.) Parata is said to resemble the dialects of both Tahiti and Fangatau, although it is closer to the former. It retains the Tahitian l\d and /h/ but sometimes substitutes /rj/ for /k/. Fangatau and Napuka dialects are, as noted, sometimes collectively referred to as "Tupitimoake," a reference to the area's stormy weather. Fangatau "seems to be close to Old Tahitian and has marked resemblances to the dialect of Ra'ivavae [in the Southern Cooks]; it preserves certain distinctions from other dialects, both in vocabulary and grammatical structure" (Stimson 1964:23). The two most divergent dialects are Napuka and Reao, remote endclusters in the network's MST. They had evidently diverged to the point of becoming distinct languages: Old Tuamotuan natives say that neither they nor their immediate ancestors could understand them.5 In Napuka the principal divergences are semantic, whereas in Reao, they are both semantic and structural, particularly the latter. The old dialect of Napuka contains many words not now found with the same or similar meanings elsewhere in the Tuamotus, and seems to have been related to Marquesan. The local traditions recount that the first colonizers came from the Marquesas, although the aberrant forms, while apparently derived from Polynesian roots, do not indicate a close affiliation with Marquesan as spoken today (Stimson 1964:23). Emory (1932:42) alludes to "curious peculiarities in [the] language, physical type, and culture" of Napuka. These differences are most pronounced in Reao: The Reao dialect raises interesting problems, both historically and morphologically. There are indications that it may be an extremely archaic form which has survived due to some unknown factor, perhaps isolation over a prolonged period, and not a congener of the other widespread and closely interrelated Polynesian speech groups. The language may contain an example of an infix - a linguistic phenomenon not recorded, to my knowledge, east of Samoa and Tonga. There are traces of other 5 Audran (1919:36) observed that "Deux Napuka aussi bien que deux Reao peuvent tres bien tenir une conversation dans leur dialect devant un autre Paumotu sans etre compris de lui."

The minimum spanning tree problem

61

anachronisms of grammatical process and morphological structure which should be verified independently. The kinship terms of Reao differ markedly from those of the Tuamotus in certain respects, and are by far the most systematic and detailed of any recorded from the archipelago, although Marshall tells me that similar systems from Rarotonga and Hawaii approach them in completeness if not in the formative method employed. Again, their counting method is the most systematic and complete of any I have recorded. It is of general interest to note that, in contrast with most Polynesians, the people of Reao are short, stocky, very strong in the arms rather than the legs, and unusually dark in color. The face has a vertically "compressed" appearance that is very distinctive (Stimson 1964:23). What are the social and numerical bases for the partitioning of the Tuamotus network into dialect groups? The dialect groups were evidently not political units, at least not uniformly or consistently so. According to Danielsson (1956:40), "the divisions seem to have been geographical and linguistic rather than political units as all available information indicates that each atoll (or atoll pair like Takume-Raroia, Napuka-Tepoto, Takaroa-Takapoto, Hao-Amanu) was completely independent and often even at war with atolls in the same division."6 Teuira Henry (1928) states that in former times the islands of Ahe, Manihi, Takapoto, and Takaroa (the Vahitu group) and the islands from Fakarava to Hao (Tapuhoe and part of the Tapuhoe-Tauaro groups) were each ruled by single kings, but there is no further mention of this. She refers to the eight islands west of Fakarava (the Mihiroa group) as "independent little kingdoms." In his ethnography of Rangiroa Atoll, Ottino (1972) describes these islanders as members of a restricted marriage network. Although Rangiroans, like other Tuamotuans, have, in Levi-Strauss's (1969) sense of the term, a complex marriage system, with great freedom of choice, they intermarry mainly with inhabitants of other atolls in the Mihiroa group: Tikehau and Mataiva, and to a lesser degree Makatea, Kaukura, Apataki, Niau, and Arutua. Mihiroa is in effect a "marriage isolate": the inhabitants of Rangiroa represent but one portion of a larger regional set of populations, who since remote times, have settled a historical, linguistic and culture area covering the western Tuamotus known as Mihiroa. The total regional population of originates (a French word that conveys best the east6 These atoll pairs are among the first independent edges to be added to the MST in Fig. 3.3 using Kruskal's algorithm. There may well be other such pairs distinguished by the algorithm but not yet identified ethnographically.

62

Island networks ern Polynesian concept of tumu which expresses the idea of originating from and belonging to such and such an atoll) or more simply "natives" originally from different islands of Mihiroa, seems to be fairly closed and forms what French demographers have called an isolat matrimonial (Ottino 1973:2).

Following Ottino, we suppose that all of the dialect groups in the Tuamotus are, like Mihiroa, marriage isolates. The concept of a marriage isolate was introduced by the Swedish population geneticist Wahlund (1928). He observed that the marriage choices of an individual are always restricted by geographical and social barriers, from which he deduced that a large population necessarily consists of smaller partial populations within which marriage is relatively unrestricted, or "panmixia" These partial populations are called "marriage isolates." Dahlberg (1948) calculated the size of marriage isolates from the expected frequency of first-cousin marriage, setting the lower and upper bounds at 400 and 1,600 and interpreting relative size as a function of the degree of urbanization. Dahlberg did not insist on these exact numbers, and it seems likely that the variation in isolate size is greater than he thought. Thus, Sutter and Tabah (1951) found that the average size of isolates in French departements ranges from fewer than 1,000 to more than 2,800 individuals. They also found that the smallest isolates are not restricted to rural areas. In human populations, marriages are obviously not panmixic or random but are regulated by culturally defined preferences and prohibitions (Sutter and Tran-Ngoc-Toan 1957). In introducing Dahlberg's work into social anthropology, Levi-Strauss (1963) emphasized the need to study structural as well as numerical aspects of marriage isolates, that is, the length of the marriage cycles. Cycle length is independent of isolate size: a small isolate may contain long cycles, while a large isolate may be made up of shorter ones. Hence the need for close cooperation between demographers and social anthropologists. Table 3.1 gives the populations of the Tuamotus from the official French Census of 1951 (Tessier 1953), with the islands arranged in dialect groups. Since figures were not available for four islands, the sizes of four dialect groups are approximations. Regarding the dialect groups as marriage isolates, the data in Table 3.1 permit us to make the following observations. 1. Marriage isolates as island networks. The populations of islands in the Tuamotus are small, in some cases less than 100, most frequently less than 200, and with few exceptions less than 300. Given the broad bilateral extension of the Tuamotuan incest taboo, inter-island marriage networks are inevitable. On Raroia Atoll, for example, marriage is for-

The minimum spanning tree problem

63

Table 3.1. Population of the Tuamotus in 1951 Dialect groups and islands Mihiroa Mataiva Tikehau Rangiroa Arutua Apataki Kaukura Niau Makatea Total

Population 1951

Dialect groups and islands

126 263 712 114 253 282 232 —

Tapuhoe Raroia, Takume Taenga Makemo Katiu Hikueru Marokau Tauere Amanu

> 1,982

Vahitu Ahe

Manihi Takapoto Takaroa Total

180 129 175 220 704

Parata

'Ana'a Hereheretue Total Tapuhoe-Tauaro Aratika Kauehi Fakarava Faaite Total

481 41

"522

— 186 206 87

>479

Hao

Total Napuka Tepoto North, Napuka

Population 1951 160 162 274 61 171 115 194 137

> 1,274 284

Fangatau

Fangatau Fakahina Total

152 124 276

Marangi Vairaatea Nukutavake Vahitahi Tatakoto Total

193 101 176

Reao Reao Pukarua Total

>470 318 176 494

Note: The census does not give the populations of Aratika, Tauere, and Vairaatea. The population of Napuka is evidently included in the figure for Tepoto North and that of Takume in the figure for Raroia. There is no figure for the native population of Makatea. Most of its 1951 population of 1,758 had come to work as phosphate miners. The population of 'Ana'a, the most fertile of all islands in the Tuamotus, was between 1,500 and 2,000 before the hurricane of 1877 (O. W. Freeman 1951). Source: Data from the official French Census of 1951, quoted in Tessier 1953.

bidden between "all persons more closely related than third degree cousins." Adoption is common, and adoptive ties count the same as those of consanguinity. With a population of a little over 100, the chances of finding a spouse on the home island are limited (Danielsson 1956:124). Even when marriage prohibitions are adjusted to population size, there is no alternative to inter-island marriage and migration:

64

Island networks The necessity of mobility derives from the rule of exogamy which prohibits marriage with persons whom one considers, even in Polynesia, as distant relatives. If one considers the demographic situation of an atoll (Rangiroa with two villages and 700 inhabitants being an exception, as most atolls do not count more than 150 to 200 persons), this rule can only increase with every new union, restricting even further for the children to come the category of marriageable persons, rapidly reaching the point of rendering every union impossible . . . In the remote atolls of the east, before marrying, the inhabitants are obliged to redefine the limit of incest and to authorize unions with second cousins, a practice which would be absolutely prohibited in Rangiroa. Nevertheless, taking into account the relative size of demographic groups, this changes practically nothing in the problem, and the only solution is the departure of part of the population (Ottino 1972:445, our translation).

2. Isolate size. The isolates range in size from just under 300 to just over 2,000 individuals which is more in line with the empirical findings of Sutter and Tabah than the theoretical expectations of Dahlberg. There is reason to believe that the isolates are demographically stable. The dialect groups are said to be ancient divisions, and available evidence indicates that the total population of the archipelago has remained about the same for the past 100 years (Danielsson 1956; O. W. Freeman 1951; Ottino 1972). In the case of Mihiroa, During the past century this regional population seems to have remained remarkably stable (as far as it is possible to judge indirectly from the genealogies, ancient habitation sites, documents, etc. fluctuating around 2,000 inhabitants). This phenomenon, unique in all of East Polynesia which experienced a rapid demographic progression after the dramatic decline of the last century, indicates that for some unknown reasons atoll demography, compared to high island demography, has its own particular characteristics (Ottino 1973:2). The populations of individual atolls may fluctuate, but this is "not very significant in itself as it expresses nothing more than a temporary change caused by the unequal distribution of births, marriages, and adoptions or merely the movement of individuals from one place to another" (Ottino 1973:2). Within each isolate the islanders' major demographic concern is to maintain adequate numbers and balanced sex ratios in local (dti groups comprising at most 30 to 50 persons. This is ac-

The minimum spanning tree problem

65

complished by various "institutions of mobility," including cognatic descent groups, nonprescriptive marriage rules, bilocal residence, and especially the widespread practice of adoption. "These institutions made it possible to compensate rapidly for the unequal statistical distribution of births by redistributing people among combinations of two, three, or four traditionally related 'dti, thus maintaining a total number of individuals and a sex ratio more suitable to familial and economic needs, including those of defense" (Ottino 1970:88). 3. Isolate structure. The one detailed study of kinship in the Tuamotus, by Ottino (1972), is primarily concerned with the structure of descent groups rather than marriage alliance. Thus, it is only possible to indicate some general features of isolate structure. The Tuamotuan marriage system is complex, in the sense that it lacks any positive rule governing the choice of spouse. The only rule is a negative one specifying the prohibited degree of marriage. As in many other "Hawaiian" systems, prohibited degree is widely extended in theory but limited in practice. Tuamotuan marriage prohibitions are qualified in three ways. First, as already indicated, prohibited degree is adjusted to isolate size. In Rangiroa (Mihiroa group, pop. > 1,982) marriage prohibitions extend to the sixth collateral degree (Ottino 1972); in Raroia (Tapuhoe, pop. > 1,274) they extend to the third degree (Danielsson 1956); while in Reao (pop. 494) they extend only to the second degree (Hatanaka 1971). Secondly, a distinction is made between prohibited and "forbidden" degree. In Rangiroa the latter extends only to the third degree of coUaterality, beyond which marriage is in fact tolerated although not encouraged. Thirdly, as in many cognatic systems (J. D. Freeman 1961a; Heritier 1981), the recognition of consanguinity depends on birthplace and residence. In Rangiroa individuals do not count as close relatives (feti'i proches, i.e., first cousins) unless they are known personally. If they were born or live elsewhere, or if one has no habitual relationship with them, they are regarded as distant relatives (feti'i eloignes, i.e., fourth cousins), even though they are biologically close. As Ottino (1972:217) emphasizes, "II s'agit de faits d'appreciation sociologique qui ne se calculent pas uniquement en degres de consanguinite." Like other complex marriage systems, the Tuamotuan system has "elementary tendencies," that is, ways of achieving more restricted cycles of alliance. In the case of Mihiroa, Ottino mentions the importance of "the locality principle, which allows marital bonds between very close relatives, provided the mates have been born and raised in separate places" (1967:472); "recurrent matrimonial alliances occurring between the same descent groups, whether 'dti or *~opu" (1967:472); and marriage, adoption, and residential changes between "combinations of two,

66

Island networks

three or four traditionally related sdti" (1970:88). He also interprets the present high rates of adoption as a partial substitute for an earlier, preEuropean practice of sister exchange: "faute de pouvoir epouser leurs proches parents [ils] les adoptent" (1972:440). We note that sister exchange, which is not repeated in successive generations, is a common practice in complex (e.g., Hawaiian) and semicomplex (Crow-Omaha) marriage systems, as attested in Oceania, for example, in the Gilbertese and Trukese systems respectively (Grimble 1989; Goodenough 1951). To summarize, all the dialect groups of the Tuamotus are joined in a single connected network, and each group is a subnetwork separated from other subnetworks by a well-defined rule of clustering in the network's MST. None of these groups is separated from its nearest neighboring group by a distance of more than 160 kilometers, the estimated limit of overnight voyaging and casual visiting in the Pacific Islands (Marck 1986). The dialect groups correspond to marriage isolates whose lower and upper bounds are about 280 and 2,000 respectively. The range is greater than would be predicted from Dahlberg's theoretical model but just as variable as the range that Sutter and Tabah found in their study of marriage isolates in modern France. With a longer time span comparable to that of settlement in West Polynesia, all of these sister dialects in the Tuamotus would have eventually become separate daughter languages. Networks such as this one offer rich possibilities for interdisciplinary research in demography, population genetics, linguistics, and anthropology, with many useful applications of graph theoretic models.

The evolution of the Lakemban matanitu An MST algorithm builds a network piece by piece. Each algorithm does it in a different way, and each can serve as a model of network growth. As an illustration, we will simulate the evolution of the Lakemban overseas chiefdom in Fiji, using the parallel-processing MST algorithm of Boruvka. This is a case in which genealogical relations, marriage alliance, and propinquity are joint determinants of network structure. The Lau archipelago numbers some 100 islands located roughly between 17° and 21° S. and 178° and 180° W. in eastern Fiji. (See Map 3.2.) The islands, about one-third of which are inhabited, are both limestone and volcanic in type. They lie in the path of the southeast trade winds, which blow between the months of April and October, the dry season and the time of most inter-island voyaging. The province of Lau consisted of three ancient divisions (Hocart 1929). The islands in the

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© Vanua Vatu

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68

Island networks

north were subjects of Thakaundrove, a district of Vanua Levu. The islands in the southwest - Moala, Totoya, and Matuku - were independent, although in the orbit of, or "facing toward," Mbau, a large and powerful chiefdom centered on a small island off the coast of Viti Levu in western Fiji. The central islands of Thithia and Naiau and all the southern islands were included in the chiefdom of Lakemba. The principal islands or island clusters of southern Lau include, besides Lakemba, from north to south, Aiwa (an uninhabited overnight stop on voyages to and from Lakemba), Oneata, Olorua (uninhabited in historical times), Mothe, Komo, Namuka, Wangava (once inhabited), Kambara, Fulanga, Ongea, and the two more distant southerly islands of Vatoa and Ono. All the inhabited islands, with the exception of Lakemba, had small populations numbering in the hundreds. The figures in 1946 (from Derrick 1951) are as follows: Mothe (400+), Fulanga (474), Ongea (about 100), Kambara (513), Namuka (317), Komo (161), Lakemba (1,883), Thithia (730), and Naiau (400). Social organization in southern Lau was based on agnation, rank and cross-cousin marriage (Thompson 1940). Patrilineal, patrilocal, exogamous clans were organized into phratries defined by ancestry, class, and, in part, occupation. The land phratry, yavusa vanua^ formed the commoner class, and four other phratries formed the noble class. The latter included the chiefs' phratry, yavusa turanga, the chiefs' carpenters, matai sau^ a carpenter phratry of Samoan origin, lemaki, and the "sea people," kai wai. Not all phratries were represented on all islands. The chiefs' phratry was found only on Lakemba, Mothe, and Kambara. The kai wai phratry was found on Fulanga and Ongea, islands whose inhabitants were major food importers and active traders, and it also included the Levukans of Lakemba, a local group of professional sailors and potters whose ancestors were immigrants from Mbau. Kambaran carpenters were evidently of the lemaki phratry (Derrick 1951). Most clans belonged to either the chiefs' or the land phratry. In the Lauan version of the myth of the "Stranger-King" (Sahlins 1985), which is ubiquitous in Fiji and widespread in Oceania (Sayes 1984; Hanlon 1988), chiefly clans are portrayed as descendants of enlightened and culturally superior immigrants led by a chief who imposed a class system on autochthonous populations about ten generations ago. Lauan traditions emphasize primogeniture and wife-taking as the bases of rank: The ranking system was founded mainly on seniority in relationship to the leader, Ndaunisai, and on success in warfare. Hence clan genealogies were important. Rank was expressed in hereditary clan titles depending on a historical division of

The minimum spanning tree problem

69

clans. Clans which descended from the immigrants (who married women of the early inhabitants and acquired land thereby) formed the chiefs' phratry or nobility (yavusa turanga). Clans which descended from the early inhabitants formed the land phratry or lower class (yavusa vanua). The highest rank was held by the high chief, who was directly descended in the firstborn line from the most powerful immigrant, Ndaunisai (Thompson 1940:214). Fijian kinship terminology is Dravidian: "Ego classifies cognates and affines of his own generation, his parents' generation and his children's generation as //they all belonged to one or [an]other of only two patrilineages which continually exchange women prescriptively" (Groves 1963:278). However, as in many Dravidian systems (Needham 1971a; Trautmann 1981), marriage rules in Fiji were variable with respect to restrictions on degree and preferences for side (Capell and Lester 1945). In southern Lau, Thompson reports preferences for both bilateral and matrilateral cross-cousin marriage. With respect to the former she says, "A man preferably marries his watchi (cross cousin, his mother's brother's daughter or his father's sister's daughter, real or classificatory) but not all marriages fall within this preferred type" (1940:53). Matrilateral cross-cousin marriage is implicit in the vasu (Tongan fahu) relation: "A man preferably takes a wife from the clan of his mother. As vasu, he has the right to demand his mother's brother's daughter. The marriage involves a ceremonial exchange of property between the clan of his father and that of his mother and more closely knits the vasu with his mother's clan" (Thompson 1940:63).7 As in Tonga (Bott 1982), genealogical seniority and the vasu relation were instrumental in establishing and maintaining inter-island hierarchies: "The vasu rights of a high chief extend to all his mother's people. For instance, the son of the sister of the high chief of Lau may, in the capacity of vasu, demand property from all the islands of the Lau group. He is called the great vasu and his demands are a form of tribute" (Thompson 1940:63). The exercise of vasu rights as a mechanism of political control is attested to elsewhere in Fiji. For example, the high chief Naulivou, of Mbau, "systematically built up a network of vasu relationships by marrying chiefly women from all those parts of Fiji that he wished to make his sphere of influence" (Routledge 1988:103-4). The connection between marriage alliance and genealogical seniority is Hocart also reports a preference for matrilateral cross-cousin marriage: "The mother's people are frequently referred to collectively as 'your uncles' and a man would be told to 'get a wife, as is fitting, from your uncles'" (1929:39).

70

Island networks

clearly shown in Lakemba's relation to Mbau: its status as a wife- and gift-giver to Mbau was "explained by saying that Mbau and Lakemba are kin, that Mbau is 'sacred blood,' that is senior branch, in relation to Lakemba" (Hocart 1929:29).8 The Lakemban chiefdom, or matanitu^ "a word which literally means 'face of the chief but which may be translated as 'state'" (Hocart 1929:22), was a hierarchically structured island network based on relations of kinship, alliance, and conquest. Thompson (1940) distinguishes three "grades of dependency between political groups" in the Lakemban matanitu. Mbatchi was an "ally, or [a group] under the protection of a power though not actually subject to it" (1940:37). Mbatchi were obliged to provide military assistance in warfare "because of kinship." Nggali was a group "subject to a power and compelled to pay yearly tribute or obliged to satisfy the demands of its chiefs" (1940:37). Nggali included conquered groups that became tributaries.9 Kai si were "chiefs' slaves" (1940:37), groups or remnants of groups completely defeated in warfare. Kambara, a noble island, was mbatchi to Lakemba, whereas Namuka and Komo were nggali to Kambara. Ongea was nggali to Fulanga through conquest, and Vatoa was nggali to Ono. All islands except Kambara were nggali to Lakemba. The Lakemban chiefdom consisted then of seven minor chiefdoms - matanitu within matanitu: Kambara, Fulanga, and Ono, each with their dependencies, and Mothe, Oneata, Thithia, and Naiau. Lakemba, Kambara, and Mothe were, as already mentioned, noble islands, that is, they had clans of the chiefs' phratry. The island of Naiau had a special relationship to Lakemba. The high chief of Lakemba traced his ancestry to Naiau, as reflected in his double title, "Roko Sau" (cognate with Tongan hau) and "Tu'i Naiau." The island was regarded as his personal possession and as a source of servants. Tribute was paid to Lakemban chiefs annually and on the occasion of rites of passage - marriage, installation, and death. Tribute gifts, collected by representatives of the high chief or by his vasu, consisted of whales' teeth, food, and craft goods, including large sailing canoes and tapa (ngatu). Gifts were given in a spirit of rivalry that enhanced the prestige of the donors, and they were substantial enough to support the 8 In consideration of the evidence from Fiji, Tonga (Chapter 4), and the Marshall Islands (Chapter 5), marriage alliance should be added to the list of strategies that chiefs may use to gain and extend power (Earle 1989). 9 According to Hocart, "The Fijian word for subject is nggali; Fison [1904] connects it with nggali (to roll string). A nggali is therefore a strand of string and means properly a group of people, as in matanggali, a clan; but it usually conveys the idea of subjection . . . There are various degrees: a people who are very much subject (nggali sara) obey (vakarorongo); the extreme is to be very serfs (kaisi sara) or a chief's very own men (tamata ndina sara ngay (1929:22).

The minimum spanning tree problem

71

high chief and his court, including his heralds (mata ni vanua)^ master fishermen {ndau ni nggali), chiefs of crops (vaka vanua), carpenters (matai), ceremonial experts, and servants. Some of the gifts, especially canoes, were given in turn by Lakemban chiefs to chiefs of other matanitus, creating a network of matanitus in the Fijian Archipelago. The origin of the Lakemban matanitu is unknown, but Thompson's hypothesis is that a number of smaller independent chiefdoms combined to produce progressively larger structures. She supposes that Lakemba and Kambara were originally heads of smaller independent chiefdoms: Rivalry between social groups led to strife. Fortifications were built on every island, but gradually the small, poor islands became dependent upon the larger, richer islands like Lakemba and Kambara. There arose small chiefdoms, within which the weaker islands stood in tributary relationship to the stronger. Finally Lakemba became the most powerful chiefdom in Lau (Thompson 1940:214-15). This scenario would be consistent with Kambara's position as head of an influential block in southern Lau and with its role in traditional history: With the exception of Ono [conquered by Lakemba], it is not known how the Southern Islands came under Lakemban rule. There is no record of wars with these islands. The inhabitants say that they "went by kinship"; that is, that they became subject to Lakemba by being kinsmen of lower rank. One informant probably preserves some of the facts when he says that Kambara formerly ruled over the Southern Islands; all the islands to the windward obeyed it; it was Kambara that brought the islands (except Ono) to Lakemba. In Viti Levu too, is a tradition that Kambara was once chief in Lau (Hocart 1929:25). The reference to kinship suggests the Tongan pattern in which junior collaterals of the paramount chief ruled over outlying islands (Bott 1982). The graph theoretic model implicit in Thompson's hypothesis is the MST algorithm of Boruvka (1926a, b). Boruvka's algorithm operates in parallel and is the most efficient one of all. It requires that the values of all edges be distinct, and hence it generates a unique MST. The algorithm proceeds by building small trees of minimum value and then joining them.10 In this procedure we need the concept of an independent set 10 See Graham and Hell (1985) for alternative versions of this and other MST algorithms.

72

Island networks

of edges, meaning that no two of them are adjacent (have a common node). ALGORITHM

3.2. Boruvka's algorithm for constructing an MST.

Given An undirected network N with distinct positive integer values (or, equivalently, positive real numbers) f(e) on each edge e of N. Wanted The spanning tree T of N with minimum edge-value sum. We will specify such a tree by building its edge set ET. Step 1. Label the edges of N by eu e2,. . . , eq such that whenever i < /, we have /"(e,-) < f(ej) and call this sequence of edges a. Step 2. Place simultaneously in ET the edge ex and all other edges e = uv such that ex and all edges e to be added form an independent set with these new edges selected in order from the sequence a. Step 3. For each component H, of ET find the smallest edge e{ joining Hj with some other component. Add all these new edges e{ to ET. Step 4. If l£Tl = p - 1, stop. Otherwise repeat step 3. THEOREM 3.2. Given a connected network N in which all the edge-values are distinct, Boruvka's algorithm will terminate with N's unique minimum spanning tree T.

Boruvka's algorithm is illustrated in Fig. 3.4, using the same network N as in Fig. 3.2. In step 1, we order the edges in the sequence a. In step 2, we simultaneously place in ET the edges be, de and af9 which are the smallest independent edges in a (Fig. 3.4a). We cannot add any other edges, because they would be adjacent with one of these three. We proceed to step 3. Each component H, of ET consists of one of these three edges. The smallest edge joining be to some other component of N is eg; the smallest edge joining de to some other component is dg; and the smallest edge joining af to some other component is fg (Fig. 3.4b). Proceeding to step 4, we see that ET -p-\-d edges, so we stop. We have the unique MST of this network, which is of course the same as the one in Fig. 3.2, and it is obtained in fewer steps because of the parallel construction. In modeling the evolution of the Lakemban matanitu, we will exclude the two southern outliers, Vatoa and Ono (a later conquest of Lakemba's), and include the formerly inhabited island of Wangava, a dependency of Kambara. The step-by-step application of Boruvka's algorithm is shown in Fig. 3.5. We begin with a set of isolated nodes representing a set of independent islands (Fig. 3.5a). In the first stage (Fig. 3.5b), we add four inde-

The minimum spanning tree problem b 12

b o

j

73

c lO

c o

o 8

(a) Figure 3.4. Generating an MST using Boruvka's algorithm.

pendent edges, representing four island pairs: (Lakemba, Naiau), (Kambara, Wangava), (Fulanga, Ongea), and (Mothe, Komo). In the first three pairs, the second member is known to have been a dependency, nggali, of the first. In the fourth pair, we suppose that Komo was a dependency of Mothe, a noble island and also the richest island and the dominant trading community in southern Lau. (See Chapter 6.) In the second stage (Fig. 3.5c), the component (edge) (Lakemba, Naiau) is joined to Thithia, forming a northern chiefdom, matanitu, headed by Lakemba. The components (Mothe, Komo), (Kambara, Wangava), and (Fulanga, Ongea) are joined to Namuka forming a southern matanitu headed by Kambera. In the third stage (Fig. 3.5d), the Kamberan matanitu adds Oneata, and "brings the southern islands to Lakemba." This would correspond to the time when, according to traditional history, the chief of Kambara "smote all the islands in Lau [and] was made Tui Oneata [and] Tui Mothe [i.e., chief of Oneata and chief of

Island networks

74 O

Thithia

Naiau

O

0

Thithia

Naiau

O

^

Lakemba

Lakemba

\>

O Oneata Komo Wangava

o

o

(7

O Oneata

o Mothe Namuka

Komo Wangava

o Mothe

&> Namuka

7

Kambara ^)

Kambara ^> Fulanga £

0 Ongea

Fulanga

^

H Ongea

(b)

(a)

C ) Thithia

C * Thithia

Naiau

m

Naiau

Vl

\ \ Lakemba

Lakemba

\>

jp Oneata

O Oneata

Wangava

F \

Kambara ^

'£> Namuka

Fulanga (c)

_i Mnthp

o Mothe

Komo

\

4—n °ngea

Wangava

* Namuka

Kambara

\ Fulanga %

$ Ongea

(d)

Figure 3.5. Modeling the evolution of the Lakemba matanitu with Boruvka's MST algorithm.

Mothe]" (Hocart 1929:25). Fulanga and Ongea were joined to the Kambaran chiefdom either as kai si, chiefs' slaves (in Kamberan memory), or as mbatchi, allies (in Fulangan memory), or as nggali. The continuing nggali relation of Wangava, Namuka, and Komo to Kambara in historical times would be a remnant of Kambara's ancient domination of the southern islands; it is the lower subtree in Fig. 3.5c with three of its edges chopped off.

The minimum spanning tree problem

75

One very interesting question remains: Why didn't Lakemba "bring the northern islands to Kambara" rather than conversely? In Chapter 6 we suggest that Lakemba eventually dominated the islands of southern Lau because of its more advantageous location in the Greater Lauan trade network.

The Renfrew-Sterud method of close-proximity analysis The minimum spanning tree problem was independently discovered in archaeology by Renfrew and Sterud (1969) as a seriation technique called the "double-link method of close-proximity analysis." The method was developed in connection with an analysis of Early Bronze Age cultures in the Aegean and has been used in Oceanic archaeology to analyze the distribution of Lapita pottery in Island Melanesia and West Polynesia (Green 1978). In treating close-proximity analysis as an MSTP our purpose is to clarify its application and introduce simpler methods of computation. Renfrew and Sterud present close-proximity analysis as a "graphical method" for ordering archaeological assemblages or types on the basis of their similarity. Like traditional methods of seriation (Robinson 1951; Brainerd 1951), it generates a classification from a similarity coefficient matrix, that is, a matrix whose entries show the degree of similarity between all pairs of units. The practical advantages of the method are its speed and its ability to handle large data structures without the use of a computer. The theoretical advantage is that it permits branching structures that reveal clustering in archaeological data: the method gives a much closer insight into the structure of the archaeological material than does a single linear series. If there are clusters, these are revealed as loops and side chains. On the other hand, the Robinson-Brainerd procedure, which is utilized by most computer seriation programs, compresses these clusters into a single linear shape (Renfrew and Sterud 1969:276). n For example, close-proximity analysis would take account of spatial as well as strictly temporal variations in pottery design. We will now go through the method of close-proximity analysis and show that it generates an MST. Notice that the method, in using the 11 Actually, the Renfrew-Sterud method calls for the elimination of loops but the retention of side chains (branches).

Island networks

76

HH

FL

LP

HH

--

(SP)

144.6

FL

183.6

LP

144.6

128.3

FG

125.9

109.6 (l81.3)

--

FG

128.3 --

D2

AC

109.6 (l84.6) 177.2

159.2

© --

113.9

143.6

150.7 (l85.6) 169.1

124.9

141.2 (l94.3) 150.4

(l 90.8 ) ( l 84.6) 143.6

124.9

--

FH

O91.5) 177.2

141.2

182.3

( l 8 2 j ) 129.2 --

136.3

AC

130.2

113.9 (l85.6) (1943) 129.2

136.3

--

FF

171.1

159.2

179.6

154.7

169.1

FF (m7)

D2

150.7

FH

125.9 (1908^ ( l 9 l 7 ) 130.2

150.4

159.8

159.8 (l79.6) 154.7

191.5 HH

190.8 >

D2

182.3

(b)

(a)

169.1/

183.6

FG-

AC-

FH

FF •

HH-

D2 •

FL

(e)

Figure 3.6. Illustration of the Renfrew-Sterud method of double-link close-proximity analysis applied to a series of Aurignacian burins. similarity coefficient matrix, adds edges of greatest value. Later on, we will show how to transform this matrix so as to add edges of least value in conformity with standard applications of MST algorithms. Renfrew and Sterud illustrate their method by applying it to a small series of Aurignacian burins (from Sackett 1966) whose similarity coefficient matrix is shown in Fig. 3.6. In the following summary of their procedure the terms "loop" and "cluster" correspond to "cycle" and "component" in graph theory.

The minimum spanning tree problem

11

Edges of N ordered by weight: AC-FG

5.7

LP-FG

18.7

AC-FF

45.3

HH-AC

69.8

HH-FH

8.5

FH-FF

20.4

LP-FH

49.3

D2-AC

70.8

HH-D2

9.2

FL-FH

22.8

FG-FF

49.6

FL-LP

71.7

LP-AC

14.4

HH-FF

28.9

LP-HH

55.4

HH-FG

74.1

FL-D2

15.4

LP-FF

30.9

LP-D2

56.4

FG-D2

75.1

HH-FL

16.4

D2-FF

40.2

FG-FH

58.8

FL-AC

86.1

D2-FH

17.7

FL-FF

40.8

FH-AC

63.7

FL-FG

90.4

(a) 14.4

5.7 FG

- AC

20.4

30.9 LP

FH

FF

15.4

9.2

8.5 HH

D2

FL

(b)

Figure 3.7. Construction of the close-proximity graph in Fig. 3.6 using Kruskal's algorithm. Step 1. Circle the two highest entries in each column of the similarity coefficient matrix. Step 2. Start with any unit, and link it to its two closest neighbors. (HH is linked to FH and to D2 in Fig. 3.6a). Step 3. Take any of the units not yet linked to its two closest neighbors and repeat step 2. (FH is linked to HH and D2 in Fig. 3.6b.) Step 4. Continue the process until looping occurs. If all units are not yet linked, start a separate cluster (Fig. 3.6c). Step 5. If two or more separate clusters are found, treat them as single units and link them as in step 2 (shown by the broken lines in Fig. 3.6d). Step 6. Where loops occur, delete the weakest link (shown by the dotted lines in Fig. 3.6e). The network in Fig. 3.6 has the form of a path. A branching structure is illustrated further on. There are two difficulties with this method. First, the procedure is unnecessarily complicated. Kruskal's algorithm for generating an MST would be much simpler. Informally stated, all we need to do is order the edges by value and keep adding edges of least value, as long as no cycles are formed, continuing until we have a connected network. We take the

78

Island networks

similarity coefficient matrix in Fig. 3.6 and subtract each edge-value from 200, so that we may add edges of least value. Then, applying Kruskal's algorithm to the sequence of edges a in Fig. 3.7a, we get the MST in Fig. 3.7b, which is identical to the close-proximity structure in Fig. 3.6e (when the dotted lines of the latter are deleted). Secondly, in the case of very large similarity coefficient matrices, it would be helpful, as Renfrew and Sterud recognize, to use a computer. The best MST algorithm for computer implementation is Prim's (1957). ALGORITHM

3.3. Prim's algorithm for constructing an MST.

Given An undirected network N with distinct positive integer values (equivalently, positive real numbers) f(e) on each edge e of N. Wanted A spanning tree T of N with minimum edge-value sum. We will specify such a tree by building its edge set ET. Step 1. Label the edges of N by el9 e2,. . ., eq such that whenever / < /, we have f(ej) < f(ej). Call this sequence of edges o\ Step 2. Take any node u of N and place in ET the first edge of a incident with u. Step 3. Add to ET the first edge e{ of a adjacent with at least one edge already in ET such that the addition of e{ does not create a cycle. Step 4. If \ET\ = p - 1, stop. Otherwise repeat step 3. THEOREM 3.3. Given a connected network N in which all the edge-values are distinct, Prim's algorithm will terminate with N's unique minimum spanning tree T.

Prim's algorithm is illustrated in Fig. 3.8. As with the preceding algorithms, we order the edges in the sequence a. In step 2 we choose node b. Then we place in ET the first edge in a that is incident with node b, namely the edge be (Fig. 3.8a). In step 3 we add to ET the first edge in a adjacent with be, which is eg (Fig. 3.8b). In step 4 we see that ET does not have p - 1 = 6 edges, so we return to step 3. We examine the edges joining nodes b, c, and g to nodes not in ET. Of all these edges, fg has the smallest weight and will not create a cycle, so we add it to ET (Fig. 3.8c). Continuing in this way, we add the edge af, followed by the edges dg and de. ET now has six edges, so we terminate the procedure. The MST in Fig. 3.8f is the same as the one obtained by using Kruskal's and Boruvka's algorithms in Figs. 3.2 and 3.4. A simple, easily programmed matrix method for Prim's algorithm is given in Wilson and Watkins (1990). We begin with a table, that is, a value matrix M of a network N. In their succinct description, "All we need to do is delete a row of the table whenever the corresponding vertex is placed in T and then choose the smallest entry in the column cor-

The minimum spanning tree problem

(a)

(b)

(d)

(e)

79

(f)

Figure 3.8. Generating an MST using Prim's algorithm. responding to the vertices in T" (1990:200). Wilson and Watkins illustrate the method as shown in Fig. 3.9. Start with a complete network N (Fig. 3.9a). Choose an arbitrary node of the matrix, and put it in T. We choose node B (Fig. 3.9b). In the matrix, delete row B and find the smallest entry in column B. It occurs in row C, so add the edge CB to T (Fig. 3.9c). Delete row C and find the smallest entry in columns B and C. It occurs in rows A and £. Choose one of these, say A, and add the edge AC to T (Fig. 3.9d). Continue in this way (Fig. 3.9e) until there are p - 1 edges. The result is the MST in Fig. 3.9f, which has the value sum of 18. Notice that the edge-values in this network are not distinct. Because there was a tie, we

80

Island networks A

B

C D

A

0

6

4

8

2

B

6

0

5

8

6 4

A

C

4

5

0

9

D

8

8

9

0

7

E

2

6

4

7

0

A B

(b)

o B

(c)

A C D E

A D

0 4 8 2

C D

6 5 8 6

4 0 9 4

A B

C

0 8 2

6 8 6

A B

(d)

E

4 9 4

E

8 9 0 7

D

2 4 7

0

E

8 0 7

C D

2 7 0

E

f8 8 9 0 7 I [_2 6 4 7 OJ

(e)

£

A 5

D fj

(f)

8

C £> E 9

0

7

E

Figure 3.9. Illustration of a matrix method for using Prim's MST algorithm (from Wilson and Watkins 1990).

The minimum spanning tree problem

81

u Q

N: 1

o 2

y

(a)

x

2

y

(c)

(b)

Figure 3.10. A network with two MSTs.

could have added edge EC rather than AC to T, which would have resulted in a different MST. We saw in Theorems 3.1, 3.2, and 3.3 that a connected network N with distinct edge-values will always have a network N consisting of a unique minimum spanning tree T. That this may not be so when edgevalues repeat is seen most readily by the simple example of a rectangle with sides 1 and 2, as in Fig. 3.10a. This network has just two MSTs (Fig. 3.10b, c): T1 lacks edge uv, and T2 is missing edge xy; both have a value sum of 4. Similarly, given a large network N in which many of the edge-values recur, there may be a substantial number of MSTs. In fact, we will see in the next section a large real-world network (called the "Cycladic cemetery network"), with repeated edge-values, that has exactly 288 MSTs. The determination of all of these MSTs can burn a lot of computer time, and in fact it did! Cycladic cemeteries Renfrew and Sterud developed close-proximity analysis as a means of classifying Early Cycladic cultures dating from 3200 to 2000 B.C. The data consist of inventories of grave goods such as pots, vases, jars, and figurines found in 21 cemeteries on eight different islands of the Cyclades. They use a presence-absence similarity coefficient matrix that has as its entries the number of types of grave goods that pairs of cemeteries have in common. The close-proximity structure that Renfrew and Sterud obtain from their matrix is graphed in Fig. 3.11. They identify two clusters in this structure, which they equate with the Keros-Syros and Grotta-Pelos cultures: The order obtained [in Fig. 3.11] brings out well an essential feature of the cemeteries, already anticipated through other archaeological considerations (Renfrew 1966, 1967). This is the separation of cemeteries of the Keros-Syros culture (namely

82

Island networks KE

KV

GL

*-

KR <

>

PA -**—>- AK

Figure 3.11. Renfrew and Sterud's (1969) close-proximity structure for the Early Cycladic cemeteries, using presence-absence similarity coefficients.

CH, KP, DO, SP, KV and part of AK) from those of the Grotta-Pelos culture [the remaining points in Fig. 3.11]. That this separation emerges so clearly from the seriation seems a corroboration both of the archaeological opinion and of the efficacy of the method (1969:274). 12 There are two problems with this analysis. First, there are some errors in Fig. 3.11. Because it is a close-proximity structure, which erases loops, either line (AP, KR) or line (AP, PA) should be deleted to avoid a cycle. In addition, according to the similarity coefficient matrix, SP should be joined to CH, not KP, and AV must be joined to PA, not PY or AK. If the dotted lines that represent erasures necessary to eliminate loops are deleted, however, the structure is not connected. To get a connected network, the following edges must be added: (PE, LE) or (PE, AK); (PO, LE) or (PO, SP) or (PO, AK); (AV, PA); and (VA, PA) or (VA, AK). 12 The explanation for AK's membership in both cultures is that two graves in the cemetery contained Keros-Syros goods and four contained Grotta-Pelos goods.

The minimum spanning tree problem

83

Secondly, because of ties in the similarity coefficient matrix, the closeproximity structure is not uniquely determined. In order to use a closeproximity structure (or equivalently an MST) as the basis for classification, it is necessary to assign all edges distinct values or else show that all MSTs of a network are clusterable in the same way. Before addressing this problem, we introduce a simple method that will permit us to add edges of least value when constructing an MST with data of this kind. To get the edge values f(ej) of N, we write down a list of artifact types - in this case, types of grave goods. Then for every unit - here, every cemetery - we write a 0, 1 binary sequence showing the absence or presence of each artifact. For every pair of units we take the elementwise product to get the number of shared artifact types. For example, Unit A Unit B

0 110 1111 10 1 1 1 0 0 1 0 0 10 10 0 1

We then take a number that is one more than the total number of artifact types and subtract the number of shared types to get the value of each edge. This avoids any zeroes on the edges of N. In our example, the total number of artifact types is eight, so the value of the edge (A, B) is 9 - 3 = 6. This is the standard procedure. The similarity coefficient matrix in Renfrew and Sterud already shows the number of artifact types shared by pairs of cemeteries. The highest number in that matrix is 10, so we subtract each entry from 11 to obtain the matrix in Table 3.2. The network of the matrix in Table 3.2 has not only 1 but exactly 288 different MSTs! This was determined by an exhaustive computer-assisted verification. Two of these MSTs, each with a value sum of 130, are shown in Fig. 3.12. The MST in Fig. 3.12a is clusterable in the way described by Renfrew and Sterud. The MST in Fig. 3.12b, however, has several edges different from the first one and is not quite clusterable in the same way. It shows that the cemetery PO can just as well be joined to the cemetery SP, putting it in the Keros-Syros cluster, while DO can be joined to LO, putting it in the Grotta-Pelos cluster. As it turns out, all 288 MSTs of the Cycladic cemetery network have as a subtree the network shown in Fig. 3.13, which has a value sum of 78. To obtain the complete MST, the remaining edges are added as follows (select one from each set): AP to : KR or PA (edge-value = 8) PE to : LE or AK (edge-value = 8) VA to : PA or AK (edge-value = 9)

84

Island networks

Table 3.2. Presence-absence similarity coefficient matrix of Early Cycladic cemeteries, based on Renfrew and Sterud (1969) and Renfrew (1972) AN GL PA PY AV KR LE ZO KP DO CH KV KE SP LO PO KO AP PE AK VA

AN GL PA PY AV KR LE ZO KP DO CH KV KE SP LO PO KO AP PE AK VA — 11 8 9 10 9 6 11 11 11 10 11 11 10 10 10 11 10 9 7 10~ 11 8 9 10 9 6 11 11 11 10 11 11 10 10 10 11 10 9 7 10

8 8 8 11 7 9 10 11 11 9 10 11 9 11 11 10 9 11 10 11

— 5 8 2 4 7 10 11 6 10 10 8 11 11 9 8 9 4 9

8 5 — 9 6 9 8 10 10 7 9 9 7 10 10 8 9 10 6 10

11 8 9 — 10 10 10 11 10 10 10 10 10 10 10 10 11 11 9 10

7 2 6 10 — 6 8 11 10 6 9 11 7 10 10 10 8 9 5 11

9 4 9 10 6 — 10 11 10 7 10 10 8 8 9 10 10 8 6 10

10 7 8 10 8 10 — 11 11 10 11 10 10 11 11 9 11 10 8 10

11 10 10 11 11 11 11 — 9 7 10 11 9 11 11 11 10 11 9 11

11 11 10 10 10 10 11 9 — 9 9 11 9 9 10 11 10 11 9 11

9 6 7 10 6 7 10 7 9 — 8 11 1 9 10 10 9 11 3 11

10 10 9 10 9 10 11 10 9 8 — 11 9 11 11 11 10 11 11 11

11 9 11 11 10 9 11 10 10 8 11 11 9 8 9 4 9 7 10 10 8 9 10 6 10 10 10 10 10 11 11 9 11 7 10 10 10 8 9 5 10 8 8 9 10 10 8 6 10 10 11 11 9 11 10 8 11 9 11 11 11 10 11 9 11 9 9 10 11 10 11 9 11 1 9 10 10 9 11 3 11 9 11 11 11 10 11 11 11 10 11 9 11 10 10 11 — 10 9 10 10 10 7 10 10 10 11 11 10 10 11 9 10 — 11 11 10 9 9 10 11 11 11 11 9 11 10 11 11 11 — 11 10 10 10 10 10 11 11 8 10 7 10 9 9 10 8 — 10 10 11 10 10 11 10 9

11 9 10 10 11 10 10 11 11 11 11 10 10 11 10 10 11 10 9

Note: Abbreviations: AK: Akrotiraki; AN: Aghios Nikolaos; AP: Apollona; AV: Avyssos; CH: Chalandriani; DO: Dokathismata; GL: Glypha; KE: Keli; KO: Kampos; KP: Kapsala; KR: Krassades; KV: Karvounolakkoi; LE: Leivadhi; LO: Louros; PA: Panaghia; PE: Pelos; PO: Polichni; PY: Pyrgos; SP: Spedos; VA: Vathy; ZO: Zoumbaria.

KE to : KO or PY (edge-value = 9) PO to : LE or AK or SP (edge-value = 9) DO to : LO or AK or CH or KP or KV or SP (edge-value = 9) These edges add 52, which gives in every case a total value sum of 130. The only MSTs that would violate the clusters identified by Renfrew and Sterud are those in which PO is added to SP or DO is added to LO. These are also edges with the highest values (weakest links). Hence there is little reason to reject Renfrew and Sterud's analysis of this particular network. In general, when using an MST for classification it is advisable, whenever possible, to assign different values to all the edges of N. This eliminates the possibility of alternative clusterings and, in the case of large networks, avoids the problem of finding all their MSTs. To finish our summary of the Cycladic cemetery MST, Renfrew (1972) observes that the geographical distribution of the Grotta-Pelos and Keros-Syros cultures is neither identical nor complementary. To test for other spatial patterns, he relabels the close-proximity structure in

The minimum spanning tree problem

85

KE

KV

SP AN

(a) KO

KV

AN

(b) Figure 3.12. Two different MSTs generated by the Cycladic cemetery matrix.

86

Island networks KO

AV

GL

KR

PA

7

AK

ZO < LE AN 0 LO

Figure 3.13. The universal subtree of the MST of the Cycladic cemetery matrix.

N

O

D ) D

N

Figure 3.14. The graph in Fig. 3.13 relabeled with island names. (Abbreviations: N = Naxos; P = Paros; D = Desphotikon; O = Antiparos; A = Armogos; S = Siphnos; SY = Syros; M = Melos.)

The minimum spanning tree problem

87

Fig. 3.11 with island names in place of cemetery names. In Fig. 3.14 we have relabeled the network in Fig. 3.13 with island names. Inspection of Fig. 3.14 reveals little in the way of geographical clustering. The Paros cemeteries are on the left, and the Armogos cemetery is on the right, but Siphnos, which lies between these two islands, is to the west of Paros. (See Fig. 6.12.) The Naxos cemeteries are distributed all over the network. Renfrew concludes, on the basis of stratigraphic and comparative evidence, that the two cultures are chronologically rather than spatially related. Grotta-Pelos has links with mainland neolithic cultures and Early Minoan I in Crete, while Keros-Syros has links with later mainland cultures, with Troy II-IV, and with Early Minoan II. The Lapita design network

Green (1978) has used close-proximity analysis to study the distribution of Lapita pottery and (by implication) the spread of the Lapita culture in Oceania. The term "Lapita" refers, in the first instance, to a highly distinctive form of pottery found in a number of sites dating from 1600 to 600 B.C. The "Lapita complex" consists of this pottery plus a variety of stone, shell, and bone tools and shell ornaments. Green defines the "Lapita culture" as a specialized adaptation of the maritime, horticultural Austronesian populations that entered Near Oceania about 4000 B.C. and settled the islands of Remote Oceania as far as New Caledonia sometime before 2000 B.C. He locates the immediate homeland for this cultural complex in the Bismarck Archipelago. In Green's view, the Lapita culture was based on an effective two-way voyaging and exchange network that connected settlements over distances of up to 600 kilometers, the probable limit of regular two-way voyaging in the Pacific (Lewis 1972). With advances in canoe design - in particular, more effective double canoes - populations making Lapita pottery expanded eastward, settling ever more distant islands. Eventually the eastern communities lost contact with the western ancestral communities, disconnecting the voyaging network that had underwritten the process of colonization. New adaptations and voyaging centers emerged. An "Eastern Lapita" adaptation, located in the Fiji-Tonga-Samoa area, was transformed into West Polynesian culture. Support for Green's model comes in part from his close-proximity analysis of the Lapita design network. The data consist of 54 well-defined pottery motifs distributed over ten sites stretching from Watom Island in the Bismarcks eastward to the Santa Cruz-Reef Islands, New Caledonia, Fiji, Tonga, and Samoa. On the basis of a similarity coefficient matrix showing the degree to which pottery motifs are shared between pairs of sites, Green constructs a close-proximity network. We

88

Island networks Watom (Fac) ^ \ ^ Reef Islands (RL-2) 35 > T v 1200-900 B.C.

O-^ New Caledonia (Vatcha)

Santa Cruz / 39 (SZ-8) / 1200-900 B.C. /

Samoa (MU-1)

\ T \

o

^ Fiji (Yanuca)

Reef Islands (RL-6) 47

650 B.C.

1300 B.C.

44

57

New Caledonia (Site 13) 1100 B.C.

1000 B.C.

57

Fiji

Tonga (TO 1-5)

(Natanuku) 1500 B.C.

800-1200 B .C.

Figure 3.15. An MST of linked Lapita pottery motifs, based on data in Green (1978, 1991). have simplified this network by eliminating all loops (cycles)13 and by expressing similarity in the standard way (the smaller the value of an edge, the greater the similarity) to obtain the MST in Fig. 3.15. This MST supports Green's model of Lapita settlement. It shows that the Lapita pottery motifs are geographically linked from west to east, and it can be clustered into an Eastern Lapita complex consisting of Tonga, Fiji, and Samoa, and a Western Lapita complex consisting of the remaining islands. Green characterizes the west-to-east trend in general terms as "indicative of distance decay in the Lapita design system from the rather ornate curvilinear and fairly elaborate rectilinear design patterns of the Western Lapita to the more simplified and generally rectilinear design patterns of the Eastern Lapita" (1978:10). The dates in Fig. 3.15 indicate a relatively rapid spread of the Lapita culture, consistent with an efficient voyaging technology and communication network. Green also notes that his graph, and the same would be true of the MST in Fig. 3.15, lends support to Frimagacci's (1975) contention that the New Caledonian Vatcha site is older than the New Caledonian 13 site. Green emphasizes that his analysis is provisional, in part, because it does not include data from Vanuatu (the New Hebrides). It does, however, suggest a Vanuatuan connection between Eastern and Western 13 When the spanning tree feature of close-proximity analysis is not made explicit, there is a tendency to add many more edges to a network than are needed. This may obscure clustering, as it does in Irwin's (1992) analysis of accessibility between islands in Polynesia. See Hage, Harary, and James (1996).

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Lapita through the slightly stronger links of Fiji with the Reef (Santa Cruz) sites than with the New Caledonia site.

On deconstructing a network Graham and Hell describe two additional types of MST algorithms besides those of Kruskal, Prim, and Boruvka. These algorithms are also greedy, but they proceed dually by eliminating longest edges first. Although they do not have any computational advantage, they do have potential applications in anthropology as models of network devolution. The following algorithm was independently discovered by Kruskal (1956) as the quantitative dual of his own algorithm, and by Kotzig (1961). We quote Kruskal: ALGORITHM

3.4. KruskaVs MST algorithm, dual to Algorithm 3.1.

Perform the following step as many times as possible: Among the edges not yet chosen, choose the longest edge whose removal will not disconnect [the network]. Clearly the set of edges not eventually chosen forms a spanning tree of G, and in fact it forms a shortest spanning tree (Kruskal 1956:49). A related algorithm was independently discovered by Rosenstiehl (1967) and Dijkstra (1960). We quote Rosenstiehl: ALGORITHM

3.5. RosenstiehVs MST algorithm.

Donnees: Une liste des aretes de G dans un ordre quelconque, et pour chaque arete son numero d'ordre et ses deux extremites. Principe: On lit la liste des aretes. Toute arete lue est "retenue." Quand l'une d'elles constitue un cycle (elementaire) avec des autres aretes "retenues", la plus grande arete de ce cycle est rejetee des aretes "retenues". Les aretes "retenues" et non rejetees constituent V* [i.e., an MST] (Rosenstiehl 1960:366). The later history of Oceania witnessed the breakup and fragmentation of once extensive inter-island networks. It should be possible to develop an algorithmic approach to these processes.

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Thought is a labyrinth; and topological thought, which sprang originally from the brain of Leonhard Euler (1707-83), gives us our best analytical approaches to the mazes of our recreation and our technology: the left-turn rule, the depth-first search. Such labels seem to announce little tinny formulas. Do not be misled, though. The formulas lift us, like the wings of Daedalus, out of everything labyrinthine, for an overview. Hugh Kenner, Mazes

In Social Stratification in Polynesia, Sahlins (1958) identified three basic forms of social organization found in Polynesian societies - the ramage, the descent line system, and interlocking organization - interpreting each as an adaptation to a specific type of island environment. Sahlins's study was directly inspired by Kirchhoff's (1955) discovery and evolutionary interpretation of an internally stratified type of descent group known as the "conical clan," which Sahlins, following Firth (1936), called the "ramage." Anthropologists, although they generally reject Sahlins's ecological interpretation, acknowledge his achievement in revealing the conical clan as the basic structural form of many Polynesian societies (Goodenough 1959; Hogbin 1959). Some archaeologists and linguists now view the conical clan genetically, as a component of Ancestral Polynesian Society (Kirch 1984a; Kirch and Green 1987; Bellwood 1978; Pawley 1982). Given the ethnographic and theoretical importance of the conical clan, it is surprising to discover that this structure has never been clearly defined. Kirchhoff characterized the conical clan only in general terms, and although Sahlins was more specific, defining it in terms of a rule of succession, his definition is imprecise and not applicable to all societies in Oceania or elsewhere in the world. 1 1 Sahlins's (1968) later attempt at a generic definition of the conical clan applies mainly to Polynesia. It would not cover the Kachin or Micronesian cases described in this and the next chapter.

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Other definitions are casual, diffuse (Fried 1967), or even self-contradictory (White 1959). Lacking an adequate formal model, the conical clan has a long history of being rediscovered in some societies while it is ignored in others. We have three aims in this and the next chapter. The first is to elucidate the structure of the conical clan, specifically, that aspect of it that concerns the ranking of individuals or titles, using the graph theoretic model of a depth-first search tree. The model, which is equivalent to a maze-solving algorithm, gives an exact, general, and intuitively appealing characterization of the conical clan in all its forms. The second aim is to present the ethnographically rare, matrilineal variant of the conical clan found in the Marshall Islands in Micronesia. As a result of Sahlins's study, the conical clan in Oceania is usually associated areally with Polynesia, topographically with high islands, and structurally with patrilineal descent. The prototypical case is Tonga. In describing the presence of the conical clan in the Marshall Islands, we show that it can occur on atolls with small populations, existing not as a historical remnant or as a rudimentary structure, as Sahlins supposed for Polynesia, but as a fully developed form of social organization. In the Marshalls, as in Tonga, the conical clan and marriage alliance were instrumental in the formation and control of island empires. From this point of view, Tonga and the Marshalls should not simply be opposed areally as Polynesian versus Micronesian chiefdoms, or ecologically as high island versus coral atoll adaptations, but should be classified together as structurally variant forms of socially stratified island networks. The third aim is to suggest, on the basis of linguistic evidence, that the Polynesian and Micronesian variants of the conical clan have a common origin in Proto-Oceanic society. As a corollary we propose, contra Murdock (1948), that the different forms of social organization in Nuclear Micronesia derive from a stratified protosociety similar in structure to that found in the Marshall Islands. For convenience and for reasons of theoretical interest, our presentation is in part historical. We review Kirchhoff's model of the conical clan because it has been almost completely neglected in kinship studies, as Fried (1957, 1967) and Service (1985) have made abundantly clear. We take special note of E. W. Gifford's work, which clearly and succinctly delineates the basic principles of Tongan social structure, including the relation between kinship rank and asymmetric marriage alliance. His monograph, published in 1929, is the acknowledged basis of most subsequent interpretations of Tongan society, from Sahlins (1958) and Goldman (1970) to Biersack (1982) and Bott (1982), and as Gifford himself hinted, and as we will see, it provides the basis for a compara-

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tive analysis of social organization in Micronesia and Polynesia. In making this comparison we shall also refer to Leach's (1954) analysis of Kachin, which confirms and broadens Gifford's analysis of Tongan social structure by presenting its structural dual, and Friedman's (1981) model of the prestige-good system, which in effect integrates the descent perspective of Kirchhoff and the alliance perspective of Leach in a regional systems analysis of social stratification in Oceania.

Independent discoveries of the conical clan Kirchhoff

In his recent history of ethnology, Service (1985) cites four independent discoveries of the conical clan in addition to Kirchhoff's: Fustel de Coulanges' (1864) Greco-Roman gens; Firth's (1936) Polynesian ramage; Leach's (1954) Kachin gumsa lineage; and Oberg's (1955) lowland South American "chiefdom."2 We will add to this list Gifford's (1929) Tongan ha'a lineage and, in Chapter 5, Mason's (1954) Marshallese clan. The most notable discoveries of the conical clan from the perspective of Polynesian studies are by Kirchhoff, who studied it theoretically, and Firth, who characterized it ethnographically. Their discoveries were almost simultaneous, since Kirchhoff's was made in 1935 but not published until 20 years later.3 In a brief, luminous paper, Kirchhoff identified and recognized the evolutionary significance of a type of descent group in which members are differentiated by kinship rank. He called this group the "conical clan" to differentiate it from the well-known "unilateral exogamous clan." Kirchhoff regarded these two forms as alternative developments from a common genetic root in "pre-clan society." The unilateral exogamous clan, as defined by Kirchhoff, may be either patrilineal or matrilineal, but in both cases it is egalitarian: "every member of the clan is, as far as clan membership goes, on an absolutely equal footing with the rest: the nearness of relation to each other or to For reasons of historical precedent and current usage, Service proposes gens, as opposed to "clan," as a generic term for this type of descent group. Etymologically, this is not a felicitous choice, because it implies that the conical clan is always patrilineal. Service (1985:126) attributes the rejection and delay in publication of Kirchhoff's paper "not [to] perversity on the part of an editor or two, but [to] widespread lack of interest in a problem that is essentially related to an evolutionary perspective." Kirchhoff's article was finally made accessible when it was included in M. Fried's (1959) Readings in Anthropology. "Fried saw its great theoretical significance, and by publishing it in his Readings he performed not only a generous act of love for scholarship, but in a real sense a creative act" (Service 1988:151).

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some ancestor being of no consequence for a person's place in the clan" (Kirchhoff 1955:4). Being completely egalitarian, with the interests of all members subordinated to the common good, the unilateral exogamous clan provides no basis for higher forms of economic cooperation that presuppose social differentiation. Thus it is an evolutionary dead end. The conical clan, by contrast, differentiates members in terms of the "degree of relationship." This "results in a group in which every single member, except brothers or sisters, has a different standing; the concept of the degree of relationship leads to different degrees of membership in the clan. In other words, some are members to a higher degree than others" (Kirchhoff 1955:6-7). From the concept of the degree of relationship Kirchhoff drew a number of conclusions concerning the structure, institutional correlates, and evolutionary potential of the conical clan. (1) The ranking of clan members stratifies the entire clan into classes, ranging from higher and lower grades of chiefs to commoners, and in some cases even slaves, but all regarded as kinsmen. (2) Since nearness of relationship counts, the conical clan is not necessarily unilateral (unilineal). Descent may be traced either through men alone or more rarely through women alone, but frequently, through either men or women, depending on which method gives the highest rank. (3) Since there are degrees of membership but no definite or fixed boundaries, the group is not exogamous. There may, however, be tendencies toward endogamy, motivated by considerations of maintaining or improving rank, including preferential marriages with parallel relatives, with brother's daughter or father's brother's daughter - possible "leitfossils" of the conical clan - or with half-sisters. (4) The principal social, economic, and religious roles are allotted to those of highest rank, and the cooperation of those in the lower ranks is ensured by the common bonds of kinship. (5) Stratification by kinship has the potential for the formation of true social classes when the interests of the aristoi - the upper strata - come into conflict with those of the lower strata. Unfortunately, Kirchhoff was not explicit about the structure of the conical clan or the meaning of the term "nearness of relationship to the common ancestor." He did, however, refer to "lines of descent" and to a "center line": Clan membership so to speak shades off the farther one is away from the center-line of the clan - the real core of the group. This core, the aristoi, consists of those who are, or are supposed to be, [nearest] descendants of the common ancestor of the clan . . .

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Island networks Genealogies, unknown and unnecessary in a unilateral clan, are here the means of establishing the "line" of descent of the nobles, this "line" being another concept unknown in unilateral clans (1955:7).

The minimum number of unilateral exogamous clans in a society is obviously two, whereas the minimum number of conical clans is one, since a single clan could be coterminous with an entire society. Thus Kirchhoff's geometrical metaphor: The one divides the tribe into a number of solid blocks with clear-cut boundary lines, each homogeneous within. The other results in a type of society which may be likened to a cone: the whole tribe being one such cone, with the legendary ancestor at its top, but within it are a larger or smaller number of similar cones, the top of each coinciding with or being connected with the top of the whole cone. The bases of these cones, representing] the circle of living members of the various clans at a given moment, overlap here and there( 1955:8). Firth

Firth's discovery of the conical clan resulted from ethnographic and taxonomic rather than theoretical interests. He noted that in many Polynesian societies the unity of kinship groups is expressed by "unilateral recognition of common descent," without delimitation by exogamy. Firth proposed the botanical metaphor of a "ramage" to capture the structural and dynamic aspects of these kinship groups that continuously fission and disperse, yet preserve the bonds of common ancestry: As a rule by historical tradition, and presumably in actual social process, [these groups] have arisen through the branching and re-branching of the family structure, acquiring greater autonomy and independence the further they move away from the parent stem. The tree metaphor is actually used by some native peoples in describing their social organization. Here, very often, great importance is attached to seniority as a principle of social differentiation. One term which might be employed to characterize such kinship groups is "ramage," for which there is literary authority, though it has now fallen out of use. This term has the advantage of suggesting immediately by its etymology the branching process by which those groups attain individuality and yet keep their connection with the parent stem. It is also consistent in metaphor with the expression

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"genealogical tree." The process can be correctly described as one of ramification (Firth 1936:370-1). Gifford In his discovery of the conical clan in Tonga, Gifford (1929) showed that kinship rank is integrally associated with marriage alliance. In an elegant analysis of Tongan society, Gifford distinguished three basic features: "To understand the system of apparent servitude which prevailed in Tonga, three features of family relationships should be kept in mind: (a) the superior rank of sisters; (b) the superior rank of older siblings; and (c) the fahu idea" (1929:115).4 It is clear from Gifford's analysis that all three features generalize to relations between groups and that they interact. In the case of the first feature, "A woman is always superior in rank to her brother regardless of seniority and it follows from this that a woman's children are always superior to her brother's children" (1929:17). It also follows that "a man marrying a woman has the same sort of social superiority to his wife's brothers that the woman herself has" (1929:117). The second feature, the superior rank of older (male) to younger (male) siblings, defines rank relations in patrilineal lineages, ha'a (the conical clan). It implies that all individuals in a lineage and all ancestrally related descent lines can be uniquely ranked and integrated in a status hierarchy by a rule of primogeniture. From bottom to top and from top to bottom of the social ladder one general scheme of family organization prevails. As the Tui Tonga is eiki (chief) to his younger brothers, so in every Tongan family the older brother is chief to his younger brothers . . . Because of primogeniture, it is obvious that the descendants of younger siblings sink in rank. The commoner is the man in a line of descent that gets further and further from the head of the lineage with each succeeding generation (1929:112). In historical times Tongan society consisted of 13 patrilineages {ha'a). Chiefs were heads of ha(a or of subdivisions of ha'a, and, together with 4 The names given to features (a) and (b) by Goldman (1970), Biersack (1982), and Bott (1982) are, respectively, "gender and seniority," "cross and parallel," and "rank and authority." Gifford's analysis seems strikingly "modern." Compare, for example, J. J. Fox's (1980) structural analysis of Austronesian societies in eastern Indonesia in terms of three elementary "categories": the "house," the "relative age of siblings" (descent), and "opposite sex" (alliance).

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their "close relatives" formed a separate class, seiki. All chiefs could trace their descent to the first paramount chief or king, Tu'i Tonga ("Lord of Tonga"), the son of a sky father and an earth mother. The commoner class, tu'a^ individuals whose kinship rank had been progressively lowered through primogeniture, constituted the majority of the population. Commoners, sometimes called kainanga oe fonua, "eaters of the land," were affiliated with chiefs on the basis of kinship and/or residence on chiefly estates.5 In between the 'eiki and tu'a were the matapule, chiefs' attendants who were descendants of immigrants from Fiji, Samoa, and Rotuma who had attached themselves to chiefs (Gifford 1929), or distant relatives of chiefs and descendants of "persons eminent for experience or wisdom" (Mariner 1818). Land tenure was based on a feudal pattern of hereditary overlapping rights and obligations between chiefs and junior relatives, which included the payment of tribute and military service: From the Rev. Thos. West we learn that "the lands were held in fief." The great landlords held them by hereditary right, but subject to the king, and they in turn subdivided them among their kinsmen and followers. It was on the great chiefs that the king depended for military support, which they willingly rendered him, as the title by which they retained their possessions. Through them also the king received a general tribute from the people. The chiefs, also in the order, claimed the service or property of their tenants. The lowest order was ground down and oppressed by that above it. The "Tuas" could not call anything they had their own. The great chiefs could seize on whatever took their fancy. Besides, the king or his representatives could assess labor upon the whole community whenever he pleased (Alexander 1888, quoted in Gifford 1929:171). Gifford described the Tongan lineage as patrilineal in theory and, largely, in practice. He viewed it as "one aspect of the Polynesian interest in genealogies." He characterized its structure by means of a botanical metaphor: The whole system of lineages may be likened to a tree with trunk, limbs, branches, and twigs. Here and there a twig develops into a branch . . .; other twigs sprout forth and die . . . Or 5 Gifford noted that "Some commoners are not aware of their lineage as such, but most are, and claim relationship to some chief, usually the one under whom they live" (1929:30). But according to Bott, "The ha'a was considered to be a thing that concerned title holders, and ordinary commoners did not speak of themselves as members of a ha'a, although all commoners were attached to some chief or other" (1982:156-7).

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perhaps a limb becomes huge and flourishing . . . , while the trunk . . . ceases to flourish. Everything points to the necessity of a line of powerful chiefs for a nucleus about which the lineage groups itself. Without such chiefs it appears to wilt and die and its membership gradually aligns itself with other rising lineages. That all the modern haa have a chief as a nucleus, or trace their descent from an ancient chief, was attested by all informants (1929:30). Gifford described branching processes in terms of the splitting of "major lineages" into "minor lineages" and the growth of minor lineages into "incipient major lineages," introducing these terms into anthropologyThe third feature, the "fahu idea," connects the first two. In Tongan the word fahu literally means one who is "above the law" with respect to another (Gifford 1929:23). The usual primary referent of the term is the sister's child-mother's brother relationship. "The sister's children are fahu (Fijian vasu) to their mother's brother: they have the privilege of taking their uncle's goods, also the goods of his children either during his life or after his death" (1929:23). But as Gifford pointed out, the term also has a political meaning: Frequently the right of fahu is even more extended and may give a chief of high rank the right to help himself to any property he wishes in an entire district controlled by chiefs to whom he is fahu (1929:25). To the demands of the fahu of a great chief, that chief and his dependants were obliged to accede. Thus all of the descendants of the Tui Kanokupolu Tukuaho are fahu to the Haa Ngata Motua chiefs and to the commoners of that lineage. In other words the whole of the district of Hihifo, Tongatapu, is obliged to yield its best to the great fahu of the chiefs of the district (1929:115). The fahu idea is correlated with matrilateral cross-cousin marriage, that is, marriage with the MBD, and accounts for the superior rank of the sister and her children. "In cross-cousin marriage it was the sister's child who held the higher rank in the marriage on account of being fahu to his spouse" (1929:190). Gifford, in fact, saw a possible genetic connection between the fahu relation and matrilateral cross-cousin marriage: "It is quite possible that cross-cousin marriage in Tonga is genetically connected with the fahu concept, for it is usually the mother's brother's daughter whom a man marries. In other words, he may make

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demand in his capacity of fahu" (1929:26).6 Interpreted in this way, to say that lineage X stands in the relation of fahu to lineage Y is to say that X is wife-taker to Y, to whom it is thus superior and from whom it may demand wives as well as material support.7 Gifford emphasized the political significance of the fahu relation and matrilateral cross-cousin marriage, noting their restriction largely to the chiefly class: The institution of fahu is established in the lower strata of society as well as in the higher, but it reaches its most extravagant development among the higher chiefs. Anciently it was not extensively practiced among commoners, for fear of punishment for imitating chiefs, and, perhaps also, for alienating goods that a chief might claim (1929:24). [Matrilateral cross-cousin] marriages were regarded as a chiefly prerogative, not to be exercised by commoners (1929:281). Gifford thought it "reasonable to suppose that the Tui Tonga habitually sought alliances by marriage with the most powerful chiefs in his domain" (1929:59-60). He also noted the value of the alliance for the wife-giver: "No small amount of the prestige of the Tui Kanokupolu's office [the hau] was due to wise matrimonial alliances. The elder daughter of the Tui Kanokupolu was usually espoused by the Tui Tonga as his chief wife [moheofo] and mother of the next Tui Tonga" (1929:99). We note that Gifford's analysis could be simplified by interpreting the superior rank of sisters over brothers as a derivative of the fahu relation, that is, of hypergamy. In a discussion of marriage and siblingship in Tonga and elsewhere in Polynesia, Levi-Strauss observes that In a hypergamic society, where the sister of a great noble ought to marry above her station (a hypothetical demand impossible to meet), one solution would be to postulate that whomever she married would be deemed to be higher in rank. The sister's superiority would thus result from a constraint of the system, which would confer the appearance of reality upon a legal fiction (Levi-Strauss 1985:161). This would give two features and a corollary. "The other form of cross-cousin marriage, i.e., to the father's sister's daughter, occurred, but was rare" (Gifford 1929:190). Gifford noted Radcliffe-Brown's psychological interpretation of matrilateral crosscousin marriage - "the mother's brother since he must indulge his nephew will more willingly part with this daughter" - (in a letter to Gifford excerpted on p. 24 of Tongan Society) but for his part emphasized the structural implications of this practice.

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Leach In Political Systems of Highland Burma, Leach (1954) described an unusual type of lineage system found among the Kachin. Kachin gumsa lineages are segmented, ranked, and stratified into social classes consisting of chiefs, aristocrats, and commoners. Lineage rank is strictly defined by a rule of ultimogeniture: the "youngest son line of a lineage" is senior to the "eldest son line," and the "ritual status of the youngest son chief and his descendants is deemed to be higher than that of the eldest son chief and his descendants": by natural right the youngest son succeeds the father in his rights to make sacrifices to the ancestral deities of the lineage, and . . . though other members of the father's lineage may on occasion succeed, they must first "purchase" the rights of office from the youngest son by making the appropriate ritual gifts. In the course of time therefore the total lineage of a chief includes a number of collateral branches tracing descent from the elder brothers of chiefs. In such a case only the "youngest son line" is, strictly speaking, du haw amyu - "of chiefly sort"; the collateral lines are inferior, they are ma gam amyu - "eldest son sort" (Leach 1954:164-5). Chiefs exercise control through a feudal system of land tenure in which they own (madu) or rule (up) the land whereas commoners only use or "eat it" (sha), and through the collection of tribute that they redistribute in large and frequent feasts. Chiefs also claim divine descent and sacrifice to spirits to preserve the prosperity of their domains. Kachin lineages are connected by a rule of matrilateral cross-cousin marriage in which wife-givers (mayu) are superior to wife-takers (dama), from whom they may demand military support and from whom they receive substantial amounts of goods in the form of bride price. Leach interpreted matrilateral cross-cousin marriage as a correlate of the lineage system: It is . . . obvious that if marriage is used to express differences of political status, then the marriage rule must be asymmetrical. If lineage A give wives to lineage B, and marriage expresses the fact that lineage A are the overlords of lineage B, then clearly lineage B cannot give wives back to lineage A. Matrilateral cross cousin marriage is thus a correlate of a system of patrilineal lineages rigged into a class hierarchy. It does not necessarily follow that the bride-givers (mayu) should rank higher than the bride-receivers (dama); but it does follow that if class difference is expressed by marriage, then mayu and dama must

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Island networks be exclusive and one of the two must rank above the other (1954:256).

In Kachin (as in Tonga), the rule of matrilateral cross-cousin marriage establishes political alliances and is observed mainly by individuals of high status: It is really only persons of high status, such as the sons of chiefs and the sons of lineage heads, who need to conform strictly to the mayu-dama rules. Their marriages become thereby marriages of state. In the Hpalang situation, when the Nmwe and the Laga lineages declared that they were mayu—dama^ this implied a relationship which in theory should continue indefinitely. But it did not imply an exclusive relationship such that all Laga males must marry Nmwe females. The relationship was adequately preserved as long as, in each generation, there was at least one marriage which conformed to the formal rule (1954:77). In describing the Kachin lineage as a type, Leach emphasized its contrast with the African segmentary lineage, its similarity to the Polynesian ramage as described by Firth, and its probable occurrence in many other societies in the world: What makes the Kachins particularly interesting from an anthropological point of view is that they have a society which is simultaneously segmentary and class stratified. In most types of lineage system that have so far been described in any detail, the process of lineage segmentation leads to a "balanced opposition" between the resulting segments rather than to a status ranking, superior and inferior. For this reason, among others, the interesting typology of political systems which Fortes and Evans-Pritchard have suggested for African societies would not cover Kachin gumsa society. The Tikopia, as described by Firth, have indeed what may be considered a "pure" lineage system associated with notions of a class hierarchy, but here the whole scale of social activities is on such a minute scale that analogy is not very useful. I think that there are plenty of societies in the world of the Kachin gumsa type, but it so happens that social anthropologists have not yet got round to looking at them. That makes it all the more difficult for me to achieve lucidity (1954:159). Clearly, if Leach had looked at Gifford's Tongan lineage rather than Firth's Tikopian ramage, he would have found a scale of social activities

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comparable to Kachin, and, as he later imagined, of Lovedu (see Krige 1975), a system that "in a salutary and thought-provoking manner seems to have much the same kind of structure in reverse," that is, a socially encompassing hierarchical lineage system based on primogeniture rather than ultimogeniture, associated with a rule of hypergamy rather than hypogamy. Even the mythologies are dual, since in some versions Kachin chiefs claim descent from an earth-father and a sky-mother rather than conversely as in Tonga, a reflection perhaps of the alternative forms of anisogamy. Unfortunately, as Service (1985) points out, British anthropologists paid as little attention to Leach's discovery of the conical clan as American anthropologists did to Kirchhoff's. The Kachin-Tonga contrast has special significance for Oceanists because these two cases represent primordial types of Austronesian social organization. In his reconstruction of ancient Indonesian society, van Wouden (1968[1935]) elucidated a "structural core" consisting of three elements: "exclusive" - that is, matrilateral - cross-cousin marriage; a clan system (with implications of stratification based on genealogical seniority); and a "sociocosmic dualism" in which wife-givers and wifetakers are associated, a la Durkheim, with the symbolic oppositions elder brother-younger brother, masculine-feminine, sacred-secular, earth-sky. In a generalization of van Wouden's analysis, Mabuchi (1960) distinguishes two types of kinship rituals in Malayo-Polynesian societies: the "Indonesian type," in which bride-giving groups spiritually predominate over bride-receiving groups and exchange "female gifts" for "male gifts," and the "Oceanian type," in which sisters and their descendants predominate over brothers and their descendants, with a similar exchange of female and male gifts. Matrilateral cross-cousin marriage is the "ideal pattern" in Indonesia, and it is found in some Melanesian and West Polynesian societies in Oceania. Kachin society (hypogamous), although it is not Austronesian, is "reminiscent of" the Indonesian type, whereas Tongan society (hypergamous) is representative of the Oceanian type. In Chapter 5 we will suggest that Marshallese and, more generally, Proto-Nuclear Micronesian society are also representative of the Oceanian type, with interesting similarities to the Indonesian type.

Social stratification in Polynesia Following "a lead provided by Kirchhoff," Sahlins (1958) identified the conical clan as a framework of social stratification in Polynesia. Sahlins specified both the structure of ranking in the conical clan - in Polynesia

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Figure 4.1. Sahlins's (1958) model of the conical clan (ramage) in Polynesia. - and the higher forms of economic cooperation referred to by Kirchhoff. Adopting Firth's metaphor in preference to Kirchhoff's, on the grounds of greater descriptiveness and wider usage, Sahlins defined the ramage as a "nonexogamous, internally stratified, unilineal - in Polynesia, patrilineal - descent group. Distance from the senior line of descent is the criterion of stratification. By this definition segments of a ramage are also ramages" (Sahlins 1958:140).8 Referring to Gifford's description of the Tongan lineage, Sahlins defined ramage stratification in terms of the rule governing succession to leadership. The rule is primogeniture: (ideally), the eldest son succeeds his father. Sahlins deduced two consequences of this rule. (1) The eldest brother outranks his younger brothers and is accorded special prestige. More generally, all brothers are ranked from high to low on the basis of birth order. (2) An elder brother's descendants outrank a younger brother's descendants. Thus a senior line of descent, consisting of a succession of firstborn sons, can be distinguished from junior lines of descent, from a common ancestor. The paramount chief is in the senior line of descent, and progressively lower grades of chiefs and commoners are in progressively more junior lines of descent. Fig. 4.1 reproduces Sahlins's model of the patrilineal ramage in Polynesia. The vertical lines of males are lines of descent, and the horizontal lines are pairs of brothers, with the elder brother on the left and the younger brother on the right. The figure illustrates the theoretical rank of ramage members and of subramages I, II, III of a single larger ram8 Later, in a general work on social organization, Sahlins (1968) substituted the term "conical clan" for "ramage."

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age. Individual A has the highest rank and is therefore the paramount chief, because he is the firstborn son in a sequence of firstborn sons descended from the ramage ancestor (the male at the top left of the diagram). This sequence is the senior line of descent. Individual X has the lowest rank, because he is the last-born son in a sequence of last-born sons descended from the ramage ancestor. He and his father are in the most junior line of descent. The model shows only male members of the group. According to Sahlins, "Every brother is ranked by the principle of seniority. Sisters may also be integrated in this hierarchy. But with few exceptions, effective leadership is vested in the male, even though he may have an elder sister. Therefore, the position of women is not treated in this general discussion" (1958:141). Sahlins showed very clearly that in many Polynesian societies the ramage had important religious, economic, and political correlates. Typically, the paramount chief was regarded as sacred, powerful, and privileged because of his line of descent. The position of the paramount chief is reinforced through deification of his ancestors, themselves former paramount chiefs, and through connecting this main line of descent to the important god or gods of the group. In consequence of divine descent, a certain sacredness and power (mana) is believed to be inherently a part of chieftainship. Often the sacredness of the chief is associated with supernatural power over the fertility of the land and people. The chief is surrounded with a number of tabus which vary in elaborateness from island to island. These tabus reinforce the position of the chief at the apex of the ramified system (1958:142). Sahlins identified two higher forms of economic cooperation associated with the ramage. The first is the regulation of land tenure through a pattern of "overlapping stewardship" in which the paramount chief, who owned the land, delegated management to lesser clan chiefs, who in turn allotted usufruct rights to commoners. The second is the regulation of production and exchange through a system of redistribution in which goods flowed up and then back down the ramage hierarchy. Some of the goods were used to support chiefly families and finance chiefly undertakings such as subsidization of craftsmen and mobilization of warriors, and some were redistributed to commoners, especially on ceremonial occasions such as chiefly rites of passage. Chiefly authority in Polynesian ramage societies varied from nominal in Tikopia to nearly despotic in Tonga. Decisions concerning punishment for offences and making peace and war were commonly arrived at in consultation with ramage heads. Compliance with chiefly orders - in

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production, for example - was ensured by the kinship obligations of commoners to chiefs and, in some cases, by the implicit or explicit threat of force. The power of paramount chiefs was constrained by the competing interests and strengths of individual ramages. Ramages and subramages were territorially based, associated with individual districts of an island, and often contested with each other for political superiority. Although all subramages were ranked, it was not uncommon for a subramage of lower rank to usurp the position of one of higher rank, depending on historical, demographic, and political circumstances. Conflict between elder and younger brothers - the heads of senior and junior lines - was, as Sahlins (1981) emphasized in a later work, an archetype of Polynesian history. By analogy with organic evolution, Sahlins proposed an ecological model of "adaptive variation" to account for the different forms of social organization in Polynesia.9 Each form is regarded as an "alternative solution to the problem of distributing surplus production" as dictated by a particular type of island environment. The ramage is an adaptation to high island environments with a scattered distribution of resources or environments with a range of crops too large to be exploited by a single household. Ramage structure permits individual households to specialize in the production of goods and exchange them reciprocally and redistributively through the extensive horizontal and vertical kinship ties of the group. Besides Tonga, examples of ramage systems are said to include Hawaii, the Marquesas, and the Society Islands. The "descent line system" consists of "local groupings of small unrelated lineages" that may be no larger than a patrilocal extended family. Rank is minimally developed: "Primogeniture is not the succession rule; in fact, there is no definite rule of succession to the familial title. Rank implied by seniority of birth is absent or not significant; rather, rank is dependent on the traditional, mythological standing of the family title" (1958:251). The descent line system is an adaptation to high island environments with a clustered distribution of resources or to environments with a range of crops too small to require household specialization in production. Thus there is no need for extensive kinship ties between groups. Examples of descent line systems are said to include Samoa, 'Uvea and Futuna. Sahlins distinguished the "form" from the "degree" of stratification. The latter refers to the "complexity of the status system, i.e., the number of different kinds of ranks, and the extent to which they confer unequal privilege in economic, social and religious life" (1958:x). He attributed differences in degree to differences in productivity or "size of the redistributive net" or, equivalently, as Orans (1966) has shown, population size.

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Atoll social organization is treated as a special case, near zero degree as far as stratification is concerned. Since the resources of an atoll are scarce and often unpredictable, the adaptive problem is not the distribution of surplus production but rather survival itself. With small and sporadic surpluses and deficiencies in local production, "a premium is put on direct or reciprocal types of distribution, not types that travel up and down hierarchies. Kinship organization on an egalitarian basis would be a selective advantage" (1958:235-6). Atoll social organization is "interlocking": it consists of multiple groups based on some combination of kinship, residential, territorial, and age-grade structures, each of which is associated with specific productive and distributive activities. Every member of the society belongs to several such groups, in order to have access to all or most atoll resources. When descent lines or truncated ramages are present on atolls, it is for historical reasons - settlement from a high island - or because the atoll environment approximates that of a high island. An example of an atoll with interlocking social organization is Pukapuka in the Northern Cook Islands, which has both matrilineal and patrilineal descent groups, age grades, villages, and atoll-wide social groups. There seems to be general agreement with Sahlins's characterization of atoll social organization as unstratified. The main objections to Sahlins's model, on the part of earlier scholars, concern the classification of resource zones and the types of descent groups found on high islands. In the case of the Society Islands, Finney (1966) considers that the distribution of resources does not require familial specialization in production. In Tahiti, all of the island's major resources are represented in a wedge-shaped area running from an interior mountain valley to alluvial flats, lagoon, reef, and seashore in an area easily exploited by an extended household. An ecologically concentric, socially radial pattern is generally true of Polynesian ramage societies, making local groups economically self-sufficient (Kirch 1984a). In the case of Samoa, J. D. Freeman (1961b) interprets the ethnographic evidence from Kramer (1902), Schultz (1911), and Mead (1930a), all supported by his own extensive fieldwork, as showing segmentary lineages, not "local groupings of small unrelated descent lines," and he points out that resources are not restricted to coastal areas and environs, as Sahlins asserts. More basically, Freeman objects that Sahlins's classification of island resource zones is much too casual to support any assertions about the environmental determinism of social structure. Later scholars, including Friedman (1981), Linnekin (1990), Thomas (1990), and Sahlins (1985) himself, have revised the characterization of East Polynesian societies as conical in structure. Hawaii, for example, lacked that nested hierarchy of lineages, titles, and property rights asso-

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ciated with the conical clan. According to Sahlins (1985:20), historical traces of it remain in the "royal cum cosmological genealogies which, beginning in divine sources and proceeding patrilaterally through senior and cadet branches, fix the dynastic relations between the several islands." But here, and more generally, genealogical seniority was only an "argument" in laying claim to high status. Other arguments were based on alternative bilateral tracings of descent, descent from primary or secondary spouses, and, similar to the Tuamotus, another East Polynesian society described in Chapter 3, the number of ancestral lines to which descent could be traced (Sahlins 1985; Linnekin 1990). Structurally minded anthropologists and archaeologists nonetheless regard Sahlins's explication of the ramage system as a major contribution to Polynesian studies. Goodenough (1959) stresses the importance of the ramage in defining rank, succession, land tenure, and the production and distribution of goods. Although one may quarrel with the details of Sahlins's model, "it enables us to see clearly the basic structural design shared by many Polynesian societies. The structural kinship between them, long felt by students of Polynesia, stands sharply revealed" (Goodenough 1959:257). Kirch (1984a) regards the ramage, now known more generally as the conical clan, as a feature of Ancestral Polynesian Society (APS). This is the society that developed in and spread from West Polynesia following the colonization of the Fiji-Tonga-Samoa area somewhere around 1500 B.C. Consistent with Sahlins's (1968) later general work on social organization, Kirch describes the conical clan in Polynesia as ambilateral (or cognatic) in practice but patrilineal in ideological bias. Kirch hypothesizes that APS was "centered around the pyramidal geometry of the conical clan." In the process of expanding into all of the primary ecological zones of an island, this geometry permitted the "segmentary branches to retain their genealogical interrelations, thus facilitating the amalgamation of large political units under the leadership of a ranking chief" (1984a:66). Even more generally, Bellwood (1995) regards the conical clan (or a kinship group "akin" to it) as an element of Proto-Oceanic or possibly even Proto-Malayo-Polynesian society. He hypothesizes that the rapidity and extent of the Austronesian expansion was due in part to a "founderfocused ideology" in which junior members of the conical clan moved into new territories, occupied or unoccupied, where they established themselves as heads of senior lines, aggrandized their resources, and justified the superior genealogical status and privileges of their descendants by claiming to be founders of kinship groups. This hypothesis is consistent with the Tongan and Kachin cases described in this chapter. The evidence for the conical clan in APS is based on the reconstruc-

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tion of Proto-Polynesian lexemes for social groups and social status. Pawley (1979) glosses *kainanga as "a land-holding descent group under the authority of a chief"; *ariki as "chief, head of a lineage; first born son in the senior line, who succeeds to the chieftainship of the lineage and has strong personal 'mana' and 'tapu.'" Lexical reconstruction also shows separate terms for elder and younger same-sex siblings: *tuakana and *t(a,e)hina, respectively. (In Tongan the reflexes are taokete and tehina.) The linguistic evidence for the conical clan in Proto-Oceanic society is discussed in Chapter 5. From a structural point of view, there are three problems with Sahlins's analysis of the conical clan. (1) It lacks an adequate formal model, and therefore precision and generality. (2) Because Sahlins followed Kirchhoff rather than Leach, his analysis overlooks the role of marriage alliance. In the case of Tonga, for example, Sahlins remarks only that "chiefs could marry as close as cross-cousins; others married somewhat further out" (1958:157). (3) It treats all islands as isolates. The second and third problems are taken up in Bott's (1982) analysis of the Tongan Empire and more generally in Friedman's (1981) model of the prestige-good system in Oceania. The first problem can be solved by modeling the conical clan as a depth-first search tree.

A structural model of the conical clan In defining the structure of the conical clan, we must observe first of all that Kirchhoff was writing in 1935, before the publication of The Nuer and Evans-Pritchard's (1940) analysis of the unilineal segmentary lineage. As J. A. Barnes (personal communication) has pointed out, had Kirchhoff written 25 years later, after the appearance of the major publications on African stateless societies, he would have been unlikely to have characterized exogamous unilateral clans in quite the same way. The cone metaphor applies equally well to an exogamous unilineal segmentary system, with its divisions into maximal, major, minor, and minimal segments, and it is precisely "nearness of relationship" that determines the oppositions and alliances of these segments. In terms of graph analysis, the distinguishing feature of the conical clan is not its shape but the ordering of its nodes. Since Kirchhoff was not specific about the structure of the conical clan, a few commentators, in particular neoevolutionists who regarded it as the missing link between tribal and state societies, attempted a clarification. Fried criticized Kirchhoff's assertion that siblings were equal in status: "It is curious to read that in the conical clan all members except siblings have different standings because in some of the most interesting

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examples of conical clans the distinctions of status between siblings are the keystones of the system, e.g., the system in Hawaii and elsewhere in Polynesia as was shown so clearly by Marshall Sahlins" (1957:5).10 Fried gives the following characterization of rank in the conical clan: The crucial thing was to establish for every member of the society his closeness of relationship with the founding ancestor. Actually, this is not the clearest way to phrase it, despite the ease with which the point is understood by people who have grown up in a ranking (and stratified) culture. It would be better to say that what must be known is the distance of relationship between any member and the highest-ranking person of his generation. This is frequently interpreted in the folk ideology as distance from the founding ancestor. The concept is made possible by adding a simple principle of seniority to that of descent, so that even in the case of multiple births, siblings can be sorted out on the basis of different moments of birth and given different privileges on this basis. In such a society the first born of a first born is considered closer to the ancestor than the first born of a second born, and so on. The line of descent is not simply the transgenerational tie that recedes toward the firstknown ancestor (for example, the sun!), but the string of first borns through time. This may be further delimited by specifying that only one of the sexes can be recognized as an authentic link (Fried 1967:126-7). This characterization is unclear and incomplete: what does "distance of relationship" mean, and what is the relation between individuals in different generations? Aside from Sahlins, White (1959) produced the only explicit structural model of the conical clan, or as he preferred to call it, the "ambilateral lineage." His diagram is reproduced in Fig. 4.2. If one ignores the redundant horizontal lines representing generation levels, then White's model is equivalent to a rooted labeled tree. White defines rank as follows: Everyone in the lineage traces his ancestry, through males or females or both, to the individual, or married couple, A, at the peak of the pyramid. The various lines of descent, and the individuals comprising those lines, are graded. The first-born child, male or female, J5, ranks highest among his or her siblings; the 10 Fried also disputed Kirchhoff's assertion concerning the genetic independence of the unilateral clan and the conical clan, preferring instead the hypothesis that the latter emerged from the former.

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Figure 4.2. White's (1959) model of the conical clan.

second-born, C, ranks second, and so on. The line of descent from a first-born child ranks above that of a second-born. But distance from the common ancestor, in terms of generations, counts also. In the line of descent of first-born children, a person in an older generation ranks above one in a younger generation: D ranks above H. But any child of the second-born child of the common ancestor, C, would rank above the great-grandchild of the common ancestor in the line of descent through first-born children only: F and G would rank above H. Also, in our diagram, the line of descent BEK would rank above CGO because B is the first born of the common ancestor, A, whereas C is the second-born child. Similarly, FM ranks above GO because F is the first born of C whereas G is a younger sibling of F. Thus we have a conical structure of kindred in which rank is determined by two factors: (1) distance from the common ancestor in terms of generations, and (2) the divergence of the lines of descent in accordance with the relative age of siblings. The line descending through first-born children only passes perpendicularly from the apex of the cone to its base. This is the highest line; the others descend in rank as their distances from the main line increase (1959:178-9).

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There are two problems with White's model. The labeling of nodes in the tree is unacceptable, since it implies the simultaneous birth of pairs of junior siblings (C and C, E and £, etc.). Here White may have been misled by Kirchhoff's geometrical metaphor. Less trivially, it cannot be true that generation overrides birth order, for this would contradict the principle that a senior line as such outranks a junior line as such, that is, that every member of a senior line outranks every member of a junior line. In White's interpretation the paramount chief, H, would be outranked by everybody in higher generations, whatever their line - however "common" they might be. In eastern Fijian societies that had "pyramidal" descent groups "approximating] the Polynesian model" (Sayes 1984) - that is, the conical clan - "a man ranks before his father's younger brother" (Hocart 1929:37). In Marshallese society in Micronesia, which had the matrilineal variant of the conical clan, a woman outranks her mother's younger sister. (See Chapter 5.) White does make one important observation on the conical clan by pointing out that the emphasis on birth order should be reflected in the kinship terminology in the form of special terms for "elder sibling" and "younger sibling." This, as already noted, is one of the main pieces of evidence cited by linguists and archaeologists who use lexical reconstruction to infer the presence of the conical clan in APS (Pawley 1982; Kirch 1984a). According to Sahlins, all individuals in a ramage are uniquely ranked: "Every individual within this group of descendants from a common ancestor holds a different status, one precisely in proportion to his distance from the senior line of descent in the group" (1958:141). The term "distance" is not defined, but if Sahlins means genealogical distance as defined by the number of parent-child links between ego and the senior line of descent, then this statement is obviously not true. For example, in Fig. 4.1, C outranks D since his ancestor was an elder brother of D's ancestor, but C and D have the same genealogical distance (= 3) to the senior line of descent, and similarly for the pairs EF, GH, EG, EH, and so forth. The same difficulty arises in comparing the distance and rank of relatives in different generations. For example, B outranks C's father, since his ancestor was an elder brother of C's ancestor, but both B and C's father are two steps from the senior line of descent. For similar reasons the common shorthand definition of rank in terms of "genealogical distance from the founding ancestor" is not satisfactory. A precise, clear definition of rank in the conical clan can be given using the model of a "depth-first search tree." What does it mean to "search" a tree? First of all, a rooted tree is always meant, and the

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Figure 4.3. The graph Ky drawn as a rooted labeled plane graph.

search always begins at the root node. Second, searching a tree means visiting each node exactly once until all nodes have been seen. Algorithms for searching a tree, and more generally for searching a graph, are similar to those for solving a maze, or labyrinth. Historically, as reported by Konig (1936), the two equivalent independently derived algorithms of Tarry (1895) and Tremaux (in Lucas 1882-94) for solving a maze apply to rooted plane trees and rooted plane graphs. This means a planar graph as drawn in the plane. This is illustrated by the graph in Fig. 4.3, which has as its underlying graph ("underlying" meaning unrooted and unlabeled) the complete graph K4. There are two differences between solving a maze and searching a tree. (1) In the former, one must always move from the present node to an adjacent node. This implies that (2) every edge must be traversed at least once. (In practice, in the algorithms of both Tarry and Tremaux each edge is traversed exactly twice.) Although each of conditions (1) and (2) may hold for certain tree searches, they do not always hold. A labeled tree with p nodes has the numbers 1,2,...,/? placed at the nodes. All 16 of the labeled trees with four nodes are shown in Fig. 4.4. n Four of them have the star K13 as the underlying tree, and the remaining 12 have the path P4.

11 Cayley (1891) proved that the number of labeled trees is Lp-pp~

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Figure 4.5. A breadth-first search of T2.

The Searching Conventions: 1. To search a given tree T means to label its nodes 1, 2, . . . , p. The search process means a visit to each node in the order of the labels 1, 2, etc. 2. Node 1 is always taken as the root and is visited first. 3. A tree to be searched is always a rooted plane tree. Breadth-first search. The labeling of the nodes in Fig. 4.5 illustrates the breadth-first method for searching the full binary tree T2. This labeling method applies to all rooted plane trees. It can be described as a Western-style reading: first from top to bottom and then within each level from left to right. An implicit anthropological example of a breadth-first search tree (BFST) is Goody's (1966) model of the "inclusive system of agnatic

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Figure 4.6. Goody's (1966) implicit BFST model of the "inclusive system of agnatic hereditary succession."

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Figure 4.7. A depth-first search of T2.

hereditary succession" in which the nodes of a genealogical tree are labeled according to the order of succession, as shown in Fig. 4.6. Depth-first search. This is the type of search that was anticipated by the maze-solving methods (algorithms) of Tarry and Tremaux. In a depth-first search, one always descends from the root in a consistent direction - say, left - as long as possible. When this cannot be done any longer, one backtracks until one can reach a new node by going down to the right, and so forth, emphasizing, as in Fig. 4.7, the tendency to move to the left. It was mentioned earlier that each edge of a graph is traversed exactly twice. This is displayed in detail in Fig. 4.8 for the search of Fig. 4.7. The reason for continuing until a return to the original starting point is to make sure that no node and no edge have been left unvisited. Thus the place where one escapes from the maze must be found. This diagram clarifies the discussion of searching a tree and solving a maze. In a depth-first search, the nodes are visited in an adjacently consecutive way; in a breadth-first search they are not.

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Figure 4.8. The details of a depth-first search. 1

Figure 4.9. A left-to-right DFST model of the conical clan in Polynesia, where rank is defined by primogeniture. A depth-first search tree (DFST) model of rank in the Polynesian conical clan is shown in Fig. 4.9. The root of the tree represents the founding ancestor and the nodes his descendants. The birth order of the sons of each father is from left to right. The founder and all his descendants are ranked and labeled 1 . . . , p in accordance with the order in which they are first visited in a depth-first search from the root of the tree. The ranking is a complete order: it is irreflexive, since no person outranks himself; asymmetric, since for any two individuals / and /, if i outranks /, then / does not outrank /; transitive, since for any three /, /, k, if i outranks / and / outranks &, then i outranks k; and complete, since for any two individuals / and /, either / outranks / or / outranks /. This is the meaning of Service's (1962) statement that in a chiefdom, that is, a conical clan society, "every individual is in a class by himself." It is also the basis of Gifford's observation on the relativity of rank in Tonga:

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Figure 4.10. A right-to-left DFST model of the conical clan in Kachin gumsa society, where rank is defined by ultimogeniture.

A chief's relatives are not chiefs in the strict sense of the word, but naturally have a higher social standing than the relatives of matapules or commoners. However, the terms eiki (chief) and tua (commoner) are used also in a relative sense. Thus, throughout the Tongan social structure at its various levels, we find people performing analogous functions, receiving similar treatment and having like terms applied to them (Gifford 1929:111-12). The DFST in Fig. 4.9 shows the full and consistent implications of primogeniture in the Polynesian conical clan. For every pair of individuals / and; in the group, / outranks / if / is an ancestor of /, or if / is an elder brother of /, or if /'s ancestor is an elder brother of ;'s ancestor, or if z's ancestor is an elder brother of /. This last condition, which is implicit in Sahlins's model, means, contra White, that ego outranks his father's younger brothers as well as his father's younger brother's sons or, more generally, that ego outranks his ancestor's younger brothers as well as their descendants. The DFST eliminates the confusion in White's model: trying to define rank both generationally and lineally, as White does, would be equivalent to trying to label a search tree simultaneously breadth-first and depth-first. The DFST completely dispenses with tortuous verbal characterizations such as Fried's. For each search algorithm, duality considerations suggest the possibility of other search procedures. One of these is left-right search duality. The right-to-left dual of the DFST in Fig. 4.9 is shown in Fig. 4.10. This

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is the graphical model of the Kachin conical clan, where rank is defined by ultimogeniture.12 It is important to note that not all social hierarchies can be modeled as a search tree.13 For example, in the Yapese tribute system, described in Chapter 2, all of the outer tributary islands are "children" of their Yapese "father," but not all children are ranked with respect to each other. Hence the nodes of this rooted tree cannot be ordered breadthfirst or depth-first. In this sense one might say that the Yapese tribute system is a "loose hierarchy."

Prestige-good systems In the most recent and influential interpretation of the conical clan, relations of descent and alliance are combined with network variables in an evolutionary model of social stratification known as the "prestige-good system" (Ekholm 1977; Friedman 1981; Friedman and Rowlands 1977). As outlined by Friedman (1981:281), in an application to Oceanic prehistory, the model consists of four elements: 1. Generalized exchange; 2. An elite monopoly over prestige-good imports necessary for social reproduction, for example, payments at marriage ceremonies; 3. An asymmetric bilineal tendency in the kinship system; 4. Asymmetric political dualism: religious versus political chiefs, original people versus invaders, female versus male, center versus periphery. The first and fourth elements are based on Blust's (1980) reconstruction of "connubium," that is, matrilateral cross-cousin marriage, and dualism as features of early Austronesian society. The second element presupposes an exchange network with few connections and long dis12 Leach is not completely consistent on the Kachin rule of ultimogeniture. On p. 205 he says, "of a group of male siblings the youngest ranks highest, the rest in order of birth," but on p. 165 he says that the "next senior line after that of the youngest son is that of the eldest son." On p. 261 he implies that only the youngest and eldest sons count, while the intermediate sons are excluded. If so, the DFST model of rank in the Kachin conical clan is, ideally, a twin binary tree. The graphs in Figs. 4.9 and 4.10 are of course ideal models. In practice, as Gifford was aware and as Leach emphasized, individuals may attempt to manipulate genealogies for purposes of social climbing. They may also be favored in succession by personal ability and affinal support, as in the case of the Tongan Tu'i Kanokupolu, or "working king" (Bott 1982). 13 We should also note, with respect to rules of succession, that more complicated systems can be modeled by search procedures. See, for example, Carre (1979).

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tances amenable to control by an elite. The term "bilineal tendency" refers to wife-taking and wife-giving relations between senior and junior lines of the conical clan or between immigrant and local kinship groups. The elements are linked in the following way: The control of prestige-goods implies an exchange situation in which wives move towards monopolistic centers and valuables (and men) move in the opposite direction. This model implies a tendency for the formation of localized matrilines and dispersed ranked patrilines whether or not they are cognized as such [Ekholm 1977]. The general dualistic structure of such systems is the result of women moving up and/or men moving down, creating a local asymmetrical dualism: low = female/ high = male (Friedman 1981:281). The prototype of the prestige-good system in Oceania is the Tongan Empire in West Polynesia as described by Guiart (1963), Kaeppler (1978) and Bott (1982). The Tongan archipelago consists of some 200 islands running from southwest to northeast over a distance of about 290 kilometers. The majority of the islands are uplifted limestone formations covered with volcanic ash. They are extremely fertile, supporting abundant and varied crops, including yams - the staple and the main item of tribute taro, breadfruit, bananas, arrowroot, and the Tahitian chestnut. Tongatapu, 440 square kilometers in area, is by far the largest and richest island in the group - Cook compared it to "the most fertile plains of Europe" - and was the traditional center of the Tongan Empire. As shown in Fig. 4.11, the main island groups from south to north are Tongatapu, which includes the island of 'Eua; Ha'apai, which consists mainly of coral atolls and islets; and Vava'u, a large, rich limestone island together with adjacent smaller islands. About 290 kilometers north and 340 kilometers northwest of Vava'u are the outliers, Niuatoputapu (a group of two islands - Niuatoputapu and Tafahi) and Niuafo'ou. Samoa is 580 kilometers northeast of Vava'u. 'Uvea, which was once part of the empire, and a source of stone blocks for the royal tombs at Tongatapu, is 850 kilometers northwest of Tongatapu. Fiji, on the border between Melanesia and Polynesia, lies about 350 kilometers to the west of Tonga. Most of the population lives on the uplifted limestone islands. Many of the islands were never settled, but some of these were exploited for resources, including timber and stone for tools, from volcanic islands, and marine products from atolls and reefs (J. M. Davidson 1979). Gifford (1929) estimated the population of Tonga at 18,500 and of Tongatapu at 8,000 in 1840. Kirch (1984a) puts the number at 40,000 at the time of contact. The north-south orientation of the archipelago permit-

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Tongatapu

Figure 4.11. A graph of the protohistoric long-distance exchange network of the Tongan maritime empire (from Kirch 1988a). ted easy two-way communication between islands during the trade wind season, and it is said that Tongan paramount chiefs and their representatives traveled regularly within the group (Gifford 1929). Tongans were famed as navigators and voyaged to Fiji in the west and Samoa in the north for purposes of trade, warfare, marriage, and sheer adventure. Anciently, the Tongan Empire consisted of the Tongatapu, Ha'apai and Vava'u groups. Sometime within the last 500 years it extended to the outliers, Niuafo'ou and Niuatoputapu, and 'Uvea (Kirch 1984a). According to traditional history, Tongan rulers were at one time established in Samoa but were expelled around 1250, and at various times they received tribute from Futuna and Rotuma. Gifford cites evidence of Tongan domination of islands as far away as Niue and the Ellice and Gilbert groups. The headquarters of the empire and the residence of the paramount chiefs was the island of Tongatapu. Traditional history recounts original rule by a single chief, the Tu'i Tonga (TT), from around 950 to 1470, after which rule was divided between two collaterally related senior and junior lines into a sacred office occupied by the Tu'i Tonga and a secular office occupied, at first, by a member of the Tu'i Ha'atakalaua (THT) line and later by a member of the Tu'i Kanokupolu

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(TK) line, established in 1610. The secular chief, referred to as the hau, which means "victor" or "conqueror" and also "the one who rules, sovereign" (Bott 1982:109), was responsible for maintaining order, protecting the sacred king, and collecting tribute.14 The TT, the THT and the TK were ranked by relations of genealogical seniority and marriage exchange. The first THT was a younger brother of the TT, and the first TK was a younger brother of the THT, giving the rank order TT > THT > TK. When the TT Kau'ulufonua Fekai relinquished secular authority to his younger brother Mo'ungamotu'a, who assumed the title of THT, part of the arrangement was that the latter would give his daughter to the former as a chief wife. Similarly, when secular power was divided between the THT and the TK, the latter became a wife-giver to the former, thereby preserving in marriage the rank order established by descent. In the process of displacing the THT, whose line finally lapsed in 1799, the TK inserted himself as wife-giver to the TT and wife-taker to the THT. On the basis of Gifford's monograph and her interviews with Queen Salote Tupou of Tonga, Bott (1982) has shown that the rulers on Tongatabu ("Land of the Chiefs") controlled the outlying tributary islands ("Land of the People") not merely by means of a good communication system permitting the ready application of military force as Captain Cook (1961, 1967) had surmised, but also through ties of kinship and marriage. Essentially, Tongan paramount chiefs or kings resident on Tongatapu defined their relation to the people of the outlying islands in the same way that they defined their relation to each other - by genealogical seniority and marriage alliance. Younger brothers (heads of junior wife-giving lines) were sent to administer the outlying islands, where they replaced the indigenous chiefs and married the latters' daughters, thereby making their own lineages fahu to the local population: Sometimes the local chiefs resisted the immigrants sent out by the Tu'i Tonga or the Tu'i Ha'atakalaua, but they usually accepted them, both because they generally came with warriors and because of their rank. A man who was a nobody at the central court on Tongatapu would be a great 'eiki on an outlying island. The usual procedure was that the Ha'atakalaua immigrant established his position by marrying the daughters of 14 In Gunson's (1979) interpretation of Tongan history, hauship was an ancient feature of West Polynesian society and referred to the principal chief, whatever his sacred or secular status. The term implies succession through victory in war. Cognates of the term are widespread in Oceania, found, for example, in Fiji in Melanesia and in Pohnpei in Micronesia.

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Island networks local chiefs; the immigrant and his heirs were then bound to be supported by the local people, according to the principle that brothers support their sisters and their sisters' children (Bott 1982:62).

The lines established by immigrant chiefs on the outlying islands were fahu and thus wife-takers to the indigenous inhabitants and wife-givers to the paramount chiefs in Tongatapu. Ruling chiefs on individual islands in the Tongan Empire were also linked by asymmetric "founding marriages,"15 as revealed in the following anecdote: "While eating, the people of Niaufoou Island turn their backs to the breeze blowing from Niautoputapu Island. This respectful manner of eating is assumed because Niautoputapu is 'chief to Niaufoou: the chiefly house of Niaufoou once supplied a wife for the chief of Niautoputapu" (Gifford 1929:17-18). Kirch's (1984a:236) synopsis of such marriages shows that Vava'u was "chief" to Niautoputapu, which was "chief" to Niaufo'ou, which was "chief" to 'Uvea. With Tongatapu as "chief" to Vava'u (and all other islands), asymmetric marriage alliance integrated the principal islands of the Tongan Empire as a complete order. The strategy of sending younger brothers to outlying islands also had the effect of eliminating potential rivals and usurpers. Thus Tongan paramount chiefs used their lineage structure to control their island network and their island network to resolve problems posed by their lineage structure. We note that the Kachin solution to potential conflicts over succession resulting from their rule of ultimogeniture was for the eldest brother to leave with a group of followers and found a new settlement (Leach 1954). The "Tongan-Kachin solution" is an alternative to the "Turkish solution" (Trautmann 1981) of killing off collaterals and potential usurpers - the Law of Fratricide - (Goody 1966), and the "Buganda solution" of having the brothers fight one another for the succession (Southwold 1966). In Hayden's (1978) model of the prestigegood system, it was the Tongan-Kachin solution - the emigration of poor nobles - that led to the rapid Austronesian expansion.16 Paramount chiefs on Tongatapu monopolized prestige-good imports essential for social reproduction. The goods included canoes, pottery, red feathers, decorated barkcloth and sandalwood imported from Fiji, and fine mats (kie hingoa) imported from Samoa. According to Kaep15 This term derives from Sahlins's (1985) analysis of asymmetric marriages between immigrant chiefs and indigenous inhabitants in the Fijian Islands. These marriages, unlike MBD marriages, are not necessarily repeated, but, once made, serve as "charters" of political relations between the two groups. 16 Another Polynesian variation is the "Arioi solution" in Tahiti, where junior collaterals of chiefs, who did not inherit titles, led privileged but unreproductive lives as members of a traveling entertainers' society (Gell 1993).

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pier, "the pre-eminent Tongan context for the use of Fijian and Samoan trade goods was and is on ceremonial occasions - especially weddings, funerals and various kinds of state and religious celebrations" (1978:250). These goods were distributed to subordinate chiefs on the outlying islands in return for tribute and wives. Friedman finds evidence of prestige-good systems, in various stages of development and decline, throughout West Polynesia, in Fiji, southern Melanesia (especially New Caledonia), and parts of northern Melanesia (the Admirality and Trobriand Islands). In the "big man" societies of the New Hebrides, for example, an earlier hierarchical political structure is indicated by elaborate graves containing wives, chiefly retainers, and long-distance imports (Garanger 1972). Friedman also notes "clear resemblances" to prestige-good systems in western and central Micronesia, "from large scale regional exchange systems (the Yap 'empire') to asymmetrical dualism in both kinship and politics" (1981:287). The disappearance of the prestige-good system is attributed to changes in network structure. In Melanesia a "long term increase in trade network density" led to a breakdown of monopolistic control and the emergence of competitive big man systems. This is the picture sketched by Spriggs (1986) for southern Vanuatu. In East Polynesia Tahiti, Hawaii, the Marquesas, Easter Island, and the Southern Cooks the prestige-good system disappeared as a result of a "quantum increase in distance between island groups." The loss of elite control of an overseas exchange system led to increased competition and warfare. Competition expressed in feasting led to agricultural intensification. Marriage preferences combined endogamy with restricted alliances between groups of equal rank. With the disappearance of asymmetric marriage alliance, bilineal kinship systems became cognatic. Kinship-based political hierarchies were easily overthrown through warfare, and titles and land were redistributed by victorious lineages. Friedman's model has been incorporated into Polynesian archaeology in Kirch's (1984a) analysis of the Tongan maritime chiefdom and in his interpretation of the Lapita exchange network (Kirch 1988a). In Kirch's view the rapid and successful Lapita colonization of the western Pacific depended on a long-distance exchange network in which environmentally vulnerable and demographically small and unstable daughter communities maintained a "lifeline" back to established mother communities based on social as well as purely material ties: A formal exchange system centering on the status enhancing acquisition of prestige goods (shell valuables, high quality obsidian, adzes, etc.) provided the social mechanism for maintaining what was in effect an essential component of the Lapita

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Island networks dispersal and colonization strategy . . . Because Lapita prestigegood exchange was essential to social reproduction (through the system of MBD marriage as suggested by Friedman) the Eastern Lapita settlers replicated the network in its essential aspects including communities specializing in the production of shell valuables (Naigami, Lakeba), and others in the production or extraction of material resources (1988a:113-14).

The following comments can be made on Friedman's model of the prestige-good system, its Tongan prototype, and its application to Micronesia. 1. Network structure. In characterizing, through graph analysis, an elite's ability to exert monopolistic control over an exchange network, it might be better to use the alpha index rather than the density of a graph. Density is concerned only with individual connections of a network (edges of a graph), whereas the alpha index is concerned with independent cycles and hence with alternative communication paths between pairs of communities. The fewer the alternative paths in a network, the greater the potential for elite control. The graph of the Tongan long-distance exchange network in Fig. 4.11 (from Kirch 1988a) is almost a tree17 with a = 13.3 percent. This compares with a = 54.8 percent for the kula ring (Fig. 7.4), the archetypal Melanesian trade network. The graph of the Yapese prestige-good network in Micronesia, shown in Fig. 2.17a, is a tree with a = 0 percent. 2. The Tonga-Fiji-Samoa network. Instead of viewing Tonga as the "apex of a three-cornered network" (Kaeppler 1978), a more global model linking Tonga, Fiji, and Samoa in a single directed cycle of exchange can be discerned by considering the most valued prestige good each society obtained from one other society. Fijians (Lauans) prized whales' teeth, tambua, which they obtained in large part from Tonga (Hjarno 1979-80), above all other objects of ceremonial exchange. "They serve as a means by which wealth may be accumulated, condensed, stored, and exchanged" (Thompson 1940:124). One of the main contexts of ceremonial exchange was marriage. In commoner marriages, whales' teeth were given by the groom's family to the bride's family, and in chiefly marriages they were given by both sides in a large number of reciprocal exchanges (Lester 1939-40). Samoans regarded the red feathers of the paroquet, Coriphilus fringillaceus, which they obtained from Fiji by way of Tonga, as a most precious good essential for the decoration of fine mats. These mats were 17

Technically, it is a cactus, a connected graph every cyclic block of which is a cycle. Every tree is a cactus.

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__^ -*

^ ^

SAMOA

FIJI

(kie hingoa) Fine mats (for Tongan marriage ceremonies)

TONGA Figure 4.12. A prestige-goods cycle linking Tonga, Fiji, and Samoa.

part of the dowry, toga, given by the bride's side to that of the groom. Chiefs sought to accumulate large quantities of fine mats through marriage to use in payment to specialists and craftsmen such as tattooers, and canoe and house builders. The husband's return gifts, 'oloa, included red feathers given as imported goods: "The most important element which a husband could give his wife's group was red feathers which would enhance the value of the fine mats when used as decoration. Most of the red feathers which were found in Samoa came from Fiji" (Hjarno 1979-80:117). Tongans placed the highest value on the fine mats, kie hingoa, that they obtained by trade and as dowry from Samoa. These mats were exchanged most conspicuously in Tongan weddings and at funerals which emphasized the bonds established by marriage (Kaeppler 1978). Each island society, therefore, provided one other society's most valued marriage (and, more generally, prestige) good, joining all three societies in a directed cycle analogous to a system of generalized exchange, as shown in Fig. 4.12. This cycle suggests that the Tongan monopoly of voyaging and trade implied in Fig. 4.11 may have been historically recent and that Tonga, Fiji, and Samoa were originally directly linked. Kaeppler is skeptical about the existence, or at least the frequency, of direct Fijian-Samoan contacts, but Haddon and Hornell (1975) and Lewis (1972) consider that Fijian navigation and voyaging have been greatly underrated: Commodore Wilkes . . . characterized the Fijians as 'daring navigators' and their canoes as 'superior to those of the other

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Island networks islands'; he described them (1845, vol. 3, p. 366) as making 'very long voyages - to Tonga, Rotuma, and the Samoan islands'. Speiser (1923, p. 250) refers to Fijian voyages to the New Hebrides (Haddon and Hornell 1975:334).

Lewis cites the diffusion of canoe technology as evidence suggestive of earlier extensive Fijian voyaging. It was the Fijian ndrua, probably inspired by Micronesian design, that introduced the changing ends system of altering direction in West Polynesia. The fact that it was the Fijians, rather than the Tongans, who adopted Micronesian ideas, suggests extensive Fijian voyaging prior to the eighteenth century . . . Some confirmation of this comes from the Polynesian-speaking island of Nukuoro in the southern Carolines. Eilers (1934:179) cites traditions of canoes from "Hiti" (Fiji) visiting or being driven to the island on several occasions. It lies more than 1800 miles north-west of Fiji and neither winds nor currents would favor drift (Lewis 1972:262). 3. Applications to Micronesia. Although Friedman's model is presented mainly in the context of Melanesia and Polynesia, it sheds considerable light on developments in Micronesia: it demystifies the Yapese Empire in western Micronesia, as described in Chapter 2, and it also contributes, by extension, to an evolutionary interpretation of social organization in Nuclear Micronesia. In the Marshall Islands the prestigegood system disappeared, but social stratification, based on the conical clan and asymmetric marriage alliance, persisted and continued to depend on elite control of exchange networks. The isolated high island societies of Pohnpei and Kosrae, on the other hand, developed along the lines outlined by Friedman for East Polynesia, whereas the isolated, unstratified atoll societies of Micronesia came to resemble those described by Sahlins for Polynesia. We turn to these developments in the next chapter.

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The contention between older and younger brothers is a celebrated condition of Hawaiian - indeed Polynesian - myth and practice. Marshall Sahlins, Historical Metaphors and Mythical Realities Children of brothers love one another, children of sisters fear one another. Marshallese proverb, August Erdland, The Marshall Islanders

In the conclusion of his monograph on Tonga, Gifford remarked that "the parallels in the social organization of Tonga and the remainder of Polynesia and Micronesia are obvious" (1929:350). Unfortunately he did not elaborate, choosing instead to discuss parallels and possible genetic connections between Oceania and Japan. 1 It is clear, however, that the sexually dual, matrilineal variant of the conical clan is found in the Marshall Islands in eastern Micronesia. It also appears that the Micronesian and Polynesian variants are genetically related, having a common origin in Proto-Oceanic society. Like its Tongan counterpart, the Marshallese conical clan was a socially encompassing, politically expansive structure, associated with asymmetric marriage alliance and implicated in the formation of island empires. Linguistic evidence suggests 1 Gifford noted parallels in descent: "In both countries [i.e., Tonga and ancient Japan] society was patrilineal; the patrilineal groups traced descent for many generations, sometimes back to a god; and each patrilineal group had a patron deity; the members of the patrilineal groups were of unequal rank. In both countries great chiefs were buried in megalithic vaults in mounds . . . There are certain general resemblances in mythology and worship: Tongan and Shinto mythologies are strikingly similar in tenor. The parallels are such as to suggest the possibility that the social organization of Polynesia, Micronesia and Japan are genetically connected" (Gifford 1929:350). Complementarily, Levi-Strauss (1985) has discussed parallels between Austronesia and Japan in terms of marriage alliance.

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that the Marshallese variant represents the ancestral form of all the differently permuted forms of social organization in Nuclear Micronesia.

The Marshallese conical clan Descent The Marshallese clan, called jowi in the Ralik chain and jou in the Ratak chain, consisted of a group of lineages called bwij, which traced descent through females from a common ancestress (Kramer 1906; Kramer and Nevermann 1938; Erdland 1914; Mason 1947, 1954; Spoehr 1949a, b; Kiste 1974).2 In Marshallese mythology all clans were descended from a pair of sisters - an elder sister who settled at Aur Atoll in the eastern, Ratak chain and a younger sister who settled at Namu Atoll in the western, Ralik chain. Clan names usually designated places of origin or residence, for example, ri kwajalein, "people of Kwajalein." As outlined by Mason, lineages were divided into three classes: bwij-iniroij, or royal lineages; bwij-in-bwirak, or noble lineages; and bwij-inkajur, or commoner lineages. Some clans consisted only of commoner lineages, but others consisted of all three classes. Every lineage had a head, almost always a male, who was, ideally, the eldest son of the eldest sister of the lineage. The head of the royal lineage was the iroij labalap - "very big chief or king." Beneath him and under his control were the iroij elab - "big chiefs" - drawn from his elder sister's family and elder sister's daughter's family, and, next in rank, the iroij erik "lesser chiefs," drawn from his younger sister's family. The bwirak nobles were said to be "distantly related" to the paramount chief. The head of a kajur lineage was called the alab. Primogeniture was reflected in the kinship terminology, which had separate terms for "elder sibling" and "younger sibling," jeo and jato respectively. The firstborn child occupied a special position with respect to both his siblings and his parents' younger siblings: The younger siblings of my parents must "esteem" me {jure, jureik id), that is to say, they may not embrace me, may not lie on the same mat with me, may not accompany me to the outer shore, may not say a disrespectful word in my presence, may not eat food touched by me, may not know of my bad deeds 2 The Marshallese term bwij, bwiji- is "from a Proto-Nuclear Micronesian root meaning 'a distinct group or cluster of something' (e.g., a school of fish, grove of trees, distinct group of people), perhaps best thought of as 'a crowd' of something. It is cognate with Trukese pwii 'group, cluster' and pwii- 'sibling of same sex' (i.e., 'my crowd')" (W. H. Goodenough, personal communication).

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and may not reproach me if they know of my errors . . . As the firstborn child, I even have the right to give orders to the younger siblings of my parents. They must obey me absolutely (Erdland 1914:85).3 In his Ph.D. dissertation concerning Bikini Atoll, at the northern end of the Ralik chain, Mason (1954) described a completely ordered primogenitural system of rank in the Marshallese clan. In effect, Mason discovered the matrilineal variant of the conical clan. Mason never generalized his discovery - he made no reference, for example, to Firth's description of the ramage in Polynesia or to Gifford's analysis of the lineage in Tonga - and he never published his work. Although his interests were purely ethnographic, his account is of general interest because it contains a numerical code he devised for describing clan rank, a code that is formally equivalent to a DFST. Mason's report from Micronesia thus lends unexpected support to the graph theoretic model we introduced in Chapter 4 to describe the conical clan in Polynesia. Mason invented his code, or "system of designation," to describe rank in the Bikini clan. Although his description refers to the office of alab, since Bikini clans lacked royal lineages, it can be understood generically as applicable to alab and iroij of all grades in Marshallese society. According to Mason, Bikinians classified the branches of a lineage, e.g., major lineages within a maximal lineage, by the relative age of the founders. The lineage of an elder sister assumed precedence over the lineage of a younger sister in community activities. Hierarchical position was indicated by the terms used to designate eldest sister's lineage (bwij-errito, "oldest bwij"), lineage of a younger but not youngest sister (bwij-iolab, "middle bwij") and youngest sister's lineage (bwij-eriklok, "youngest, smallest bwij") (1954:220). Each lineage had a head, alab, who was (ideally) the eldest surviving son of the woman who founded the lineage. The most senior and most deferred to alab was the head of the eldest sister's lineage. Mason's code consists of assigning numbers to the female members of a lineage depicted in a genealogical chart, as shown in Fig. 5.1. The number assignments define the rank of individuals and lineages: A decimal system of designation is utilized in order to cope with the complexities that accompany any description of the All translations from the German of Erdland, Kramer, and Nevermann are from the Human Relations Area Files.

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Bilib

Married; no daughter Ditto; no further issue anticipated.

—CHI}-

Ruban

HUD-

Jamwel

Lewoj Lajurilik Berro Jajua

- Numbers Females in a descent group (lineage). - Names • /Wafc: Oldest son of a female founder of a lineage. Her number is at left of his name.

Jakob

-{73327]

-Q35]

Figure 5.1. Mason's (1954) coding of rank in the Bikini clan. ramifications of Bikinian lines of authority. Each number in the chart represents a female who was either a real or potential founder of a lineage. Her lineage may be designated simply by number (her name is not essential), followed by the appropriate clan name if necessary, e.g., Lineage 111 of Makaoliej

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[clan]. The number itself supplies the relative position of the lineage within a maximal lineage, or subclan. Thus, Lineage 111 was bwij-errito in first line of descent for two generations from the founder of a maximal lineage; on the other hand, Lineage 1311 was bwij-errito for two generations from bwijeriklok of a maximal lineage. Obviously, the former lineage possessed a higher status within the community structure . . . The relative position of any alab is easily indicated by reference to the lineage number of his mother, since he was ordinarily the eldest surviving son of the founder of the lineage whose interests he represented (1954:229, 233). The sequences of numbers in Fig. 5.1 show a unique rank for every lineage and for every female lineage member. Thus individuals 111, 112, and 113 are eldest, middle, and youngest sisters ranked from high to low, as are the lineages 111, which the eldest sister continues, and lineages 112 and 113, which the middle and youngest sisters potentially found. In the terminology of Sahlins's Polynesian model, the sequence 11111 is the "senior line of descent" - a succession of firstborn daughters (instead of firstborn sons). The number sequences show the continuous ramification or, to switch to the biological metaphor that LeviStrauss (1969:247) employs for the Kachin, the "continuous parthenogenesis of lineages." We note in connection with Mason's code that the Marshallese themselves may use numbers to rank lineages. On Arno Atoll in Ratak, "Branches of a lineage which are descended from several sisters are ranked according to the relative age of the latter. The ranks of branches of the same lineage may be specified by referring to them by . . . numbers (e.g., bwij kin kar juon 'branch number one'; bwij kin kar ruo 'branch number two'; etc.)" (Rynkiewich 1972:44). The term "maximal lineage," in Mason's description, comes from Evans-Pritchard (1940), who borrowed it from Gifford (1929). The irony of this connection between Oceanists and Africanists cannot go unremarked. Mason worked out a notational system to handle rank in the Marshallese conical clan, but he also used Evans-Pritchard's model of the Nuer segmentary lineage, which distinguishes "maximal," "major," "minor," and "minimal" lineages, as a conceptual aid to describe lineage branching. Meyer Fortes relates that Evans-Pritchard had found help for his Nuer model in Gifford's account of the Tongan lineage: Evans-Pritchard states in his review of my Dynamics of Clanship . . . that the suggestion of how to handle the data of Nuer descent groups came from a conversation with RadcliffeBrown in 1931 . . . I was present on this occasion. Evans-

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Island networks Pritchard was describing his Nuer observations, whereupon Radcliffe-Brown said, as he stood in front of the fireplace: "My dear Evans-Pritchard, it's perfectly simple, that's a segmentary lineage system, and you'll find a very good account of it by a man called Gifford . . . " Thereupon Radcliffe-Brown gave us a lecture on Gifford's analysis of the Tonga system (Fortes 1979:viii, quoted in Kuper 1982:72).

Recalling our review of independent discoveries of the conical clan in Chapter 4, it is clear that the analysis of social organization in Oceania would look very different today if Leach (1954) had read Gifford (1929) instead of Firth (1936), if Sahlins (1958) had read Leach instead of Kirchhoff (1955), and if Mason had read Gifford instead of EvansPritchard. Marriage alliance would not be unduly neglected in favor of descent relations, and Micronesia would not be treated as an "other," third category standing outside of the traditional Melanesian-Polynesian contrast. Oceanists might have accepted Spoehr's (1952) suggestion that Micronesia and Polynesia should be viewed as a single region, "Micro-Polynesia," for purposes of studying large-scale historical problems. In our criticism of White's model of the conical clan, in Chapter 4, we noted that primogeniture and generation are mutually contradictory principles of rank. An illustration is provided by the following account from Arno, a less traditional society than Bikini: It is quite clear to Arno Marshallese that the third and fourth principles which determine rank within the lineage, i.e., seniority of lineage branch and seniority of generation, may be contradictory in specific instances. For example, if a lineage is divided into several branches, and if males of a junior ranking branch are generationally superior to males in the senior branch, it is uncertain which has the right to succeed. However, in the aristocratic lineage, there is some indication that males of the senior branch of the lineage have the right to succeed before males of the senior generation but junior branch (cf. Mason 1947:54 whose data indicate same). Thus, within the aristocratic lineage, there is an expectation that males of junior branches might never succeed to the chieftaincy (Rynkiewich 1972:67).4 In Marshallese society, gradations in kinship rank were spatially expressed in a concentric residential pattern, symbolically elaborated to 4 The first and second principles that Rynkiewich discusses are that only men may succeed to the paramount chieftaincy and that elder brother succeeds before younger brother.

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include a series of correlated oppositions in which center : periphery :: chief : commoner :: cultivated : wild :: public : private :: silence : noise, and so forth. The correspondences recall those in Fiji: First it should be mentioned that the rank of the individual kin groups, as well as of the different lineages within the same kin group, is indicated by where they live on the island, whether on the lagoon side, farther in the interior of the island, or not far from the outer shore. The soil on the lagoon side of an island is the most fertile. Besides, canoe traffic on the lagoon starts from the lagoon shore, and all the traffic can be observed from there. Because of all these advantages, settling closest to the lagoon is a birthright of the families of high chiefs. The lower the rank, the farther interior is the place of residence. The lowest families live closest to the outer shore, there where the ground is covered with stony debris, where shrubs that require the least to grow, like the salt-water bush, are neighbors of equal status, where the surf breaking on the outer reef pounds loudly, and where, finally, the outer reef, as the place where everybody, rich and poor alike, relieves himself, does not exactly smell like narcissuses and roses, particularly at ebb tide. The low families are therefore likened to wild pandanus trees, which likewise thrive on the outer shore. When the word lik (outer shore) [outer side] occurs in the name of a kin group, it is evident that this group is an inferior one, usually the lowest lineage of a kin group or set of kin groups (Erdland 1914:263). The religious, economic, and political correlates of the conical clan in the Marshalls were similar to those of the conical clan in Tonga. Chiefs, and their near relatives, were regarded as sacred by reason of their descent: The power of an irodj [iroij] is based upon the assumption that he is provided with mana (. . . debbo) by virtue of his descent. Although it comes to the chief from the original ancestors through the women of the kin group, he himself can strengthen it, for example, by successful wars or the completion of his tattoo (Kramer and Nevermann 1938:279). The paramount chief and his immediate . . . relatives were customarily surrounded with an aura of spiritual force which was considered dangerous to commoners. Although Bikinians did not support an iroij-lablap on their own atoll, they retained a sense of awe bordering on fear of the royal personage whenever he visited Bikini. This fear was due not entirely to the

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As in Tonga and other conical clan societies, Marshallese land tenure was based on a pattern of overlapping stewardship. As outlined in Mason (1947), land was owned by royal lineages and administered by paramount chiefs, iroij labalap, who delegated its management to lesser chiefs, iroij erik, or to local headmen of commoner lineages, alaps. The iroij erik, or the iroij labalap if there was no intermediary, assigned specific parcels of land to the alaps, who in turn granted usufruct rights to individuals or families. A district of an island or an entire island was headed by an alap if there were no resident chiefs, as in Bikini, by an iroij erik if authority had been delegated and by an iroij labalap if it had not. Erdland mentions that noble, bwirak, lineages, might also own land, in which case they were regarded as iroij by their subjects. When two or more clans resided on the same island, authority was either divided between them, as on Arno and Majuro, each of which at one time had a pair of independent paramount chiefs or kings, or else the clan chief with the strongest following dominated, exacting tribute from the other clans. Land changed hands frequently as a result of warfare between different royal lineages and competing branches of the same royal lineage. Commoners, kajur, had evidently not lost their kinship connections to chiefs as they may have done in Tonga, but they were nonetheless basically tenants on chiefly estates. Commoner lineages paid tribute, including firstfruits, mats, and cordage, to royal lineages. More generally and informally, commoners were expected to provide labor, servants, and all food - fish, pandanus, breadfruit, arrowroot flour - required by chiefly families, who were thus freed from subsistence tasks. It was said that "the paramount chief has three stomachs: one for food, one for storing peoples' gossip and one as a storehouse of goods for the people" (Rynkiewich 1972:65). As in the more developed Polynesian chiefdoms (Kirch 1984a), the emphasis on the flow of goods seems to have been more on consumption by chiefs than on redistribution to commoner producers. In Tonga, as Gifford noted, chiefly rank was affected by marriage: The ideal combination for chiefly rank is that both parents be of chiefly blood. The offspring of a chief and a commoner are spoken of as half-chiefs, or, as the Tongans express it, "halfshell" (ngeesi taha). A chief may chide a half-chief by saying,

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iroij (nejin ak)

iroij

iroij

ik

ib

u O

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bwirak bb bwirak elap

A

bi A

ki bwirak egmouij

kb

bk

jibtok (bwirak)

kk

jiblok (bwirak)

kajur

Figure 5.2. A product graph of intra- and interclass marriage and status of the children in Marshallese society.

"Ah, you half-shell." (The term vaataha - one branched - also refers to a person whose parentage is half-chief, half-commoner) (Gifford 1929:123). A similar relation obtained in the Marshalls: If an iroij marries a kajur woman, their children are really commoner by birth, but have more status as bwirak erik, more commonly called bwirak-in-egmouij. In using the term egmouij) the Marshallese make an analogy to the animal world: among the shorebirds (kotkot) is a red-feathered bird referred to by natives as a "chief" and called mirlep; in bird society is one with mixed red and white feathers known as egmouij. Thus bwirak-in-egmouij are those persons just below chiefs (Mason 1947:58-59). Given three social classes, iroij, bwirak, and kajur, the category bwirak-in-egmouij is one of nine possible marriage types. All nine can be represented by forming the product of two graphs, G1 = P3 and G2 = P 3 , each of which represents the three social classes. The products of intra- and interclass marriage are shown in the graph in Fig. 5.2. The first letter in each combination designates a female parent and the second a male parent. The names by each marriage combination give the status of the children. If both parents are iroij, the children are iroij and are re-

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ferred to as nejin ak, "frigate bird children," by analogy to the bird regarded as "der Konig der tropischen Meere" (Erdland 1914). A child is chiefly, iroij, as long as the mother is iroij. A child's status is raised if the father is iroij and the mother is bwirak or kajur, but this holds only during the father's lifetime. In the long term the status of one's male ancestors apparently did not count for much: "When Kabua [a paramount chief of Ralik] dictated the genealogical tree of his ancestors to me and mentioned only women, I asked him to name the men too as is generally customary in Polynesia. He laughed and said that they were completely irrelevant, and therefore he did not know them" (Kramer 1906:431). An excellent summary of Marshallese social stratification is given by Spoehr: In former times, the social distinctions that underlay this class system were very real indeed. The paramount chief was possessed of autocratic powers that were shared to a lesser extent by the nobility. The paramount chief and his nobles were the leaders in war and in sailing expeditions. They controlled the land and the fruits thereof. They provided the primary leadership of the community, and in turn enjoyed the privilege of being fed and supported by the commoners. Of the noble class, the paramount chief himself was accorded the greatest respect. His position involved the hereditary acquisition of magical power, somewhat similar to Polynesian mana. He was approached only in the most deferential manner; in his presence persons walked stooped over, or moved on their knees. Of all the members of the community he was supposed to command the best information on the affairs of the Marshallese world. In recompense for his inherited responsibility, he received the best of the food produced on the land or caught in the sea. He lived in the most favored location. His lineage had its own cemetery. No restrictions were placed on the number of his wives and he had access to all commoner women. Over his people he exercised autocratic powers. The commoners were the workers of the land, the fishermen, the sailors, and the ordinary fighting men. With the possible exception of the alabs - the heads of the commoner lineages the commoner class were the workers (ri-jerbal) in every sense of the word. Their tribute supported the nobility. Their houses were built in less favorable parts of the island. Nor were they permitted the distinctive tattooing and the finer dress of the nobility (Spoehr 1949a:77, quoted in Oliver 1989).

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Marriage alliance Spoehr, who worked in Majuro, a large atoll in the southern part of the Ratak chain, reports a Marshallese preference for bilateral cross-cousin marriage. The preference is reflected in "the use of a special referential term for the cross-cousin of opposite sex and the special pattern of privileged familiarity between these relatives. It is also reflected in the fact that children of cross-cousins of opposite sex use brother-sister terminology and cannot marry, even though their parents are not married to each other" (Spoehr 1949a:204). Mason, who worked in Bikini, an atoll that because of its relative isolation at the northern end of the Ralik chain "may have retained certain features of traditional Marshallese social organization," mentions a preference for matrilateral cross-cousin marriage. A man may not marry "the daughter of his father's brother or his mother's sister since that relationship is regarded the same as that of brother and sister, and an incestuous union would result. The daughter of the mother's brother is the best choice he could make" (Mason 1947:27). Mason specifies a rule of subclan rather than clan exogamy. He also states that matrilocal residence was the "aboriginal ideal," with avunculocal residence common for prospective successors to lineage headship. It is difficult to know for certain what the marriage rule was in traditional Marshallese society, since that society was only a distant memory even in Erdland's (1914) time. One could argue for bilateral crosscousin marriage, on the grounds that matrilineal societies, being relatively unstable, prefer the short cycle (Levi-Strauss 1969). On the other hand, matrilateral cross-cousin marriage would be consistent with the "harmonic" - matrilineal, matrilocal - character of traditional Marshallese social structure.5 P. E. de Josselin de Jong (1951) makes this argument in the case of Minangkabau, an Indonesian society that, like Marshallese society, is matrilineal and matrilocal, with primogenitural succession to titles, a class division into nobles and commoners, and a Generational-type kinship terminology. In Minangkabau, bilateral cross-cousin marriage is the norm, but the "really ideal marriage" is with the MBD. "There we have a matrilineal regime harmonique - as a 5 In Levi-Strauss's (1969) theory of kinship, "harmonic" regimes have similar rules of descent and residence - patrilineal and patrilocal or matrilineal and matrilocal whereas "disharmonic" regimes have opposing rules - matrilineal and patrilocal or patrilineal and matrilocal. Disharmonic regimes lead to restricted exchange, whereas harmonic regimes "announce" generalized exchange. Generalized exchange leads to anisogamy, i.e., marriage between spouses of different status. For a modified application of the harmonic-disharmonic contrast to marriage systems in South Indian societies, see Dumont (1957).

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matter of fact Minangkabau is one of the 'extremely rare' examples of this type of social organization. A system of echange generalisee would therefore be expected, and this does indeed agree with our conclusions as to the ideal type of marriage"6 (1951:185). Matrilateral cross-cousin marriage would also be consistent with the system of ranked kin groups that we have already described for Tonga and Kachin (harmonic societies with patrilineal descent and patrilocal residence). As Levi-Strauss (1982) has observed, in societies in which rank is determined by birth order and "proximity to the common ancestor," marriage is ineluctably anisogamic, the only choice being between hypergamy and hypogamy. Asymmetric marriage alliance is implicit in certain usages of Marshallese kinship terminology, and it is explicit in an upper-class marriage strategy described by Erdland. In the affinal terminology, a sister's husband (ZH) is called "big father," jema lallab, and a wife's brother (WB) is called "my comrade," au mamman. Spoehr offers the following psychological interpretation: The practice of a man's referring to his sister's husband as a "big father" can be related, I believe, to the fact that the sister's husband is the father of a man's nephews and nieces. The latter are a man's heirs to his position, and will in time be the principal guardians of the land rights of the maternal lineage. The father of these nephews and nieces is by virtue of his paternal relation to a man's heirs a very important person. The "father" term is applied out of consideration of the sister's husband as father of a man's heirs, not as a special sort of "father" to ego. Conversely, ego's wife's brother is the head of the wife's lineage and eventually this headship will pass to ego's children. So for the sake of his children and his relation to their lineage, a man treats his wife's brother with due consideration and applies a special term to him (Spoehr 1949a:190). Perhaps. But one could more easily give a structural interpretation based on an original alliance meaning of marriage: "big father" refers to ZH in his role as chief, and "my comrade" refers to WB in his role as military supporter established by the wife-giving relation. Such alliances are frequently mentioned in traditional Marshallese history, as described in the next section.7 6 In his comparative study of Micronesian social organization, Stillfried (1953) notes "frappante Aehnlichkeiten" between Minangkabau and Micronesia. Pak (1993:89) describes Minangkabau as having "le connubium asymmetrique fonde sur le rang social par rapport au roi et Pechange generalise, fonde sur les relations de parente." Pak also hypothesizes that the system is in the process of changing from an elementary to a complex structure. 7 In another context Spoehr (1949a:110) mentions that the relationship between sister's

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Further to this point, Rynkiewich (1972) mentions that a man can refer to his sister by an alternative term, inu, "mother," while she can refer to him as manu, "child." This is consistent with the designation of ZH as "father" and his alternative designation of WB as neju emman, "my male child." We note that the use of kinship terms to denote alliance relations occurs in a number of Indonesian societies. Among the Ema of Timor, for example, who have a rule of exclusive MBD marriage, superior wife-givers are called "maternal uncles and ancestors" or "mothers and fathers," while inferior wife-takers are called "sisters and their children" or "paternal aunts and their husbands" (Clamagirand 1980). In the Tanimbar Islands "one's male wife-givers are collectively conceptualized as one's 'brothers and uncles' (ura-nemi) or one's 'masters' (dud) and one's female wife-takers are collectively conceptualized as one's 'sisters and aunts'" (McKinnon 1991:115). Of particular interest is the respect accorded the sister, which is similar to that accorded her husband: "The relationship between male Ego and his 'big father' is like that which exists between Ego and his sister. The sister's husband is 'very greatly respected in one's eyes'. . . Requests by sister's husband may not be refused" (Rynkiewich 1972:53-4). The elevated position of the sister would be consistent with the rule of hypergamy that we described in connection with Tonga. More direct evidence of asymmetric marriage alliance, at least in the form of founding marriages, emerges very clearly in the context of Marshallese inter-island networks or empires. Marshallese empires

The Marshalls consist exclusively of low islands - 29 atolls and 5 raised atolls - strung out in two parallel chains, Ralik and Ratak, running from southeast to northwest for some 1,000 kilometers. Twenty-two of the islands were permanently settled. There is no single island in either chain comparable to Tongatapu in Tonga, which dominates all others by a quantum difference in size, population, and wealth. The population at contact is estimated at 15,000 (Mason 1947) with individual island populations ranging from less than 100 to 1,500. The principal ecological contrast is between the drier northern atolls and the wetter and richer southern atolls. Crops include pandanus, coconut, arrowroot, and breadfruit. Arrowroot flour produced in the north was traded for breadfruit paste produced in the south (Pollock 1975). The atolls are basically husband and wife's brother "seems to be connected with the more inclusive relationship between the two lineages involved," but his interpretation is psychological in the manner of Radcliffe-Brown - extreme respect as a means of avoiding conflict - rather than structural - marriage as a means of establishing alliance.

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Island networks G: Dry islands

Bikini

Rongelap Utrik

Mejit

Elder sister clans

Younger sister clans

Mili Namorik

Ratak Graph Ralik Graph

Ebon

Wet islands

Figure 5.3. A graph of the Ralik-Ratak voyaging network.

like those described by Sahlins for Polynesia, with one significant difference: all the atolls in Sahlins's study are either relatively isolated - Ontong Java, Pukapuka, Tongareva - or consist of small clusters - the Tokelaus, Manihiki-Rakahanga - whereas all those in the Marshalls (except for the two outliers in Ralik) were part of a regular voyaging and trading network, and in this respect similar to the islands in Tonga. As in Tonga, the north-south orientation of the islands, perpendicular to the trade winds and westerlies, facilitated easy two-way communication. A graph of the Marshallese voyaging network constructed from

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data in Kotzebue (1821), Erdland (1910, 1914), Winkler (1901), Davenport (1960), and Pollock (1968) is shown in Fig. 5.3. The solid lines represent voyaging links within each chain and the dotted lines less frequently used but significant links between the two chains for purposes of visiting, trade, and warfare. In the Marshalls as in the Carolines (Gladwin 1970), east is the cardinal direction - the direction of the sunrise, the trade winds, and the dominant swell. Consistent with this physical asymmetry, the clans descending from the elder and younger sister founding ancestresses are located in the eastern (Ratak) and western (Ralik) chains respectively. In comparison to the rich elevated limestone islands of Tonga, the atolls of the Marshalls provided few resources. Unlike Tonga, land was scarce, and this was a major cause of inter-island warfare and residential mobility. While there were occasional contacts with the Caroline Islands to the west and the Gilbert Islands to the south, the Marshall Islands were not part of a larger interregional exchange system. Political organization in the Marshalls was on a smaller scale and less stable than in Tonga, with empires confined to each of the two chains or parts of each chain of the archipelago.8 Genealogical depth and historical traditions are much shallower as well as sketchier in the Marshalls, going back only to the early nineteenth century, not the tenth century as in Tonga. But in the Marshalls as in Tonga, the conical clan and marriage alliance defined the basic order of social stratification. In the Marshalls a small number of independent royal lineages (iroij) and some noble (bwirak) lineages owned all the land. A royal lineage might own strips of land on several islets of a single atoll or on several atolls, or it might own the land of an entire atoll or island.9 The set of dominating royal lineages varied depending on the vicissitudes of politics and demography. With its headquarters located at one island and its resources, including land and subjects, scattered over a number of islands, the paramount chiefs of the Marshalls maintained control of their island netWe do not wish to exaggerate the stability of the Tongan Empire. On this point see Bott (1982). Some measure of instability seems to be an inherent characteristic of chiefdoms: "with their few high status positions [they] are inherently competitive in their political dynamics. A centralizing tendency as individuals seek to concentrate power and eliminate the opportunities for rebellion is opposed by a fragmenting tendency as local leaders seek to establish their independent authority" (Earle 1989). In actual practice a similar situation obtained in Tonga: "In theory, the Tu'i Tonga held all land and its people and could dispose of both as he pleased. In practice, certain areas belonged traditionally to certain titles, or the descendants of certain aristocrats, and such land was handed down from father to son or from brother to brother. Most of the great 'eiki had estates in several parts of the kingdom. The personal estates of the Tu'i Tonga were, of course, the largest" (Bott 1982:109-10).

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works by periodic visits for the collection of tribute, by the threat or application of military force, and by the delegation of authority to junior kinsmen. In conformity with the matrilineal structure of the Marshallese conical clan, the paramount chief put his sisters' sons in charge of districts or islands (Mason 1947). Ideally, he chose his younger sister's sons, iroij erik, lesser chiefs and distant potential successors, rather than his elder sister's sons, iroij ^higher chiefs and immediate successors. The paramount chief thereby established control over outer islands and at the same time separated heads of potentially rivalrous junior and senior lines of his lineage. In contrast to Tonga, political rivalry in the Marshalls centered on the relation between elder and younger sisters' sons: "In the Marshalls, children of sisters may wage war against one another. Rank and inheritance give rise to quarrels. The old chiefs Loeak and Kabua whose mothers were sisters were enemies till shortly before death" (Erdland 1914:108). It is clear from Rynkiewich's (1972) brief historical account of Ratak that chiefly political and military success depended on marriage alliance. Chiefs who prevailed in the rivalry between junior and senior lineages for the office of paramount chief and who established or maintained their rule over other islands were assisted by their wives' relatives, who served as their warriors and "war leaders" (leatoktok). The pattern is evident in the success of the paramount chief La Mari, of Arno Atoll, who early in the nineteenth century subjugated all the islands of Ratak. La Mari came from Majuro Atoll and married a woman from a RiBikarej lineage on Arno. "The marriage included an alliance with warriors from the RiBikarej and other Arno lineages, i.e., they became La Mari's 'war leaders'" (1972:82-3). La Mari's brothers, it is said, "maintained the chiefdom through strategic marriages and alliances with war leaders and warriors who were commoners" (1972:85). Wife-giving and tribute relations were directly associated, as illustrated by Erdland's account of one of the marriages of Kabua, a nineteenthcentury paramount chief of Ralik: The landed property of a buirak [bwirak] may be much more extensive than that of an iroj [iroij]. In order to gain tributary rights over such estates of a buirak, the iroj have always endeavored to form connections with wealthy libuirak [female bwirak]. Because Nimokwa was the richest landowner of the Ralik group, Kabua, who already had many wives, took her in marriage. Despite this marriage, however, Nelu remained the sole owner of all the territories of his mother, Nimokwa, but he paid annual tribute to his cousin and stepfather. Nelu's father and Kabua's mother - Kaibuiki on the one side and Looj [Libokean according to Mason, n.d.] on the other - were siblings.

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141 bwirak lineage

iroij lineage

f wives "V- tribute Loojj

6

Kaibuki A

=

Nimokwa Q Ni

Kabua

6

6

A Nelu

Figure 5.4. Wife-giving and tribute relations between noble and royal Marshallese lineages. Moreover, as the firstborn of a firstborn [i.e., the eldest son of the eldest sister] Nelu was particularly respected (Erdland 1914:72-3). The genealogical details are interesting. Kabua repeated the marriage of his predecessor and mother's brother Kaibuki by marrying the latter's widow, thereby maintaining the connection to a wife-giving, tributepaying lineage, as shown in Fig. 5.4. Kaibuki, it is said, had married Nimokwa, a local princess, while serving his mother's brothers, who resided on Ailinglaplap Atoll, as iroij erik on Ebon and Jaluit Atolls (Mason 1947). When, in the latter part of the nineteenth century, Kabua vied with his mother's younger sister's son Loeak for control of the islands in Ralik, Kabua was aided by Nelu and Loeak was aided by Litoku, a lesser chief whose sister he had married. Marriage with the mother's brother's widow in connection with matrilineal succession was evidently a Marshallese, as well as a Pohnpeian, custom (Mason, n.d.; Hambruch and Eilers 1936). Malinowski (1932:114) mentions a similar practice in Trobriand society, an Austronesian matrilineal chiefdom in Melanesia. Although Powell doubts that true widow inheritance occurs in the Trobriands, he does emphasize that a "successful leader repeats his predecessor's [i.e., his mother's brother's, mother's mother's brother's or elder brother's] marriages," thereby maintaining an alliance hierarchy based on tribute payments and hypergamy (Powell 1960, 1969a, 1969b). Lounsbury (1965) suggests that widow inheritance in the Trobriands was a practice unique to persons of chiefly status. In general, according to Erdland, "through marriages and inheritance each iroj has one or another buirak who pays tribute to him. The sub-

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chief puts his subjects at the high chief's disposal for free labor, fights for him, and supports him in the most important matters" (1914:73).

The devolution of social organization in Nuclear Micronesia Linguistic evidence suggests that hereditary chieftainships in the Marshall Islands and Tonga had a common origin in Proto-Oceanic (POC) society. Commenting on Douglas's (1979) purely typological hypothesis that hereditary chieftainship in Polynesia and in the Melanesian societies of Fiji and New Caledonia is more easily explained on the basis of common origin than borrowing or convergence, Pawley (1982) points out that a proof would require linguistic evidence of cognate terminologies: It is necessary to establish the original terminology for leaders and associated institutions and to show that the daughter language communities in question retain a significant part of the common ancestral terminology. In the case of Polynesian, Fijian and New Caledonian languages, the immediate common ancestral stage is Proto-Oceanic or an interstage close to ProtoOceanic (1982:39). Pawley provides such evidence by showing that a pair of contrasting terms, *qalapa(s), "Chief," and *qadiki, "Firstborn son of the chief," "Chief-to-be," are reconstructible "as far back as Proto-Eastern Oceanic (PEO), an interstage ancestral to the MN [Melanesian] languages of the Southeast Solomons, most if not all New Hebrides (Vanuatu) languages, Fijian, Rotuman and probably Nuclear Micronesian. It is . . . likely that PEO was no more than an eastern dialect or dialect region within a POC dialect chain, and that both words are of POC antiquity" (1982:41 ).10 Some OC languages (e.g., Arosi in San Cristobal) have retained both terms to refer to the institution of chieftainship, while others have retained only one term. Thus Polynesian has retained *qadiki ('eiki in Tongan), while Fijian has retained *qalapa(s). In Arosi, a-ri'i, literally "Little One," or "The Little Person," designates the firstborn son and successor to the chief, a-raha, literally "Great One," or "Great Person." In Polynesian the term *qadiki became the term for "chief" itself, and the term *qalapa(s) was no longer used. According to Pawley "the verbs *lapa(s) 'be big, great' and *diki 'be 10 Lichtenberk (1986) modifies the terms to *ta-la(m)pat and *qaariki and gives the more conservative glosses "leader" and "oldest child."

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small' are of course widely reflected. What is of more interest is that *lapa(s) commonly has reflexes meaning 'genealogically senior, older' e.g., Woleai [in the Caroline Islands] tame-lap (lit. 'great father') 'eldest male of a family'" (1982:42). We can supply additional evidence from Micronesia. In Marshallese lap means "big," as in the name Ailinglapalap, "very big atoll." As already mentioned, iroij labalap, the head of a royal lineage, means "very big chief" or "king"; iroij elab, those next in rank, means "big chief"; and alab refers to the head of a kajur lineage. Also, as noted earlier, the superior status of the sister's husband is reflected in the term jetna lallap, "big father." In Pohnpeian the two highest chiefly titles are Nahnmwarki and Nahnken (Riesenberg 1968), or Nahnmariki and Naniken, in Bascom's (1965) orthography. The term lap appears in numerous chiefly titles: for example, Nahnid Lapalap, "Great Lord of the Eel"; Souwel Lapalap, "Great Master of the Forest"; Isolap, "Great Honored One," and so forth. The services performed for chiefs are of two kinds: taulap^ "Great Work," and tautik, "Little Work." In the kinship terminology, wahwahlap is "great maternal nephew" and whalaptik is "little maternal uncle." Aklapalap means "proud, conceited," and aktikitik means "humble" (Peterson 1982). In Lamotrekan, a dialect of Woleaian, the term malalap denotes a senior representative of a lineage. One of his rights concerns marriage alliance: "a young man will have his first marriage arranged for him by a senior male member of his own lineage, usually his mother's brother or malalap (big man) . . . The malalap can not only arrange the first marriage of a man but he can also prohibit any such marriage he does not approve for he is not looked upon as just an elder but as the representative of the lineage or descent group for that individual" (Alkire 1965:55-6). The role of the mother's brother in arranging the marriage of his sister's son in matrilineal Lamotrek is analogous to that of the father's sister in arranging the marriage of her brother's son in patrilineal Tonga (Rivers 1910). Marshallese, Pohnpeian, and dialects of the Trukic Continuum (e.g., Woleaian and Lamotrekan) belong, together with Kosraen, Gilbertese, and probably Nauruan, to Nuclear Micronesian (NM) (Bender 1971; Pawley 1972), a family that is most closely related to the languages of the New Hebrides and Banks Islands in Melanesia (Grace 1955, 1964). A schematic map of languages in Micronesia, from Bender (1971), is shown in Fig. 5.5. (The dashed line in the map refers to an immigrant community of Carolinian speakers in Saipan.) According to the age-area hypothesis, the probable homeland of a language family is an area of greatest linguistic diversity. This would lo-

160

130E

170

-4-

20 N

Utirik

Eniwetok 10 - -

Ujelang C

MARSHALLESE

YAPESE PALAUAN •

^

PONAPEAN Ngulu Pingelap Ngatik

A Sonsorol

KUSAIEAN

Mortlocks > NUKUORO

Tobi

• KAPINGAMARANGI

• NAURUAN Arorae 5S

Figure 5.5. A schematic map showing languages in Micronesia (from Bender 1971).

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cate Proto-Nuclear Micronesian (PNM) in eastern Micronesia, a region which includes the islands of Pohnpei, Kosrae, the Marshalls, and the Gilberts. On the basis of this hypothesis, Bender (1971) places the "historical focal point [of NM]" at "the eastern edge of the Carolines." According to Dyen (1965:55), "an east-to-west movement [of the NM languages] is clearly discernible starting from the Kusaie-Ponape-Marshalls-Gilberts area; the Carolinian languages from Truk to the west stem from the Ponape area." Our conjecture is that PNM society looked very much like Marshallese society - an island network based on the encompassing structure of the matrilineal conical clan and asymmetric marriage alliance. Variations derived from this prototype consist of changes in ranking from primogenitural to age-based hierarchies, and changes in marriage alliance - from asymmetric to symmetric and from elementary to semicomplex and complex systems. The variations include analogues of asymmetric marriage alliance and diverse expressions of social and political duality. There are also shifts from matrilineal to double, patrilineal, and cognatic descent. We now describe these variations, together with their associated ecological, demographic, and network conditions. In effect we treat Nuclear Micronesia as a "field of ethnological study" (J. P. B. de Josselin de Jong 1935; P. E. de Josselin de Jong 1980), giving an account of the differences (transformations) as well as the similarities between linguistically and historically related cultures occupying a common geographical area. Our analysis is consistent with (although it does not require) the hypothesis that eastern Micronesia was settled by members of a stratified Lapitan society (Bellwood 1978). It is also consistent with the intuition of some Micronesianists (e.g., Alkire 1988) that PNM society was stratified, and with the technological argument that overseas voyaging presupposed a sophisticated culture: So viel konnen wir feststellen . . . , dass die ursprunglichen Einwanderer [in Mikronesien] in ihrer Kultur eine gewisse Hohe erreicht haben mussten, da sie unter anderem auch Hochseefahrzeuge besassen. Daraus geht weiter hervor, dass sie den Aufstieg zu jener hoheren Kultur nicht auf den Inseln selbst, sondern vorher im sudostasiatischen Raum durchgemacht haben mussen. Wenn heute also gewisse Inseln primitivere Kulturziige besitzen, so haben wir es eindeutig mit einer Degeneration zu tun (Stillfried 1953:117).11 11 Translation: "So much can be ascertained . . . that the original immigrants (in Micronesia) must have reached a certain height in their culture because among other things they possessed ocean-going craft. It follows from this that they must have gone through the ascent to that higher culture not in the islands themselves but earlier in

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We do not necessarily assume that the Marshalls were the first islands to be settled but only that PNM society was best preserved and represented there. Blust (1984-5) conjectures that the Gilberts were settled first, followed by the Marshalls, Kosrae, Pohnpei, and the Carolines, in that order. Goodenough (1957) presumes the same sequence (without distinguishing Kosrae and Pohnpei from the rest of the Carolines). On the grounds of linguistic divergence, Jackson (1983) speculates that Kosrae was reached first. Conceivably, much of eastern Micronesia was settled almost simultaneously, since the islands lie relatively close together and the original settlers had a highly developed voyaging technology (Bender and Wang 1985). Pohnpei

A diarchic variant of PNM society, based on symmetric marriage alliance and competitive exchange, developed in Pohnpei (Ponape), an isolated high island in the eastern Carolines. The precontact population, estimated at more than 15,000 (Peterson 1982), was divided into three and later five independent states or tribes across which were distributed a number of exogamous, internally stratified matrilineal clans. "Clans are conical in the sense of Kirchhoff and Sahlins (1958) that is, relative status declines as one moves from the line of eldest daughters and their sons towards the younger cadet branches" (Peterson 1982:120). Each tribe was headed by two paramount chiefs, the Nahnmwarki and the Nahnken, of different subclans, who held the highest titles in two lines of ranked titles: Al5 A2, . . . , A12 and Bl5 B2, . . . , B12. Tribes were "organized on a feudal basis and subdivided into a number of sections (kousapw) whose heads {koun or soumas) are appointed by and formerly held their fiefs as vassals under tribal chiefs. The sections are further subdivided into farmsteads whose relation to section heads was likewise a feudal one. In theory all land formerly belonged ultimately to the Nahnmwarki and Nahnken who received regular tribute and whose rule was absolute" (Riesenberg 1968:8). In theory tribal titles were held by the 12 highest-ranking men in each subclan, with succession upward through each line, but in practice titles were awarded for service, including warfare and, later, competitive feasting: the southeast Asian area. If, today, certain islands possess more primitive cultural traits, we clearly have to do with a degeneration." See also Hayden (1978:127), who argues that the "construction, ownership and the sailing of long distance boats . . . was probably one of the keys to the maintenance and perhaps to the origin of the unusually strong social stratification on islands occupied by Lapita and Polynesian groups."

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Movement within a line of titles is ostensibly determined by genealogy. The closer one's genealogical ties to the senior line of a matrilineage, that is, to the founding ancestress and the line of eldest daughters descending from her, the more "mana" (manaman) one has. A chief has the right and the ability to rule because of his ancestors; their collective mana is invested in him. In practice, succession upward through a line of titles depends upon one's service to the Nahnmwarki, Nahnken and the community at large. Ponapeans say this service used to be rendered primarily in battle. Since the end of the 19th century it has been measured in terms of production for feasts, provision of "first fruits" (nopwei) and personal gifts to the chiefs (Peterson 1982:17). Section titles were also awarded for feasts. It appears that in the course of Pohnpeian history, titles proliferated to the point where virtually all men, commoners included, held titles of some sort awarded by tribal chiefs and section heads for success in competitive feasting: In Net, which has a total population of some 450, a list of 210 issued tribal titles was collected, and there are undoubtedly more . . . It is clear that the majority of mature men must possess tribal titles and that among them must be a large number of commoners. Informants state that formerly there were fewer tribal titles, that new ones have been invented and issued in order that chiefs might profit by the title-payment feasts, and that most commoners in times past had only section titles (Riesenberg 1968:76). Feasts were given in large community houses, symbolically partitioned into chiefly and commoner, male and female, right and left, front and back, upper and lower sections. Feasts consisted primarily of competitive gifts of yams presented to and redistributed by chiefs or section heads. Success depended on giving yams of great size and on introducing new varieties. A man was closely identified with his yams and had the privilege of naming new varieties, usually after himself. The A-line chiefs were referred to as "royal fathers" and the B-line chiefs as "noble sons." The former held a semisacred status, while the latter were charged with the conduct of practical affairs, analogous to the relation between the Tu'i Tonga and the hau in Tonga, and the alii (chiefs) and tuldfale ("talking chiefs") in Samoa. Ideally, the two lines intermarried exclusively. Bilateral cross-cousin marriage was also preferred in the general population, with a similar "tendency toward the pairing of matrilineal clans" (Bascom 1965). Residence was matrilocal

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for all but the highest titleholders, for whom it was evidently avunculocal (Fischer 1956). There is an obvious contradiction between the kinship designations of the royal and noble lines and the marriage rule, since with matrilineal descent the men of each line would have their fathers and sons in the opposite line. Pohnpeians offer a metaphorical explanation, based on the privileged familiarity permitted members of the B-line. They say that the Nahnmwarki is "an indulgent 'father' and permits his 'sons' to take liberties that no one else would dare. Not so with members of the A-line; such men though potential heirs to the Nahnmwarki's position must be meek and humble in demeanor for they have no fiction of being favored children to support any untoward behavior" (Riesenberg 1968:50). Riesenberg, sensing perhaps the fit between a system of ranked lineages and a rule of asymmetric marriage alliance, suggests a historical explanation. In an independent invention of the Lane and Lane (1959) theory of Crow-type kinship terminology, he conjectures that the kinship terms designating the royal and noble lines would be consistent with a previous regime of matrilateral cross-cousin marriage. In a Crow terminology, as in Pohnpei, patrilateral cross-cousins are raised (FZS = F) and matrilateral cross-cousins are lowered (MBS = S). If in the past the junior B-line was a wife-giver to the senior A-line, the men of B would have their fathers in A, and the men of A their sons in B. Whether or not this is a true conjecture, we have already noted that parent-child kinship terms are widely used in Micronesia to designate asymmetric political relations, which, as in the Marshall Islands, may imply a wifetaking-wife-giving relation as well. Kosrae The social organization of Kosrae (Kusaie), a second isolated high island in the eastern Carolines, was similar to that of Pohnpei. The population at contact is estimated at about 5,000 (Peoples 1990). Although missionization and depopulation had obliterated much of the traditional culture by the end of the nineteenth century, it appears that a paramount chief or king, the Tokosa, held the highest title in a series of 18 ranked titles that were divided into two sets of nine high and low titles (Sarfert 1919-20; Wilson 1968). Ethnohistorical sources give the terms iros or urosse, cognates of Marshallese iroij, for the highest social ranks (Ueki 1990). Succession upward was genealogically regulated, in theory - by rank in a matrilineal conical clan - but, as in Pohnpei, it also depended on success in competitive feasting. Land tenure was based on a feudal pattern of overlapping rights, with tribute flowing upward from

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commoners to local chiefs to the paramount chief. Bilateral cross-cousin marriage was permitted, and residence was matrilocal. Nauru A vestigial form of PNM society in which the conical clan persisted but without its characteristic functions was found in Nauru, a completely isolated raised coral island in southeastern Micronesia. In comparison to Pohnpei and Kosrae, Nauru is a relatively poor island with crops limited to coconuts and pandanus. The language is distantly related to Gilbertese. Nauruan society was divided into 12 exogamous, internally stratified matrilineal clans. "The most important man in the clan is the eldest son of the woman who traces her descent back through a line of eldest daughters to its original foundress. Such a woman and her children are spoken of as being temonibe. People who belong to the junior branches of the clan are amenengame" (Wedgwood 1936:377). Temonibe and higher-ranking amenengabe were the principal landowners. A third social class, itsio, or "serfs," was composed of landless individuals - prisoners captured in warfare, refugees, and castaways - who lived under the protection of a temonibe, or "war leader." Temonibe evidently had few privileges and powers. Wedgwood could find no evidence that they had any recognized judicial functions. The office of district chief was a European innovation. Homestead land was inherited by daughters, but coconut and pandanus land, trees, and fish ponds were inherited and jointly owned by all children. Canoes and weapons passed to sons. In the case of a widower survived only by sons, all property passed to them, "with the eldest son enjoying the priority which is usually accorded to an eldest daughter" (Wedgwood 1936:21). In Wedgwood's opinion the mixed matrilineal and patrilineal features are unusual for Oceanic societies but were a characteristic "compromise" of traditional Nauruan society. A similar compromise is found in Trukese society, as noted further on. According to Wedgwood, sister exchange was "favored highly" and (bilateral) cross-cousin marriage was "warmly approved." Residence remained matrilocal. The northern Gilberts (Butaritari-Makin) A minimal, analogical variant of PNM society developed in the northern Gilbert Islands. The Gilberts (now part of the state of Kiribati) lie just below the Marshalls in southeasternmost Micronesia. The archipelago consists of a chain of 17 atolls and coral islands running from southeast

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to northwest on either side of the equator and covering a distance of some 800 kilometers. The primary ecological contrast is between the wetter, richer northern islands and the drier, drought-prone southern islands. Gilbertese social organization has been described as "aristocratic" in the north and "democratic or gerontocratic" in the south (Grimble 1989). According to Lambert (1978:82), the two northernmost islands, Butaritari and Makin, "either developed or retained a stratified society on the Micronesian pattern culminating in a high chief who reigned over both islands." By the "Micronesian pattern" Lambert refers in particular to the Marshallese system of hereditary chieftainship and feudal land tenure. In Butaritari-Makin three status levels were distinguished: the high chief, his siblings and children (uea); aristocrats (toka); and commoners, referred to (as in the Marshalls) as "workers" (taani m'akuri). Commoners were responsible for cultivating crops and providing hospitality and food to aristocrats, and aristocrats for collecting tribute for the high chief and watching out for his interests while presiding over local councils. As in the Marshalls, commoners could not be arbitrarily evicted from estates and were free to move if dissatisfied with their treatment. High chiefs and aristocrats belonged to "quasi-patrilineages" that Lambert equates with Sahlins's (1958) Polynesian ramages (conical clans).12 Succession was by primogeniture - "only the eldest son of a chief possessed the full supernatural powers of chieftainship." Chieftainship in Butaritari-Makin was a small-scale version of the Polynesian-Micronesian model: The high chief's powers were sharply restricted by the size of the society - Butaritari and Makin had a combined population of about 2,000 in the middle of the nineteenth century - and by the authority of the autonomous councils of village elders. Nevertheless, his privileges and responsibilities, and the taboos that separated him from ordinary folks, all recall practices associated with chieftainship on a grander scale elsewhere in Polynesia and Micronesia (Lambert 1978:82). Chiefly privileges included exemption from working, signs of special respect, polygyny, and elaborate life-crises rites celebrated by distributions of food and goods. Localized segments of "ambilineal ramages," utu, held title to individual estates (Lambert 1966).13 Commoner ramages held conditional 12 Lambert is reluctant to call these groups patrilineal, "for though most ramage founders were men (and the most remote socially relevant ones only five or six generations from living descendants) some subdivisions derive from women" (1971:149). 13 The scholarly terminology for kinship groups in Oceania is bewildering. For example,

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title, whereas aristocratic ramages founded by siblings of the paramount chief held permanent title. The paramount chief, who resided in Butaritari, was the ultimate owner of most estates on both islands. In the frequent succession battles between younger and elder, real and classificatory brothers, especially between a man and his father's younger brothers, the victor was able to seize and re-allot estates. The Gilbertese marriage system was complex in the sense that its only rules were prohibitions defined by genealogical distance and its alliances were therefore probabilistic. As in other complex systems, sister exchange was commonly practiced. Unions were prohibited between lineal relatives, descendants of a common ancestor who were not in ego's generation, and collaterals as far as second cousins, as expressed in the Gilbertese saying that "the fourth generation goes free" (Grimble 1972). In Butaritari-Makin, however, chiefs were permitted and encouraged to marry first cousins, as a means of consolidating families and family lands. Grimble emphasizes that "marriages of this kind have no connection in the Gilbertese mind with the cross-cousin idea" (1972:60). Chiefly marriage was thus "close" in the sense that Kirchhoff assumed for the conical clan. The structural equivalent of the "cross-cousin idea" was expressed in the fosterage relation. In the Butaritari-Makin fosterage relation a child from one family was sent to be raised by another family while remaining a member of his natural family. The foster family received conditional rights to land (taro plots) from the child's natural parents, in return for which they provided certain services, including attendance at life-crises celebrations of the foster child. They also agreed that their lineage would rear one of the foster child's descendants, ideally an eldest child of the foster child, in each succeeding generation. It was stipulated that the foster child should stand in the relationship of classificatory grandchild to his foster grandparents - ideally the great-grandchild of a parent's sibling or the great-great-grandchild of a grandparent's sibling. High chiefs did not act as foster grandparents and did not raise their own children but gave them instead to aristocratic lineages for fosterage. Aristocrats in turn gave their children to fellow aristocrats and free commoners. Foster grandparents were subordinates of natural parents and served as important allies in times of crisis. Sabatier (1977) thought the fosterage relation an "odd exchange of presents to uphold a friendship," but Lambert has shown its political significance, noting in particular its analogies to "sib" is sometimes used for "clan" (Murdock and Goodenough 1947) (an etymologically unfortunate choice, as R. Fox [(1983)1967] points out); "kindred" for "cognatic descent group" (Emory 1934); "ramage" for both "conical clan" (Sahlins 1958), and "ambilineal descent group" (Firth 1957).

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matrilateral cross-cousin marriage as described by Leach (1954) for the Kachin: Both institutions connect any descent group with a limited number of others on a semi-permanent basis, since it must provide wives or foster-grandchildren for the same lineages or ramages in each generation . . . A Kachin does not marry a woman of a lineage to which his own gives wives, nor do Makin ramages exchange foster-grandchildren. The possibility of placing a different value on rights over people, goods, and services makes both asymmetrical fosterage and asymmetrical cross-cousin marriage consistent with a system of social stratification. Child-givers may be of the same rank as, or of higher rank than, child-rearers in Butaritari-Makin, but never of lower, just as wife-takers never outrank wife-givers among the Kachin and Batak (Lambert 1964:257). The southern Gilberts The southern Gilberts were relatively unstratified. Land tenure was regulated by cognatic descent groups, ooi, but political and ceremonial life was dominated by the maneaba system. This system, named after its architectural embodiment in a community meeting house, consisted of a set of nonunilineal title-holding descent groups called boti {bwoti) (W. H. Goodenough 1970, personal communication).14 Membership was regulated by inheritance of specific plots of land.15 Inheritance was usually, but not exclusively, patrilineal. An individual might have rights to membership in several different boti but could activate membership at any given time in only one such group. Much as in the Samoan fono, men in senior lines of descent were favored in succession to a boti title. Speakerships in boti meetings were held by men with the next-highest, not the highest-ranked, titles. Boti chiefs or elders had considerable power in their own communities but not beyond. Maude and Doran (1966) attribute the absence of stratification in the southern Gilberts to the relative poverty of the islands: Even the luxury of a High Chief (or King as they are usually termed in the early literature) was only possible where the pressure of population on resources permitted a surplus production 14 An alternative characterization of the boti is given in Maude (1963a). 15 Goodenough (personal communication) has pointed out that in cases such as this it is necessary to consider property rights in the characterization of descent groups. We have no objection. Our aim has been to present a model for defining genealogically based rank in descent groups.

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to be devoted to the needs of a largely unproductive aristocracy and its warrior and female dependents. This was preeminently possible on fertile Butaritari and Makin, and made possible on Abemama and its satellites by a deliberate reduction of the population to under half in the latter years of Tern Baiteke's chieftainship. But in the southern sub-group, and notably on Nonouti and Tabiteuea, the meagerness of the physical resources, accentuated by periodic drought and consequential famine, kept a large population on the very margin of subsistence and resulted in political fragmentation and government by independent local councils of old men (1966:275).

Truk

In the high island complex of Truk (Chuuk) Lagoon in the central Carolines, the conical clan disappeared. Where primogeniture persisted its significance was primarily symbolic. In some communities marriage alliance retained elementary bilateral tendencies, while in others it changed to a semicomplex structure. In the following description of Trukese social organization, from Murdock and Goodenough (1947) and Goodenough (1951), the statements concerning succession and marriage apply mainly to Romonum Island. Trukese society is divided into 40 matrilineal, exogamous clans or sibs called einang, a reflex of POC *kainanga. Each sib consists of a number of politically independent lineages. Succession within a lineage and a descent line is determined by age rather than primogeniture: It is important to note that succession within a lineage is not strictly matrilineal in the sense that headship stays within a single matrilineal descent line passing to an own younger brother and thence to the eldest sister's son, as it does for example on Puluwat [an atoll in the central Carolines]. Many of Romonum's lineages have two or more distinct descent lines going back several generations. In no case is headship confined to one of them as a senior line. Each descent line has its own mwaanici ["head"] and whichever of them is senior in birth order is mwaanici of the lineage. Succession is based exclusively on a rule of seniority [i.e., age] not on nearness of kinship (Goodenough 1951:76). According to Murdock and Goodenough, although initially "Trukese social organization appears to be a typical matrilineal and matrilocal

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system," closer examination reveals "noteworthy patrilineal and patrilocal features" (1947:333). Patrilocal residence occurs whenever there are too few matrilineally related women to form a viable extended family and as a common alternative in inter-island marriages. Patrilineal connections can also be used to shift matrilineage affiliations. An unusual and distinctive patrilineal feature is the special relationship between the members of a matrilineage and the children of its men, collectively referred to as its ofokiir (jefekyr in Goodenough 1951). The ofokiir have some rights of inheritance, including the right to inherit land when a matrilineage becomes extinct. This relationship is "reflected" in Crow-type kinship terminology: all members of the father's lineage are terminologically classed with the father and his sister, irrespective of generation, and all ofokiir including the children of mother's brother or mother's mother's brother, are called "child." The result is a kinship system of the Crow type, despite the characteristic Malayo-Polynesian paucity of distinctive denotative terms (Murdock and Goodenough 1947:339-40). The Romonum Trukese kinship system is semicomplex in the sense that the marriage rules apply to classes of relatives, as in an elementary system, but since the rules consist entirely of prohibitions the alliances are probabilistic, as in a complex system. In Truk marriage with a sibmate is disapproved [while] marriage with a subsibmate is forbidden. It is equally forbidden with a member of one's ramage [i.e., "the members of a subsib who have residence in the same political district, but who are organized into more than one corporation"], lineage and descent line. Marriage prohibitions are likewise extended bilaterally. It is forbidden to marry anyone in one's futuk [kindred] in fact to marry anyone who is recognized as a consanguineal relative. It will be noted that this prohibition includes the members of one's father's lineage among the taboo relatives, a fact which makes it impossible for two lineages to intermarry regularly generation after generation and excludes cross-cousin marriage as a possibility (Goodenough 1951:120). There are, however, as Heritier (1981) has shown in the case of other semicomplex (Crow and Omaha) systems, elementary tendencies in the Trukese system. These include sister exchange, alliances between different lines of prohibited lineages, and (evidently) the possibility of renewing alliances by limiting bilateral marriage prohibitions to three or four generations:

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Arranged marriages sometimes included brother-sister exchange, the parents arranging that both a son and a daughter will marry the daughter and son of another couple (Goodenough 1951:121). The prohibition against marriage into one's father's lineage does not extend to members of his ramage, subsib or sib. In fact, if his lineage is divided into two well organized descent lines, each with its own separate house, marriage into a descent line other than the father's may occur . . . The extension of incest taboos bars any kind of marriage with first cousins and with at least half of one's second cousins as well (Goodenough 1951:120).16 As a further indication of more restricted and repeated alliances Goodenough (personal communication) specifies a "marked tendency" for men to marry into their FF's lineage or their MF's lineage provided the individuals do not belong to the same kindred. From a Trukese perspective the purpose of such marriages is to "circulate land given out of a matrilineal lineage estate to children of its men back to someone in that lineage or in another local lineage of the same clan." Politically, Truk has been described as an "almost unified society" located on an "almost atoll" (Alkire 1977). About 10,000 Trukese inhabit a complex of small high islands lying within a large lagoon encircled by a barrier reef. In contrast to Pohnpei, Kosrae, the Marshalls, and the northern Gilberts, Truk is relatively unstratified, divided into numerous independent districts. In traditional times districts were joined in networks or "leagues" based on political and military alliance. Each district had a dominant lineage headed by a chief who had "a sort of eminent domain" over the cultivable land and the right to collect tribute in the form of gifts and firstfruits from subordinate lineages. The district chief was traditionally the war leader but lacked judicial powers. The subordinate lineages of a district stood in the relation of jefekyr, "children," to the dominant lineage. In Goodenough's descentist interpretation, the social composition of a district is revealed as a group of lineages which are in theory, at least, the patrilineal descendants of the chiefly lineage, which is in the position of founding "fa16 Marshall (1981) has called attention to the importance of "sibling set marriages" in Truk and Greater Trukese society: "Sibling set marriage takes the form of 'sibling' exchange in which a set of natural or lineage siblings marries a set of natural or lineage siblings of a different clan" (1981:213). In Romonum, Truk, two-thirds of all marriages by natural siblings, and 90% of all marriages by classificatory siblings in the same generation of the same matrilineage take this form. Marshall, however, does not distinguish between same-sex and cross-sex sibling set marriages.

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Goodenough goes on to say that "Truk presents the only known instance to the writer wherein one rule of descent is used to affiliate individuals while the opposite rule of descent is used to affiliate the resulting kin groups with a larger kin group" (1951:138). An alternative, alliance interpretation, consistent with the definition of the jefekyr relation and with our interpretation of the "big father" relation between sister's husband and wife's brother in the Marshalls, is that the subordinate lineages of a district originally stood in the relation of wife-giver to the dominant lineage. It was marriage alliance rather than mixed-descent metaphors that cemented interlineage relations in Trukese districts. We note with interest that Liep (1991), in his devolutionary model of Austronesian society in the Massim region of Melanesia, describes the relation between chiefly wife-taking and commoner wife-giving lines as one between "father" and "child." We note further that Trobriand "cluster" chiefs received tribute (urigubu) in their capacity as wife-takers to subordinate subclans in their district (Powell 1969b). In many parts of the Truk Lagoon other than Romonum Island there were preferences for first- and second-cousin marriages motivated by land transfers, in particular by the return of land given as gifts by fathers to their children (Caughey 1977; Parker 1985). There was also a residue of primogeniture, expressed in an unusual variant of the MicroPolynesian pattern of dual chieftainship. In a forthcoming book Goodenough distinguishes between "symbolic" and "executive" chiefship. The symbolic head of a lineage or district, referred to as "chief of food" (somwoonun mwenge) succeeded by a rule of primogeniture as the senior male in the senior female line of a lineage. He had the right to ceremonial presentation of firstfruits. The executive head, however, referred to as "chief of talk" (somwoonun kkapas) was the oldest competent male of a lineage. In the ideal scheme of things the two were expected to

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cooperate in managing district affairs with the symbolic head taking over both aspects of chiefly office upon reaching full maturity. Western Caroline atolls

The conical clan persisted, or reemerged, on a reduced scale in some of the western atolls of the Carolines. The atolls were probably settled from Truk (Alkire 1977) and were all part of the Trukic dialect chain (Quackenbush 1968). According to local tradition, the low islands were settled from the east, from Truk. The Trukese in turn look eastward to Ponape and Kusaie, from whence the first woman arrived pregnant, sailing on a coconut frond. Linguistic evidence seems in this case to support tradition, pointing to an original settlement a few thousand years ago on Kusaie or the nearby Marshalls and spreading out from there (Gladwin 1970:4). With small populations of from less than 100 to about 800, and extreme vulnerability to the devastating effects of tropical storms, the islanders ensured their survival by active participation in a voyaging and trading network capable of redistributing people as well as goods (Alkire 1965). Some of the more centrally located, economically and politically dominant atolls in the network were socially stratified. On Lamotrek Atoll, which once received tribute from neighboring islands, Lineages are status ranked within subclans just as the latter are ranked within clans. The status of a particular lineage when compared to another of the same subclan depends on the seniority of the women from whom they trace their descent. A lineage is always traced from a known ancestor and her status when compared to any siblings of the same generation is known so that respective lineages which originated with these women are ranked as they themselves are (Alkire 1965:41-4). The kinship and political organization . . . emphasizes a system of status ranked chiefly and nonchiefly matriclans which can be divided into component subclans, lineages and descent lines, each with particular kinds of rights to land and titles (Alkire 1965:73). Rank is a "master value" of Ifaluk society, with individuals, matrilineages, and matriclans all ordered by a rule of primogeniture. To Burrows and Spiro (1957:183) it "seems probable that the chieftainship

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complex of Polynesia and that exemplified by Ifaluk are historically related." With a few notable exceptions Carolinean marriage systems have not been described in any detail. In the atolls of Greater Trukese Society (the Mortlocks, Namoluk, Losap, Nama, and the Hall Islands), there is a marked preference for bilateral cross-cousin marriage (Marshall 1981). In Lamotrek members of chiefly clans intermarried exclusively and established interdistrict alliances. Isolated atolls

On the isolated atolls of Nuclear Micronesia the conical clan disappeared or persisted only as a historical remnant, and matrilineal descent gave way to double or patrilineal descent. On Pingelap in the eastern Carolines, which was probably settled from Pohnpei, unsegmented matrilineal clans regulated marriage, but the ownership and inheritance of land was in the hands of patrilocal extended families (Damas 1979, 1981). Chiefly titles, of which there were 15, were held by patrilineages and transmitted by primogeniture. The paramount chief was called the Nahnmwarki, as in Pohnpei, but there was no system of competitive feasting or any analogous practice that influenced accession to titles. Damas assumes with Murdock (1949) that matrilineal institutions are inherently unstable and that double descent represents a "half-way house" in the transition to patriliny. Because of their isolation, matriliny in Pingelap, as well as Mokil and Ngatik, was left with few functions other than exogamy: I would argue that the comparative vitality of matrilineal emphasis in the Yap and Truk districts is closely related to regular reinforcement of those ties through the process of clientship, trade relations, and (in the case of the Yap regions) a system of tribute which operated largely within a matrilineal context. By contrast, relative isolation of the atolls and islands in the eastern Carolines appears to have promoted conditions which served to weaken matriliny (Damas 1979:192). There are two outliers in the Marshall Islands - Eniwetok and Ujelang. The islets of each atoll were divided into patrilineal moieties headed by chiefs, iroij, who, however, had few real powers. Land was inherited bilaterally with some patrilineal emphasis. With small populations of less than 200 and virtual isolation, there was little scope for retention of the stratified system found elsewhere in the Marshalls: Chieftainship on Enewetok Atoll was formalized and support-

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ed by traditional sanctions. However the institution was not as highly formalized or structured nor as feudalistic as in the rest of the Marshalls. This is probably due to the small population, the closely interrelated kinship ties of all the people and the absolute localization of the political authority of the two chieftainships to a single atoll. The operational situation is similar to that described by Murdock for Truk: "Local and lineage chiefs exercise considerable authority . . . The influence they exercise is in essence that of oldest brother over his younger siblings. Their status is [unofficial and personal and does not have the effect for example of elevating their families above the general run of the population [Murdock 1965:15]" (Tobin 1967:130). Moieties were characterized by economic cooperation and competition. The chiefs of the Eniwetok moieties at the time of Tobin's visit were cross-cousins, suggesting perhaps a general pattern of marriage alliance. Network determinants of social stratification

Social stratification in Nuclear Micronesia was clearly tied to network variables as well as to demographic and economic conditions. The most stratified societies were found in the Marshall Islands in eastern Micronesia, the presumed locus of PNM society. Marshallese chiefdoms based on the conical clan and asymmetric marriage alliance were genetically related to similarly structured chiefdoms in Melanesia and western Polynesia. Although the Marshall Islands were not part of a long-distance prestige-good system like that in Tonga, chiefly power depended on the control of inter-island networks. Chiefs exercised control through a feudal system of land tenure, strategic marriage alliances, and the delegation of authority on outlying islands to junior kinsmen. As a result, much of the economic exchange in the islands took the form of tribute payments rather than trade: A disadvantageous influence on the development of indigenous trade, despite the seamanship of the natives, has probably been the circumstance that the nobles generally possessed landed estates in the north and in the south and everywhere merely collected tribute from their subjects who were given no opportunity to conduct trade independently (Kramer and Nevermann 1938:228). Chiefs also exerted control over the means of communication, includ-

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ing the navigators themselves. Like their Tongan counterparts, toutai vaka^ who were members of the matapule class, Marshallese navigators, ri medo ("people of the sea") were members of a privileged class of "distinguished persons," leatoktok, who received hereditary fiefs from chiefs: Distinguished persons (leadakdak) [leatoktok] are elevated kadjur [kajur] who have received fiefs from irodj [iroij] or burak [bwirak]. These fiefs are hereditary and they return to the irodj or burak only if the leadakdak is killed or punished for an offense by having his land and house taken from him. The leadakdak have usually acquired their position by virtue of their mental ability and prominence as navigators, . . . astronomers, sorcerers . . . , ceremonial experts, distributors of food . . . , or simply as learned and wise men . . . This knowledge is usually inherited. Lagediack, for example, who gave Kotzebue information on navigation, was in all probability a leadakdak (Kramer and Nevermann 1938:282). Finally, paramount chiefs chose as their headquarters centrally located islands - Namu in the Ralik chain and Aur in the Ratak chain - from which they could have some control over communication between the economically interdependent northern and southern islands. (See Chapter 6.) The least stratified societies were the isolated atolls - Pingelap, Mokil and Ngatik in the eastern Carolines, and Eniwetok and Ujelang in the Marshalls - and the atolls of the southern Gilberts. The isolated atolls of Nuclear Micronesia had the sort of "interlocking" social structures, including moieties and double-descent systems, which Sahlins (1958) described for the isolated atolls of Polynesia. The poorer, unstratified southern Gilberts were part of a voyaging network but one which lacked an economic basis for trade. They contrast with the richer, stratified northern Gilberts - the mini-inter-island empire of ButaritariMakin, with the moderately stratified Caroline atolls, which were joined in networks based on vital trade and tribute relations, and with the highly stratified Marshallese atoll networks. Significantly, inter-island travel in the Gilberts was "extremely limited in comparison with either the Marshalls or the central Carolines" (Alkire 1977:79). The isolated high islands of Pohnpei and Kosrae retained the conical clan system of PNM society but were in the process of unraveling. Their fate appears similar to that described by Friedman (1981) for the isolated high islands in eastern Polynesia. The loss of monopolistic control of an overseas exchange system led to increased competition, agricultural intensification, competitive feasting, and endemic warfare, resulting in

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frequent redistribution of titles (but evidently not of land) and general political instability. Asymmetric marriage alliance gave way to symmetric alliance with a strong tendency toward endogamy, expressed in Pohnpei in the pairing of matrilineal clans and subclans. The Nahnmwarki system may in fact have been one of several different competing and less centralized polities in Pohnpei (Peterson 1990). Alkire (1977) attributes the low level of stratification in Truk, in comparison to the high islands of Pohnpei and Kosrae and the surrounding atolls of the central Carolines, to network and economic variables: the dispersed nature of the islands of Truk made it difficult under aboriginal conditions for one district and kin group to extend anything approaching permanent control over an area larger than a single district, and there seem to have been fewer environmental imperatives to form permanent political unions in Truk than in the more exposed and typhoon-prone atolls of the central Carolines. At the time of European contact each district of Truk was primarily an autonomous unit balanced in a state of precarious peace, engaged in periodic feuding and warfare, and maneuvering in shifting alliances with surrounding districts (Alkire 1977:59). Caughey (1977:140) has also commented on the relative isolation of Truk: "Unlike the inhabitants of the outlying low islands, the people of Truk rarely made long journeys on the open sea, and the art of navigation was not highly developed." The semicomplex marriage system of Romonum Island in Truk resembles the systems of dispersed alliance that Bowden (1983) has described for certain societies in New Guinea that have Omaha-type kinship structures and that McKinley (1971a, b) has described for societies with Crow and Omaha types in general. In these systems, nonrepeated marriage alliance is carried on by other means. In McKinley's model (1971b:411-12), these systems are found in tribes and low-level chiefdoms characterized by the following structural features: 1. Exogamous unilineal descent groups. 2. Dispersed, intermittent marriage alliance. 3. Maintenance of alliances between groups once linked by marriage. This is accomplished by periodic economic exchanges, ritual services, or a reliance on ties of complementary filiation. 4. "Tribal completeness," that is, political organization is at the level of an independent multi-clan tribe, district, or region. The third feature is linguistically expressed by the intergenerational skewing in Crow/Omaha kinship terminologies that is said to immortal-

162

Island networks

ize or "freeze" the time dimension of a single nonrepeated marriage alliance. The reflectionist interpretation of Trukese kinship terminology given by Murdock and Goodenough, and the jefekyr relation as described by Goodenough, would be prime examples of this feature.

Search trees and the organization of genealogical knowledge Kirchhoff, in his discovery of the conical clan, emphasized the importance of keeping genealogies for purposes of establishing rank. In a recent essay on Polynesian chiefdoms, Marcus (1989) aptly characterizes a chiefly genealogy as a "cognitive aid to rule" and as a "sort of systems model of a realm." By way of example he cites the phenomenal genealogical knowledge of Queen Salote of Tonga: Exclusive knowledge, and particularly special genealogical knowledge, has always been a source of chiefly power in Polynesia . . . Although genealogical knowledge has been quite variably preserved among the contemporary population, it was cultivated and comprehensively developed, in line with the dynasty's version of history, by Queen Salote (Tupou III) during her long reign (1918-1965). She selected particular nobles for training in traditions and genealogy. Placing all persons and groups in a grid of historic chiefly genealogies, as well as monitoring the assignment of clergy in Tongan churches to congregations throughout the Kingdom, enabled Salote to identify all of her subjects (a population of about 40,000 by the middle of her reign). She was in fact famous for this capacity and astounded her subjects by her intimate personal knowledge of them (Marcus 1989:201). We do not know how Queen Salote accomplished her feats of memory, but it is quite likely that she used efficient searchlike procedures for organizing and retrieving genealogical information. Suggestive evidence on this score comes from Robin Fox's (1978) account of kinship on Tory Island, a cognatic society off the coast of northwestern Ireland. Like many Polynesians, Tory Islanders have a keen interest in genealogy, motivated by problems of inheritance and land tenure rather than succession to titles, but serving similar functions in the preservation of customs and historical traditions. They also have professional genealogists (sloinnteoiri) whose virtuosity would rival Queen Salote's. These individuals, as Fox discovered, have their own way of doing things. His attempts to use the standard genealogical method in which one works

Search trees: II

163

up from an ego to ramifying ancestors led only to confusion and forgetting of exact links and persons. An islander therefore suggested that Fox take down the genealogies in the manner in which they are recited: one takes an "old one" (seanduine) or "ancestor" (sinsear) and traces out all his descendants (clann). Technically speaking, one reconstructs a stock. There were two common methods of doing this. In the first method, an original ancestor would be chosen and the names of his children given. "Then, one of the children would be chosen and his children named, and so on until all the present descendants were reached, or at least until that line became exhausted. The narrator would then go back to one of the other children of the original ancestor and repeat the process" (Fox 1978:32). In the second method, one would take "all the children of the ancestor, then all their children, all their children's children, and so on. If the group was small and had relatively little segmentation, this method was favored. It treated the whole group, in effect, in the same way as a 'segment' of a much larger group would be treated" (1978:32). In the case of an extremely large group of great generational depth the recital would proceed segment by segment. "This is very important because it illustrates the islanders' own conception of the nature of genealogy and the structure of these groups" (1978:31-3). Fox illustrates these methods with the graph in Fig. 5.6. Assuming that the genealogy consists only of the ancestor Aa and his descendants, the recital in the first method would go in the following order: A,, 1,2,3 1,4,5 2, 6, 7, 8 3, 9, 10 9,11,12 10, 13, 14 The recital in the second method would go Ai

1,2,3 9, 10, 6, 7, 4 11,12,13,14,8,5 Assuming that this was only a segment of a larger genealogy headed by A2, the recital would first name the descendants of A1 and then those of A3The second method, for shallower genealogies, uses a breadth-first

164

Island networks

-etc

Figure 5.6. The "ideal scheme for reciting genealogies" among the Tory Islanders (from R. Fox 1978). search, while the first method, for longer genealogies, combines breadth-first and depth-first procedures in a manner that intuitively seems more efficient. A strictly breadth-first search of a tree of any height, that is, a genealogy of any depth, would require much more backtracking than a depth-first search. We suppose that Queen Salote and other Polynesian genealogists used a combination of breadth-first and depth-first search procedures. In these kinship structures we have an interesting case in which the methods of real and artificial intelligence coincide.

Centrality

Mon centre cede, ma droite recule, situation excellente. J'attaque. Marshall Ferdinand Foch, "Message to Joffre"

One island may enjoy a structural advantage over other islands in a network by virtue of its more central location. Several Oceanists have relied on the concept of median centrality ("short-path connectivity") in their analyses of trade networks (Brookfield and Hart 1971; Irwin 1974; Hunt 1988; Kirch 1988b). However, there are many different, empirically applicable ways of defining centrality in a graph (Buckley andHarary 1990). In the following presentation we use the concepts of degree centrality, median centrality, and betweenness centrality to account for the emergence of trade and political centers in the Lau Islands, Fiji. Our analysis complements Thompson's (1949) ecological model of trade in southern Lau, offers a network alternative to Sahlins's (1962) theory of relative agricultural fertility as the primary determinant of trade centers in Fiji, and develops Reid's (1977) network explanation for the rise of Lakemba as a political power in eastern Fiji. We use the prototypic graph theoretic definition of centrality, which is based on the eccentricity rather than the distance sum of a node, to account for the location of political capitals and mythological centers in the Ralik and Ratak chains of the Marshall Islands. We introduce the concept of betweenness in a rooted graph to evaluate Harris's (1979) hypothesis concerning the relation between trade and subsistence practices in the western islands of Torres Strait. We then go on to consider all the islands of Torres Strait - western, central, and eastern - as nodes in a single network joined to communities on the coasts of New Guinea and Australia. It turns out that centrality, trading success, and social stratification are associated in the islands of Torres Strait much as they are in certain other Melanesian networks. We conclude by describing an application of median centrali165

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Island networks

ty to a network in a very different area and historical period, the Archaic Aegean (Davis 1982), which has interesting parallels with Melanesia. Our presentation includes matrix methods for studying centrality in graphs and weighted graphs and a characterization of networks whose graphs are self-centered.

Southern Lau, Fiji: "A natural trade area" In an early contribution to cultural ecology, Laura Thompson (1949) characterized the islands of southern Lau in eastern Fiji as a "naturally balanced self-sufficient trade area." (See Map 3.2.) The system as she depicted it consists of six islands, divided into two types: volcanic islands with good garden land and abundant crops, especially yams (Dioscorea) but scarce timber, and limestone islands with poor garden land but abundant timber resources, including the hardwoods mbau (Pittosporum brackenridgei), makota (Dysoxylum richii), and - most valuable of all for its use in canoe manufacture - vesi (the "greenheart" of India, Intsia bijuga). Three of the islands are limestone - Fulanga, Ongea, Namuka - and three are partly or wholly volcanic - Mothe, Komo, Kambara. Trade was based on a complementary exchange of food from the volcanic islands for craft goods - canoes, wooden bowls, mats, and barkcloth - from the limestone islands.1 Trade was socially regulated by a form of competitive ceremonial exchange, solevu, which joined villages on different islands, stimulated production, and expanded distribution. Solevu exchange also joined villages through the exchange of feasts, entertainments, and athletic contests, and it provided a context for the exchange of gifts and barter between individuals.2 The implicit structural model of this sytem is the complete bigraph K33 (Fig. 2.10), in which the two node sets V1 and V2 represent the limestone and volcanic islands and the edges represent the trading relations between them. It is clear, however, from Thompson's (1940) earlier monograph on southern Lau, that the number of trading partners was greater, that trade was not restricted to pairs of volcanic and limestone islands, and that the network was centralized. Thompson distinguishes three rather than two types of islands: volcanic islands - Mothe and Komo; limestone islands - Wangava, Fulanga, Namuka, Ongea, and Vatoa; and composite volcanic and limestone islands - Lakemba, Kambara, Oneata, and Ono. The composite islands have both garden 1 See Chapter 3 for population figures for these islands. 2 Sahlins (1962) distinguishes several varieties of inter-island exchange in Lau, including solevu, kerekere (a solicited gift to be reciprocated at a later date), and tauvu (licensed appropriation of goods between kin groups).

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167

land and timber resources in varying degree, as does the limestone island of Namuka. The most important industry in southern Lau was canoe making, and the leading manufacturers were Kambara and, secondarily, Fulanga and Ongea, the only islands (aside from uninhabited Wangava) with abundant growths of greenheart timber. Canoes made in southern Lau were traded throughout Fiji and in Tonga. Thompson (1940:210-11) gives a detailed account of inter-island trade, the highlights of which may be briefly summarized. The main exports of Kambara, besides oceangoing canoes, were wooden bowls, mats, and Tongan barkcloth, items that were distributed all over Lau and elsewhere in Fiji. Kambara also exported timber for house posts and digging sticks to the volcanic islands, and coconut oil and vetau bark for hair dye. Kambara imported tobacco, kava, and food - including yams, taro, and breadfruit - from Mothe, Komo, and Namuka, basaltic stones from Mothe and Komo, and barkcloth from Namuka. Namuka's primary export was Tongan barkcloth, most of which was traded to Lakemba in return for yams and taro. Some barkcloth was also traded to Kambara for mats, kava, and food bowls. Smaller amounts were traded to all the islands in southern Lau except Ono, which produced its own. Slit gongs made in Namuka were distributed throughout Lau. Namuka exported manioc, breadfruit, yams, and bananas to Fulanga, Ongea, and Kambara in exchange for mats and bowls. Mothe exported manioc, yams, and bananas, tobacco, kava, sikethi for dye, volcanic stones, and reeds to Fulanga, Ongea, and Kambara. Mothe also exported large amounts of Fijian barkcloth to Lakemba and smaller amounts to all the neighboring islands. Mothe imported canoes and timber for house posts, adz handles, wooden bowls, digging sticks, and tapa boards from the southern islands. Some of these manufactured articles were traded on to Lakemba. Mothe imported banana shoots and taro plants from Lakemba, paper-mulberry plants from Namuka, and sandalwood seed from Oneata and Lakemba. Fulanga and Ongea exported canoes, headrests, mats and mat sails, timber, wooden bowls, and digging sticks to Mothe and Namuka in return for food. Komo exported food, basaltic stones, vunga leaves for hair dye, and red ocher for barkcloth stencils to neighboring islands and imported canoes and timber, among other items. The general pattern is clear: all islands traded with each other directly or indirectly - "down the line" - and individual islands were distinguished by their economic specializations and by the relative scope of their trading activity.

168

Island networks

Two different hypotheses could be used to predict the site of a trade center in southern Lau, one based on an island's agricultural potential and the other on its network location. The first hypothesis is that of Sahlins (1962) and applies to Fiji in general. It holds that "Interisland trade receives critical impetus from and is decisively structured by the differential distribution of agricultural potential. The key centers of trade have been the places of greatest and least fertility, provided that the latter have enjoyed some local resource that could be turned to account by trade" (Sahlins 1962:419). As illustrations Sahlins cites Moala, an island of greatest fertility, and Kambara, an island of least fertility.3 Although Moala may have been a trade center in eastern Fiji, Kambara was not the key trade center in southern Lau. The second hypothesis predicts that the trade center will be a centrally located island. Three different graph theoretic measures of centrality, from L. C. Freeman (1979), are applicable to the southern Lau network: "degree centrality," "closeness (median) centrality," and "betweenness centrality." Degree centrality refers to the number of edges incident with a node. The degree center of a graph is the set of nodes of greatest degree. In the present context, degree centrality refers to the number of direct trading partners of an island. We now introduce the adjacency function of two nodes by defining a v i b vk) = 1 if and only if vt and v^ are adjacent and a(vh v^) = 0 otherwise. Then the measure CD(vk) of degree centrality is defined by a(vhvk)

(6.1)

1=1

The maximum possible size of CD(vk) is p - 1. Hence the relative degree centrality of a node, irrespective of graph size, is the ratio

This measure is illustrated in Fig. 6.1a. Recall that the distance between two nodes v{ and Vj in a graph, denoted dfy is the length of any shortest path (geodesic) that joins them. In a connected graph G, the distance between any two nodes is a positive integer. For each node w, the distance sum or status of w, written s(u), is the sum of the distances between u and all other nodes (Harary 1959). Hocart (1929:5) describes Moala, Totoya, and Matuku as the "most fruitful" of all islands in Lau. Derrick (1951:330) says that the "soil of Moala is the most fertile in Lau."

Centrality

169

(c)

(b)

(a)

CD(vk) 4

1

4

1

2

.33

2

3

.75

5

.8

.5

.08

3

3

.75

5

.8

.5

.08

4

2

.5

6

.67

0

0

6

.67

0

0

1

5

2

.5

.62

.29

Figure 6.1. Illustrations of degree, closeness, and betweenness centrality in a graph.

Thus 7=1

The median of G, also called its distance center', is the set of all nodes u of G such that s(w) is minimum (Buckley and Harary 1990). Closeness (median) centrality refers to the sum of the lengths of the shortest paths (geodesies) between one node and all other nodes of a connected graph. In our application, closeness centrality refers to the nearness of an island to all other islands each of which is a potential source of trade goods. As Freeman points out, the measure of closeness is actually a measure of node decentrality or inverse centrality, since it grows as nodes become farther apart. As usual in graph theory (Harary 1969), we denote the distance function by d(vh vk) - the number of edges in a geodesic joining Vj and vk. Then the closeness centrality of a node vk is defined by the equation

170

Island networks (vhvk)

(6.3)

The minimum possible sum of distances from a node to all other nodes in any connected graph is p — 1. The relative closeness centrality of a node vk is defined by v» Vk)

p-\

(6.4)

p-\

Freeman interprets this formula in the following way, writing pk for the &'th node (point): Since the sum in this expression is based on the distances from pk to the n-\ other points CQ (p^) may be understood as the inverse of the average distance between pk and the other points. But since n - 1 is also the minimum sum of distances for a point that is adjacent to all other points - CQ (pk) m a v also be interpreted as the inverse of the ratio by which pk exceeds its minimum distance. Thus, CQ (pk) is a direct measure of distance based point centrality. It takes a value of unity when pk is maximally close to all other points and shrinks as the average distance between pk and other points grows (1979:226). This measure is illustrated in Fig. 6.1b. Betweenness centrality refers to the frequency of occurrence of each node on the geodesies between all pairs of nodes in a graph. The betweenness center is the set of nodes that fall on the highest proportion of such paths. Betweenness centrality indicates the importance of an island as an intermediary in trade between other pairs of islands. Let gjj be the number of v{ - Vj geodesies, and let g;;(t>&) be the number of these geodesies containing vk. Then the betweenness value of vk with respect to the node pair vh Vj is the ratio bij(vk) = —

,

(6.5)

which is of course the probability that a randomly selected /-/ geodesic contains node k. In terms of these probabilities the partial betweenness value of vk (independent of any one pair of nodes) is defined by the formula

Centrality

171

CB{vk) = X I bif(vk).

(6.6)

Whenever vk lies on every /'-; geodesic, the node pair /, / contributes 1 to the sum CB(vk). The particular case that there is a unique path (and hence geodesic) joining each pair of distinct nodes occurs only when G is a tree. When there are alternative geodesies, CB(vk) grows in proportion to the frequency of occurrence of vk among these alternatives. Among all p-node graphs, the maximum possible betweenness value of a n o d e , CB(vk),

is (p - l)(p - 2)12 = (p2 - 3p + 2)12 and is attained by

the central node of a star. The relative betweenness CB(vk) is the ratio of CB(vk) to this maximum expression, so that 2CB(vk) pi3P + 2'

{6J)

This measure is illustrated in Fig. 6.1c. For each of these three centrality concepts Freeman defines an index of graph centrality. Intuitively, a star Kln is maximally centralized, and a cycle Cn is minimally centralized, while most other graphs fall somewhere in between. Thus some trade networks are more centralized than others. Freeman's indexes of graph centrality are based on the "degree to which the centrality of the most central point exceeds the centrality of all other points." The indexes are expressed as a "ratio of that excess to its maximum possible value for a graph containing an observed number of points" (1979:227). Where v* is any node in the graph having the largest value, the given index measures of the degree centrality CD(G), closeness centrality CC(G), and betweenness centrality CB(G) of a graph are:

CD(G)

- »

p 2 3 p

+2

(6.8)

Cc(G) =

CB(G)= ^

—

(6.10)

These three indexes are illustrated in Fig. 6.1. The graph of the southern Lau network is implicit in Thompson's (1940:212) list of trade routes connecting all the islands from Lakemba to Fulanga and Ongea. The main routes are:

172

Island networks Lakemba 1 2 Aiwa

3 Oneata

6 Mothe

Wangava 7

Kambara 9

Figure 6.2. The graph of the southern Lau trade network.

1. 2. 3. 4.

Fulanga and Ongea to Mothe, directly or via Namuka. Fulanga to Namuka, directly or via Mothe. Kambara to Mothe via Wangava and Namuka. Kambara to Lakemba via Wangava, Komo, Olurua, and Aiwa, returning via the same route or via Aiwa, Oneata, Komo, and Wangava. 5. Mothe to Lakemba via Oneata and Aiwa.

Thompson's description of inter-island trade also implies direct links between Komo and Mothe and Komo and Namuka. The graph of this network is shown in Fig. 6.2. Thompson identifies two trade centers in southern Lau - Mothe and Namuka. The former appears to have dominated as a trade center and entrepot; the latter had apparently lost its voyaging traditions and had

Centrality

173

Table 6.1. Relative centrality of islands in the southern Lau trade network Island 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.

Lakemba Aiwa Oneata Olorua Komo Mothe Wangava Namuka Kambara Fulanga Ongea

CD(vk)

Ch(vk)

1 3 3 2 5 6 4 5 1 2 2

0.1 0.3 0.3 0.2 0.5 0.6 0.4 0.5 0.1 0.2 0.2

Cc(^)"1 32 23 18 21 16 15 19 18 28 23 23

C&(vk)

CB(vk)

Ci(vk)

0.313 0.435 0.555 0.476 0.625 0.667 0.526 0.555 0.357 0.435 0.435

0.000 9.500 11.667 2.333 10.167 18.833 10.000 2.500 0.000 0.000 0.000

0.000 0.211 0.259 0.052 0.226 0.419 0.222 0.055 0.000 0.000 0.000

Note: CD = 0.355 C c = 0.413 CB = 0.316

to rely on others to ship its exports and imports (even though Tongans built canoes there - Derrick 1951). Mothe clearly seems to have overshadowed Kambara in the scope of its trading activity: Mothe, situated at the junction of trade routes from the north, south, and west is a trade center. The natives of Mothe trade with all their neighbors. They act as middlemen in exchanging goods especially from southern Lau northward as far as Lakemba. Namuka is also a trade center. Both Namuka and Mothe are visited by trading canoes from all the southern islands. The natives of Namuka own no large sailing canoes and all the exports from Namuka are carried by canoes from other islands. The natives of Kambara make trading journeys to Namuka, Komo, Mothe and occasionally to Lakemba (Thompson 1940:211-12). The figures in Table 6.1 reveal the relative centrality of all islands in the southern Lau trade network. Mothe has the most trading partners, it is closest to all other islands, and it lies on the greatest number of (direct) trade routes joining all pairs of islands; it therefore ranks first on degree, closeness, and betweenness centrality. The network itself is moderately centralized on all three measures, CD, C c , and CB(G). Although Mothe ranks only slightly higher than Komo and Namuka on the first two measures, it ranks significantly higher on the betweenness measure than any other island, indicating its major importance as a trading intermediary. Namuka, a "relatively fertile limestone island" (Thompson

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Island networks

1940:25) and an importer and exporter of both craft goods and food, ranks second and third on degree and closeness but next to lowest on betweenness centrality. Fulanga, a "famine island," and Ongea are islands of least fertility whose inhabitants were food importers and canoe exporters. Fulangans and Ongeans were active traders, known for their navigational skills, but their islands were too peripherally located to become trade centers. Kambara, a composite limestone and volcanic island, had both timber resources and some garden land, including land on the adjacent uninhabited island of Wangava. Although a major canoe exporter, Kambara was one of the most peripherally located islands in southern Lau and did not become the principal trade center of the region. Mothe, a volcanic island with rich garden land, like Moala, was a major food exporter and craft-goods importer. Here it would appear that the fertility and network hypotheses converge, but an examination of the larger inter-regional network suggests that location, not relative fertility, was the principal determinant of trade and correlatively of political centers in Lau.

Power centers in the Greater Lauan trade network In accounting for the rise of Lakemba as a political power in eastern Fiji, Reid (1977) emphasizes that it was "neither the most fertile of the Lau Islands nor the best endowed with natural resources." It did, however, have an advantageous location: Lakeba lay at a cross-roads of the sea. It was a natural meeting place for the sea-going canoes of previous centuries island-hopping from southern Lau, from the northern states of Vuna and Somosomo on Taveuni and from the western chiefdoms via Gau and Moala. Here they foregathered with canoes from Tonga Matanisiga (i.e. towards the Sunrise, the east). To an astute leader each new arrival represented an opportunity for increasing influence and enhancing authority - through hospitality, the exchange of gifts, the conclusion of a marriage alliance or the promise of support in diplomacy and war. It was these opportunities and the use made of them by the chiefs of Lakeba which, by the early 19th century, had created a maritime state of considerable significance in the South Pacific (Reid 1977:3). The idea of a natural meeting place as a "crossroads of the sea" - an island on the way to other destinations - is captured by the concept of betweenness centrality. In order to assess the advantageous location of

Centrality

175

Lakemba, we will place it in the context of what might be called the Greater Lauan trade network, which once joined the Lau Islands with the Moala and Lomai Viti groups, western Fiji, and Tonga. (See Map 3.2.) We offer the following somewhat speculative reconstruction of this network. Lakemba traded with the islands to the south, as previously described, and north, and also with Moala and Totoya to the west, giving barkcloth and coconut oil for mats (Hocart 1929). Moala traded with the islands of Lomai Viti to the north and southern Lau to the east and had connections extending as far as Mbau and Rewa in western Fiji. Moala imported tapa, yellow ocher, and mats from Matuku; coconut oil, sandalwood, tapa, and mats from Totoya; canoes, sennit, and tapa from Kambara, Fulanga, Mothe, and other islands in Lau. Through intermediaries Moala obtained famous fine mats, tabu kaisi, from Ono-iLau and sail mats and fine tapa from Mbau. Scented cinnamon woods (macou) and mats were imported from Gau, Koro, and Nairai in the Lomai Viti group (Sahlins 1962:422-3). Tongans traded with Lauans, providing mats and whales' teeth in exchange for canoes, pots, sandalwood, and barkcloth. They sailed as far as Taveuni in northern Lau to obtain the red feathers that they later traded to Samoa. Tongans also sailed via Lau to western Fiji. Referring to Cook's list of islands known to Tongans, Sharp (1956:152) notes that the "nearest islands to Tonga on the west are Tuvana-i-Tholo, Tuvana-iRa, and Ongea. Other islands on the way to Mbau on the eastern side of Viti [Fiji] are Mothe, Wanggava and Moala." Mothe was a "point of departure of sailing canoes for Tonga" (Thompson 1940:7). In most general terms, the overseas chiefdoms of Lakemba, Cakaundrove, Mbau, and Tonga were part of a single trade network based in large part on Lakemba's export of canoes produced in southern Lau (Young 1982). Fig. 6.3 is a graph of the Greater Lauan trade network. The dotted lines represent geographically longer distances than can be shown in the diagram. Figures for the relative betweenness of each island are given in Table 6.2. Certain uninhabited islands were used as voyaging stopovers. Lakemba, in its rise to political preeminence in Lau, had to contend with two rival centers of power. Kambara, as noted in Chapter 3, was the head of a maritime empire in the south. According to historical traditions, Kambara "was formerly a rich and powerful chiefdom . . . that. . . held the highest rank in Lau before the rise of Lakemba" (Thompson 1940:38).4 In the east, Lakemba was opposed by Moala, which once claimed two of the islands in the Lakemba state - Naiau, whose leading chiefs were mainly of Moalan descent, and Thithia. At the time of contact, 4 See also Hocart (1929).

176

Island networks 33 Taveuni

Qamea, Laucala

34 32 aitaumba 29 Vanuambalavu (Exploring Isles)

27 Tuvutha 23 ° Viti Levu

22 ° <\ 19 Gau Bau

10 H

Y. l l \ O n g e a

Matuku 17

12

Tuvana 14

-'o Tongatapu 15

Figure 6.3. A graph of the Greater Lauan trade network.

the network of reefs and islands between the 178th and 180th parallels had come under three centres of influence - Taveuni, Lakeba and Moala. Moala claimed an ancient suzerainty over the islands to her northeast - Nayau [Naiau], Cicia [Thithia] and Mago. These formed a buffer between the archipelago, visible from the heights of Taveuni and under its domination, and the Lakeba State which . . . had expanded to the south. The coming to power of the Vuanirewa brought Nayau within the ambit of Lakeba and it is possible to detect from now on a continuous thread of policy aimed at bringing Moala and its

Centrality

177

Table 6.2. Relative centrality of islands in the Greater Lauan network Island 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34.

Lakemba Aiwa" Oneata Olorua Komo Mothe Wangava Namuka Kambara Fulanga Ongea Vatoa Ono Tuvana Tongatapu Totoya Matuku Moala Gau Naairai Koro Bau Viti Levu Vanua Vatu" Naiau Thithia Tuvutha Mango Exploring Isles Kanathea Yathata Naitaumba Qamea, Lancala Taveuni

CB(vk)

Ci(vk)

Rank

250.397 139.097 109.547 8.983 23.633 158.726 100.603 20.841 101.403 24.218 26.218 23.270 1.500 8.564 37.064 25.235 0.000 165.235 120.000 32.000 0.000 32.000 0.000 118.467 216.500 50.000 125.000 84.333 53.167 1.000 40.000 21.000 0.000 0.000

0.474 0.263 0.207 0.017 0.045 0.301 0.191 0.039 0.192 0.046 0.050 0.044 0.003 0.016 0.070 0.048 0.000 0.313 0.227 0.061 0.000 0.061 0.000 0.224 0.410 0.095 0.237 0.160 0.101 0.002 0.076 0.040 0.000 0.000

1 5 9 25 21 4 11 24 10 20 18 22 27 26 16 19

29 = 3 7

17 = 29 = 17 = 29 = 8 2 14 6 12 13 28 15 23

29 = 29 =

Cc(^)"1 112 120 130 138 132 126 135 143 127 153 151 178 207 181 152 125 151 120 144 174 206 174 206 123 126 146 144 164 165 190 189 190 218 218

C6(vk) 0.295 0.275 0.246 0.239 0.250 0.262 0.244 0.231 0.260 0.216 0.219 0.185 0.159 0.182 0.217 0.264 0.219 0.275 0.229 0.190 0.160 0.190 0.160 0.268 0.262 0.226 0.229 0.201 0.200 0.174 0.175 0.174 0.151 0.151

Rank 1

2= 8 10 7

5= 9 11 6 16

14 = 20 25 21 15 4

14 = 2= 12 = 19 = 24 = 19 = 24 = 3

5= 13

12 = 17 18

23 = 22

23 = 26 = 26 =

Note: CB = 0.367 C c = 0.164 a. Uninhabited island.

associated islands, whence the dynasty derived its roots, under Lakeba control (Reid 1977:19-20). Our hypothesis is that Lakemba had an advantage over both Kambara and Moala through its more central location and hence greater access to wealth, influence, and diplomatic and military support in the Greater Lauan trade network. Hocart (1929:4) observed that the Moala group "formed a link between the windward [Lau] islands and 'Lomai Viti' the main Fiji Islands," but as Table 6.2 shows, Lakemba was much more advantageously located with respect to betweenness than Moala

178

Island networks

fl = 0.474 vs 0.313) and all other islands in the network. The second most "between" island, Naiau, was eventually brought under Lakemban control. Lakemba is more than twice as between as Kambara, which ranks a distant eighth among inhabited islands (C# = 0.192). Mothe, the fourth most between island, was not a major power center in the network, but it was a trade center of southern Lau and a "noble" island with kinship connections to Lakemba. Table 6.2 also shows the relative closeness of all islands. Lakemba again ranks first, although all islands are arranged in a finely graded continuum. On this measure the network, in spite of its appearance, is relatively uncentralized, with CC(G) = 0.164 compared to CB(G) = 0.367. The most significant locational advantage of Lakemba was its "crossroads" position, that is, its relative betweenness in the Greater Lauan trade network.

Political and mythological centers in Ralik and Ratak In Chapter 5 we described how Marshallese paramount chiefs controlled their overseas empires by exploiting kinship and marriage ties, by monopolizing the means of communication, and by choosing as their headquarters centrally located islands from which political and military force could be exerted most effectively. By "central" location we mean the "center" of a graph as defined by the "eccentricity" of its nodes. The eccentricity, e(v), of a node v in a connected graph G is the maximum distance d(u, v) for all u. The diameter of a graph G is the maximum eccentricity of a node, that is, the maximum distance between two nodes of G. The radius r(G) is the minimum eccentricity of the nodes. A node v is a central node if e(v) = r(G), and the center of G is the set of all central nodes. Fig. 6.4 shows a graph G with the eccentricity of each node. The diameter of G is 4, the radius is 2, and the center consists of nodes c and f. In a communication network the center has minimum distance to any location, while the median has minimum total distance to all locations. The center and the median of a graph may be disjoint. In the graph of Fig. 6.5, for example, the center is d but the median is f. At various times in their history the Ralik and Ratak chains were each brought under the control of paramount chiefs. Since rebellion was always a threat - Ratak and Ralik were each united in the first part of the nineteenth century only to break up in the second part - we assume that paramount chiefs chose as their capitals islands from which any other island could be reached in the fewest steps (island hops), that is, the cen-

Centrality

179

b (3)

d (4)

c (2)

(4) a

e (3)

8 (3)

/ (2)

Figure 6.4. A graph and the eccentricities of its nodes.

a o-

b -o-

*

7

Figure 6.5. A graph in which the center and the median are disjoint.

ter of the graph of each network. The center of the Ratak graph in Fig. 5.3 consists of three nodes, Aur, Maloelap, and Wotje, with e(v) = 3, and the center of the Ralik graph consists of a single node, Namu, also with e(v) = 3. Notice that the centers of these two graphs are not identical with the medians: the median of the Ratak graph is Wotje, with s(u) = 14, and that of the Ralik graph is Kwajalein, with s(u) = 20. Although the capital of Ratak was Aur (the Russian explorer Kotzebue met the Ratak paramount chief La Mari at Aur), the capital of Ralik is usually designated as Ailinglaplap (Alkire 1977), contrary to our hypothesis. It appears, however, that the real capital was Namu: Legendary sources and Captain Moore's report in 1804 indicate that Namu was not only the traditional home of the Namu people but also was the eonhut or capital of the Ralik

180

Island networks chain. Informants say that whereas the Iroij (meaning Lejolan Kabua, and his father, Jemata Kabua, before him) set up a "royal" household on Bwoj, Ailinglablab, they have always looked to Namu as the central point to be defended. As a mark of this special relationship between the people of Namu and their Iroij they are allowed the special favour of etetal in Bojar (which in free translation means "to walk upright before the highest chiefs") in contrast to the usual custom which dictates that a subject shall crawl before his chief (Pollock 1970:29).

The symbolic importance of Aur and Namu was their identification in Marshallese mythology as the "mother of all clans" in each chain: Legend relates that long ago only seven clans existed in Ralik, all of them originating on the atoll of Namu. There, in the village of Bojar, stands a pillar of basalt which the natives call Luatonmur, "the mother of all clans." The child which was born to this mythical mother is remembered as the first chieftain of Ralik. Annually, the natives of Namu still congregate at the pillar for feasting and dancing in memory of Luatonmur. A similar pillar of basalt, called Liribribju, exists on the atoll of Aur in Ratak. The natives say that these two stones are like sisters, and that from these stones came the people of lajirik clan, who consider themselves better than others because of their sacred origin (Mason 1947:32).

Expeditions in Torres Strait The islands of Torres Strait, which join the southern coast of Papua New Guinea and the northern coast of Australia in stepping-stone sequences of some 160 kilometers or less, are theoretically appealing to biogeographers because they provide an ideal situation in which to study the flow of plant and animal species through a well-defined network (Walker 1972). They are of interest to anthropologists because they provide an unusual situation in which to study the origins of agriculture. Although all the islands were joined in a single trade network based on overlapping social and cultural ties, some were primarily horticultural, like societies in New Guinea, while others were primarily foraging, like societies in Australia. Harris (1979) has proposed that differences in the economies of the western islands in Torres Strait can be accounted for by differences in their network location: islands that emphasized horticulture over foraging occupied "critical" locations in the trade routes leading to Papua

Centrality

181

New Guinea. The ethnography of Torres Strait is sufficiently comprehensive, thanks to the Reports of the Cambridge Expedition (Haddon 1901-35), that it is possible to evaluate Harris's "trade-horticulture hypothesis" and to study the effects of location on trade and social organization in the larger networks of the region.5 The Torres Strait islands (Map 6.1) are divisible into four types, located in four different regions (Harris 1979; Beckett 1987): 1. High, fertile, volcanic islands, in eastern Torres Strait: Murray Islands (Mer, Dauar, Waier), Erub (Darnly Island), and Uga (Stephens Island). 2. High, less fertile islands, part of a former land bridge joining Cape York and the Papuan coast, in western Torres Strait: Muralug and adjacent islands, and Moa, Badu, Mabuiag, Dauan, Nagir, and Yam. 3. Coral islands with poor soil but often rich marine resources, in central Torres Strait: Waraber, Paremar (Coconut Island), Aurid, Masig (Yorke Island), Damut, Tudu (also called Tud and Tutu). 4. Alluvial islands, swampy, with limited garden land, off the Papuan coast: Saibai and Boigu, in the northwest. The eastern islanders, or, more accurately, certain communities in the eastern islands, emphasized gardening - sweet potatoes, yams, and bananas. Some western islanders emphasized gardening and others foraging. The people of Nagir, for example were primarily gardeners, whereas the people of Muralug were seminomadic foragers. The central islanders were seminomadic fishermen and foragers who exchanged seafood - fish, dugong, and turtle - shells, and other trade goods for garden produce. The languages, like the economies, were mixed. The language of the eastern islands, Miriam, is a member of the Trans-Fly family, indicating a Papuan origin, whereas the language of the western and central islands is structurally Aboriginal, with some Melanesian features suggesting a merging of Australian and Papuan populations (Beckett 1987). The most significant cultural, social, and economic connections of the islanders were with Papua New Guinea rather than Australia, although connections with the latter should not be underestimated. Lawrence (1991) is critical of the data in the Cambridge Reports, based as they are on a combination of fieldwork and documentary sources. He objects in principle to the idea of fixed trade routes connecting islands of Torres Strait and argues instead for "fluid links . . . open to manipulation." He nonetheless speaks of "customary exchanges." Any network model will be a simplification of reality, but hopefully an enlightening one. Lawrence approves of Harris's ecological model of exchange, but that model is based precisely on the analysis of specific connections between islands.

182

Island networks

Erub Is.

Murray Ids,
144'

I

Map 6.1. Torres Strait (adapted from Beckett 1987). Haddon (1935) found a social division of the islands into clusters of named, allied groups, within which marriage was frequent and between which it was rare. There were three groups in the western islands: Saibailaig (Saibai, Boigu, Dauan, and Daru); Mululaig (Mabuiag, Badu); and Kauralaig (Moa, Muralug). Haddon gives Kulkalaig as the name for the group in all the central islands but distinguishes as a subgroup Tutulaig (Tutu, Yam). He designates the Murray Islands as Miri-

Centrality

183

am-le (but this term apparently refers to one of four groups of clans on Mer, the largest of the Murray Islands). Social organization in Torres Strait was based on patrilineal exogamous descent groups with sister exchange the preferred mode of marriage.6 With one apparent and notable exception, to be described later, chieftainship was absent. Headmen, informal nonhereditary leaders, are described as polygynists, owners of large canoes and sometimes of good garden land, and most especially as bold and successful warriors. Warfare, in which headhunting figured prominently, was a salient feature of inter-island relations, always a threat even between allies. "All the islanders preserved the skulls of those they had slain and many of the Western Islands and the Yam-Tutu and Nagir men made forays for the avowed purpose of getting heads. The skulls and jaws so obtained were retained as trophies and sometimes traded or given to mourners" (Haddon 1935:348). The western and central islands were joined by bonds of kinship, marriage, common cult membership, and trade. The eastern islands were joined directly to some of the central islands and indirectly to all of the central and western islands by trade relations. The basis of trade for all islands in the Strait was the purchase of canoes from Papua New Guinea. Lacking timber of their own, the islanders were completely dependent on Papuan communities for the canoes they used in fishing, warfare, communication, and trade. The canoes, or rather the canoe hulls, which the islanders themselves fitted with double outriggers, were purchased through intermediaries located on trade routes joining buyer to seller. It is the position of individual islands on the routes in western Torres Strait that forms the basis of Harris's trade-horticulture hypothesis. The trade-horticulture hypothesis

In his reconstruction of subsistence patterns in the mid-nineteenth-century western Torres Strait, Harris treats each of the social divisions reported by Haddon (modified slightly to include the southwestern central island of Nagir with Muralug and Moa and to exclude the northeastern island of Daru from Saibailaig) as an ecologically balanced community consisting of one primarily horticultural island that exchanged or supplied its products to one or two primarily foraging island or islands. The horticultural islands were the small high islands of Nagir, Mabuiag, and 6 Upon encountering this custom Haddon had a shock of recognition: while he was a student at Cambridge, he and his best friend had exchanged sisters as dates for May Week, subsequently marrying them. As noted by Haddon's biographer, "Here also [at Tudu] he first met with 'sister-exchange in the raw, like a case not unknown to us,' which he describes in a letter home" (Quiggin 1942:84).

184

Island networks

Table 6.3. Estimated pre-European (ca. 1840) populations and population densities of the western Torres Strait islands Area

Population

Muralug Other islands of the Muralug group Muralug group Nagir

204.9 88.9 293.8

Moa

170.5 465.3

100 150 250 200 500 950

Muralug-Moa—Nagir community Badu Mabuiag Badu—Mabuiag community Dauan Saibai Saibai-Dauan Boigu Saibai-Dauan-Boigu community Western Torres Strait islands

1.0

104.4 8.3

112.7

670 300 970

Density (km2 0.5 1.7 0.8

200.0 2.9 2.0 6.4

36.1 8.6

33.3

106.4 109.4 85.1 194.5

100 500 600 350 950

772.5

2870

3.7

3.0

4.7 5.5 4.1 4.9

Source: Data from D. R. Harris 1979.

Dauan, each of which is estimated to have had a much denser population than its foraging partners, as shown in Table 637 Since there is no evidence for superior soil and water resources on Nagir, Mabuiag, and Dauan, Harris attributes their emphasis on horticulture and resulting denser populations to their strategic location in the trade network through which the islanders purchased their canoes from communities in coastal Papua: The trade in canoes, which originated at the mouth of the Fly River, passed from Dauan in the northwestern community to Mabuiag in the mid-western community and thence to Badu and Muralug, while another route reached the southwestern community via Tud and Nagir. Thus each of the three small, "horticultural" islands was situated at a critical point in the inter-community trade network where canoes and other goods passed from one community to another. The populations - permanently or temporarily resident - on the three islands are likely to have been disproportionately large in relation to island size - as the demographic data indicate - because they 7 J. Beckett (personal communication) considers the evidence for a Muralug, Nagir, Moa community persuasive but doubts that Mabuiag was primarily horticultural. (See Nietschmann and Nietschmann 1981.)

Centrality

185

supported and were in turn supported by the trade network (Harris 1979:104-5). Harris provides ethnohistorical evidence (Macgillvray 1852) that the people of Nagir specialized in horticulture and the manufacture of trade goods, and he infers such specialization for Mabuiag and Dauan. As evidence for a direct connection between horticulture and trade he cites Landtman's (1927:215) comment that "Custom requires every seller of a canoe to provide it with food to be used on the journey." The islanders purchased canoes primarily with shells - arm-shells and bailers - which were highly prized in New Guinea. Payments also included stone axes, skulls, harpoons, and sundry other items. For islanders distant from New Guinea, an order for a canoe was transmitted through a succession of intermediaries until it reached a Papuan seller, who then sent the canoe back along the same path. Intermediaries were usually paid for their services by adding on to the purchase price. The two canoe-purchasing routes given by Harris are segments of routes given by Wilkin (in Haddon 1904). Beyond Dauan lies Saibai, and beyond Saibai lie Mawatta and an east-west sequence of coastal Papuan communities through which canoes passed from Kiwai in the Fly River delta. Beyond Tudu lies Mawatta and the same sequence (Landtman 1927). "Most of the friendly contacts between the [western and central] Islanders and New Guinea people concentrated at Mawatta, the people of which were well aware of this important role and of their opportunities" (Laade 1968:153). These two routes are shown as a rooted graph in Fig. 6.6, in which the root, Mawatta, is regarded as the primary Papuan canoe source for the islanders. Harris defines a critical point in this network as an intermediary, but he does not specify the degree to which a point is critical. Hence there is no basis for differentiating the horticultural from the foraging islands. In the graph of Fig. 6.6, Dauan and Mabuiag are structurally indistinguishable from Saibai and Badu. If, however, we count the number of paths that pass from the root of the graph southward through each node v^, then we would have a measure of the intermediary activity of each island in the network. Let us call this number the betweenness value of a node in a rooted graph. Four such paths pass through Saibai: (Mawatta, Saibai, Dauan; these three islands plus Mabuiag; these four islands plus Badu; these five islands plus Muralug). Three paths pass through Dauan, two through Mabuiag, one through Badu and Nagir, and none through Muralug. Two paths pass through Tudu and one through Nagir. Harris's hypothesis receives partial support: the horticultural islands of Mabuiag and Nagir are more critically located (between) than the foraging islands of

186

Island networks Mawatta

Tudu

Nagir /

\ Muralug

..•'

Figure 6.6. A rooted graph of the western Torres Strait-Papua New Guinea canoe-purchasing trade routes implicit in Harris (1979). Badu and Muralug, but Saibai, a foraging island, is more critically located than Dauan, a horticultural island. Actually Harris's version of the canoe-purchasing network of western Torres Strait is incomplete. It omits two islands, Boigu and Moa, as well as several other routes. According to Haddon, the Muralug Islanders could purchase canoes through Moa as well as through Badu: If a Muralug man wanted a canoe he would communicate with a relative at Moa who would speak to a friend of his at Badu; possibly the Muralug man might himself go to Badu. The Badu man would cross to Mabuiag to make arrangements and a Mabuiag man would some time or other proceed to Saibai, or at all events let a Saibai man know about it. If there was no canoe available at Saibai word would be passed on along the coast that a canoe was to be sent down. The canoe would then retrace the course of the verbal order, and ultimately find its way to Muralug . . . Another channel of the canoe trade was from Mawatta to Tutu and thence to the Central Islands or via Nagir to Muralug (Haddon 1904:296).

Centrality

187

Boigu 3

Saibai

() Mawatta /--\ ^A y

Dauan / > 4 Tudu

Mabuiag 5 J/

Badu 6

IxJ(\j

1

\Y Moa / Nagir

9 Muralug Figure 6.7. A rooted graph of the western and central Torres StraitPapua New Guinea canoe-purchasing trade routes based on Haddon (1904, 1935). And the inhabitants of Badu and Moa could obtain canoes through Tudu as well as through Mabuiag. "Badu and Moa sent human skulls to Tutu to exchange for canoes, one head would purchase an ordinary canoe and a lower jaw a small canoe" (Haddon 1935:65). Also, Dauan is not a necessary stopover between Saibaig and Mabuiag. Only Wilkins (in Haddon 1904) mentions it as a stop, and then not consistently. By adding these other islands and routes we have a second network, depicted by the rooted graph G in Fig. 6.7. To accommodate all the canoe-purchasing routes mentioned by Haddon, we count the number of paths of length 2 or greater that pass from the root through each node vk. For convenience the nodes of G are numbered in such a way that every canoe-purchasing path consists of an increasing sequence of integers, for example, (0, 1, 2), (0, 1, 5), (0, 1, 5, 6, 9) but not (0, 1, 5, 7, 4). As shown in Table 6.4, not all the horticultural islands are more critically located than all the foraging islands. Mabuiag, a horticultural island, is more critically located than Badu, a foraging island, but Dauan and Nagir, horticultural islands, are less critically located than Saibai and Moa, foraging islands. The most critically located island in the entire network is Saibai. The trade-horticulture hypothesis is not confirmed. The analysis does, however, suggest an interesting relation between crit-

188 Island networks Table 6.4. The position of the western Torres Strait islands in the canoe trade with Papua New Guinea Number of canoe-purchasing paths from New Guinea (Mawatta) on which islands lie as intermediaries in the graph of: Communities

Islands

Harris's network

Haddon's network

Saibailag

Dauan (H) Saibai (F) Boigu" (F)

3 4

8 16

Mululaig

Mabuiag (H) Badu (F)

2 1

12 9

Kauralaig

Nagir (H) Muralug (F) Moa* (F)

1 0

1 0

0

6

Abbreviations: H = horticultural economies; F = foraging economies. Note: a. Not included in Harris's network.

ical location, trading dominance, and social stratification in the larger Torres Straits network. Centrality, trade, and stratification

All the islands in Torres Strait - western, central, and eastern - were connected to each other and to communities in New Guinea and Australia in a single trading network. In eastern Torres Strait, instead of a division into horticultural and foraging islands there appears to have been a division of individual islands into horticultural and fishing/trading districts occupied by different clans. On Mer, the largest of the Murray Islands, the eastern side of the island, inhabited by the horticultural Miriam-le, was the "garden" of the western side, inhabited by the fishing and trading Komet-le.8 Similar divisions are reported for Uga (Ugar) and Erub (Laade 1969). Haddon describes two trade routes between the eastern islands and New Guinea. The longest route was via Tudu and passed through a succession of islands and Papuan communities in which the islanders had hereditary trade partners: The trade route from Mer was by Erub, Ugar, Damut and Tutu, Daru, thence to Mawata, Turituri, Parama and Kiwai. After leaving Ugar there was a change of language, but they 8 There were also other groups of maritime foragers on Mer - the Meuram-le and the Gem-le, which exchanged their products with the Miriam-le (Laade 1969).

Centrality

189

were able to communicate by means of an island and Kiwai jargon trading-language, which was mutually understood. On all these islands and at the villages on the mainland that they visited every Eastern man had a friend, tebud /e, who regarded him as a brother. These friendships once formed were never broken; they were hereditary, having come down from past ages from father to son (Haddon 1935:183). The shortest route was via Erub directly to New Guinea - to Parama and thence on occasion to Mibu and Kiwai. Laade (1969) mentions a slightly longer route via Erub and Uga to Parama. Whichever route was taken, it is "doubtful whether a Murray Islander would ever go across [to New Guinea] without voyaging in an Erub canoe or at all events accompanied by Erub men" (Haddon 1935:350). Although the Fly River people sometimes brought the canoes to the eastern islands, the usual method of delivery was through the same series of intermediaries that had forwarded the payment. The price of a canoe was a shell armlet, with other shell ornaments and food added on as recompense for the intermediaries. The intermediaries used the occasion to exchange competitive gifts with each other, thereby joining the islands in "chains" reminiscent of those that ethnographers have described for Highland New Guinea (Meggitt 1974; Strathern 1971). Fig. 6.8 is a graph of the canoe-purchasing network of the entire Torres Strait. The graph is doubly rooted, at Mawatta and Parama, the two immediate sources of canoes in Papua New Guinea for all the islands in the Strait. An enumeration of all the paths from the roots (done in the same way as for the rooted graph in Fig. 6.7) shows that Tudu lies as an intermediary on 17 canoe-purchasing routes, Saibai on 16 routes, Mabuiag on 12, Badu on 9, Dauan on 8, Moa on 6, Uga on 4, Damut and Erub on 3, and Nagir on 1. Tudu is a foraging island that must be considered in relation to Yam, a horticultural island. Tudu is barely more than a small sandbank less than half a mile in length, dependent for garden products on the neighboring island of Yam. The population of Tudu was estimated at 200 in 1871 (Haddon 1890). The people of the two islands, as already noted, constituted a single community, the Tutulaig. According to Haddon (1904:173), the "natives of Tutu and Yam live at different periods of the year at either island." Although the two islands formed one community, Tudu was nonetheless regarded as the dominant one: Tutu lies in the center of very large reefs which afford prolific fishing grounds for dugong, turtle, numerous kinds of fish, and especially shellfish - the pearl shell Conus, Melo (for bailers and saucepans) formed important articles of trade. Thus the is-

190

Island networks Boigu 3

0 Mawatta

16 ^T\ Parama

18 Erub Murray 19 Islands

15 Muralug Figure 6.8. A doubly rooted graph of the western, central, and eastern Torres Strait-Papua New Guinea canoe-purchasing trade routes, based on Haddon (1904, 1935).

land was a valuable possession and it is not surprising that the natives were skillful sailors and noted warriors. It is significant that all accounts of fighting refer to Tutu and not to Yam, and it is probable that war parties usually started from Tutu (Haddon 1935:74-5). Yam and Tutu, with their complementary garden and marine resources, formed an ecologically balanced community in the same way as Muralug and Nagir, Mabuiag and Badu, Dauan and Saibai. The people of Dauan, a horticultural island, lived for part of the year on Saibai, a foraging island, but like Tudu in relation to Yam, Saibai was recognized as the dominant trading community: "The natives of Saibai were largely the 'middlemen' between the Western Islands and the Daudai [coastal Papua New Guinea] villages" (Haddon 1904:295). Dauan was the garden of Saibai as Yam was the garden of Tudu. To fully appreciate the dominant position of Tudu as a trading community, we may consider more general aspects of trade between the islands in the Strait and between the islands and communities on the Papua New Guinea coast and Cape York in Australia.

Centrality

191

Eastern islanders traded with central islanders, giving garden products for seafood and shells - sea shells for necklaces and other ornaments, turtle shells for masks - as well as other items that the central islanders obtained from the eastern side of Cape York: The old folks say that in earlier days Aurid was a bartering centre for the Miriam-le and thus occupied an important position. The Miriam-le came in their canoes at certain seasons of the year bringing arm-shells which they exchanged for stones for clubs, ochre for painting themselves and their zogo stones, turtle grease, and other products. These articles were obtained by the Aurid men as well as by those of Masig, Damut, and Paremar, when they visited the islands off the east coast of North Queensland, particularly the Sir Charles Hardy group, and the Forbes islands, whither they resorted every south-east season to live for a while and to barter. The stone for making stone-headed clubs was obtained from the Forbes islands (Haddon 1935:88). Red paint was especially prized - "even canoes were exchanged for it" (Haddon 1935:77). In western Torres Strait, the people of Muralug traded with communities on the western side of Cape York, obtaining javelins and spears that they traded northward. Trade in Torres Strait was based on the purchase of canoes from New Guinea, but there were also inter-island exchanges of imported and locally manufactured goods. Bows and arrows, drums, cassowary and bird of paradise feathers obtained from New Guinea were traded between islands, as were some of the canoes owned by island headmen. Dance masks made of wood, shells, and feathers were traded. Haddon, for example, purchased masks at Nagir and Yam that had been made at Tudu. Skulls and tobacco were also traded. A number of island communities manufactured and exported craft goods. Tudu produced the finest white shell armlets (from Conus millepunctatus); Yam apparently made or polished stone axes for export; Muralug was famous for its dugong harpoons; Moa, Yam, and Nagir exported bamboo knives, bows, and containers to Muralug and to the central islands. Muralug imported leglets, and Tudu imported coconut palm arm guards. "Feathers, shells, ornaments, of all descriptions, weapons and in fact any of the goods and chattels were continually being bartered and exchanged throughout the islands of Torres Strait" (Haddon 1904:294). A graph of the pan-Torres Strait network insofar as it can be reconstructed from data in Haddon (1890, 1935), Laade (1969), Landtman (1927), and McCarthy (1939) is shown in Fig. 6.9. The relative centrality of each island in this network is given in Table 6.5.

192

Island networks Kiwai Mibu

Mawatta

Daru

-o

o^*

Parama

Tureture

Erub Murray Ids

Muralug East Cape York West Cape York Figure 6.9. A graph of the Torres Strait-Papuan coast-Cape York trade network. Tudu ranks first in closeness centrality, which measures ease of access to all other communities, and first by far on betweenness centrality, which measures its importance as a trading intermediary. The position and characteristics of Tudu invite comparison with trade centers in three other Melanesian networks: the hula ring in the Massim (Hage 1977; Hage and Harary 1991), the Mailu network of southeastern coastal Papua New Guinea (Irwin 1974, 1978), and the Vitiaz Strait off the northeastern coast of Papua New Guinea (Lilley 1985). In all three networks the dominant trading community, Tubetube in the kula ring, Mailu Island in Mailu, and the Siassi Islands in the Vitiaz Strait, was a very small but centrally located island or cluster of islands that lacked sufficient resources to support its population and was therefore a food

Centrality

193

Table 6.5. Relative centrality of locations in the Greater Torres Strait-Papuan coast-Cape York trade network Cc(^)- 1

Island 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27.

Muralug Moa Badu Mabuiag Dauan Boigu Saibai Nagir Waraber Paremar Aurid Masig Damat Tudu Yam Mawatta Uga Erub Murray Ids Tureture Daru Parama Mib Kiwai Mt. Adolphus East Cape York West Cape York

80 65 65 80 96 121 79 68 71 68 67 62 65 53 64 64 76 76 80 79 90 87 111 111 91 116 105

C6(vk)

Rank

0.325 0.400 0.400 0.325 0.271 0.215 0.329 0.382 0.366 0.382 0.388 0.419 0.400 0.491 0.406 0.406 0.342 0.342 0.325 0.329 0.288 0.299 0.234 0.234 0.286 0.224 0.248

10 = 4= 4= 10 = 14 18

9= 6= 7

6= 5 2

4= 1

3= 3= 8= 8= 10 = 9= 12 11

16 = 16 = 13 17 15

CB(vk)

Ci {vk)

Rank

29.714 32.079 32.079 32.776 25.000 0.000 21.319 60.190 1.286 4.717 14.417 61.181 18.807 146.983 19.671 60.852 31.414 21.957 2.033 27.962 17.762 56.800 0.000 0.000 25.000 0.000 0.000

0.091 0.099 0.099 0.101 0.077 0.000 0.066 0.185 0.004 0.015 0.044 0.188 0.058 0.452 0.061 0.187 0.097 0.068 0.006 0.086 0.055 0.175 0.000 0.000 0.077 0.000 0.000

9

7= 7= 6

11 = 21 = 13 4 20 18 17 2 15 1 14 3 8 12 19 10 16 5

21 21 11 21 21

= = = = =

Note: Cc = 0.308 CB = 0.382

importer and trading intermediary. Each was in possession of a well-developed maritime technology with which to exploit its favorable location. Tudu, the dominant trading community in Torres Strait, with its central location and navigational expertise and its garden in the neighboring island of Yam, is a variant of this general economic pattern. Like Tubetube and Mailu, which manufactured and exported pots, Tudu exported shells and ornaments. And similar to Tudu, Tubetube was an important intermediary in the canoe trade. Correlative with their position as trade centers, Mailu Island had a "complex" form of social organization (Irwin 1985) while Siassi and Tubetube had a system that combined the achieved status of Melanesian big men with the ascribed status of Polynesian chiefs (Lilley 1985; Macintyre 1983; Hage and Harary 1991). Tudu, alone among all societies in Torres Strait, apparently had a system of hereditary chieftainship:

194

Island networks Owing doubtless to the barrenness of their island and the necessity for fishing on neighboring reefs, the inhabitants of Tud were noted seamen and warriors. I believe they were greatly feared throughout the Strait on account of their ferocity and their continued raids on various islands. This warlike tendency has left its impress on the social conditions of the people; for example, as far as I could learn, this is the only island in which a district chief is recognized. Fighting men require a leader and apparently only in Tud was this position hereditary; in fact I do not believe that real chiefs existed elsewhere (Haddon 1890:408). The chief of Tud was also the chief of Yam; when residing in Tud, he put in a locum tenens in Yam but when he visited Yam, he at once exercised his own power. This I believe is quite an exceptional case (Haddon 1890:329).

Haddon does not discuss chieftainship at Tudu in detail, but he does say that there were "at least two lines of chiefs" who were heads of the Kursi and Kodal clans. His genealogies show patrilineal succession for five generations in the Kursi clan (with adoption, in one generation, due to lack of an heir). One of these chiefs, Kebisu, was made chief of both clans. "'King' Kebisu was the much dreaded chief of the warlike Tuduans, father of Haddon's informant Maino" (Laade 1968:141). Tuduans were apparently well traveled. They were, for example, frequent visitors to Mawatta, where they participated in local ceremonies. Kebisu and Maino both had Papuan wives. Assuming that Haddon is correct, we would attribute stratification on Tudu to a combination of island physiography and network location, with an emphasis on trading success rather than warfare.

On the position of Delos in the Archaic Aegean network A network can be treated as a graph G in which the edges in effect all have a value of 1 or, where data permit and circumstances require, as a network N in which the edges have different numerical values based on actual distances between nodes. An archaeological example similar in certain respects to networks in Melanesia is Davis's (1982) analysis of the position of Delos as a religious and trade center in the Aegean network of Archaic Greece. Davis shows that the most applicable model for defining centrality in this network is a "measurement of accessibili-

195 v

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I

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I

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A(G)

6

v

\

V

2

"3

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1

Dis(G) =

1 1

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1 1

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1

Figure 6.10. A graph and its adjacency, reachability, and distance matrices. ty" based on geographical distance. In the language of graph theory, this refers to median centrality in a network N. The median of a network is defined analogously to the median of a graph. Both properties can be found using matrix methods, in the first case by finding the "distance matrix" of a graph and in the second case by finding the "cost matrix" of a network. The adjacency matrix A(G) of a graph G is the p xp matrix [tf/;], with aif = 1 if v-Uj is an edge of G and atj = 0 otherwise. The reachability matrix R(G) of a graph G has the entries r/; = 1 if there is a path between v{ and Vj and r,; = 0 if not. Thus r/; = 1 if nodes v-t and z/; are in the same connected component of G. The /, / entry in the distance matrix Dis(G) gives the distance between v{ and t/;- and is defined as infinity (oo) if there is no path between Vj and Vj. The median of a graph is the node whose row sum in Dis(G) is least. In Fig. 6.10 the median is v5. The following theorem and its corollary from Harary et al. (1965) give the construction of the distance matrix. THEOREM 6.1. The i, j entry a{tfof An is the number of walks of length n between w{ and v,.

196

N:

Island networks 0

2

oo

oo

oo

1

2

0

1

oo

4

oo

1

0

2

4

oo

oo

oo

2

0

3

°°

00

4

4

3

0

3

1

oo

oo

oo

3

0

C=

Figure 6.11. A network N and its cost matrix C.

COROLLARY 6.1. The entries of the reachability and distance matrices can be obtained from the powers of K as follows:

1. For all *, rH = 1 and du = 0. 2. r/; = 1 if and only if for some n, a^ > 0. 3. d{vhVj) is the least n (if any) such that a{ wise.

> 0, and is oo other-

The distance matrix was used to find the median (closeness) centrality of the nodes in the graphs of the Lauan and Torres Strait networks. 9 In the cost matrix C of N, the diagonal entries cH are 0, the entry cif- is oo if there is no edge v-v^ and c/; is the cost value of edge v-Vj when this edge is in N. Fig. 6.11 shows a network N and its cost matrix C. The cost distance of a path is the sum of the cost values of its edges. A u-v cost geodesic is a u-v path with minimum cost length, and this value is the cost distance from u to v. In other words, the cost distance from u to v is the minimum cost among all u-v paths. The cost distance from each node to every other node in N can be found by matrix operations analogous to those used in Theorem 6.1 for finding distances in a graph. Let us denote by f{i the cost distance from v{ to Vj. The cost-distance matrix F of a network N is a square matrix whose entries are the cost distances fa. We begin with the cost matrix C of the network but use a modified arithmetic 10 to obtain "powers" C[2\ O3\ . . . of C. In this arithmetic, "modified multiplication" of numbers is given by a*b = a + b and "modified addition" by a + b = min(<2, b). Thus, for example, 2 * 3 = 5 and 2 + 3 = 2. To show how to obtain C[2] from the cost matrix C of Fig. 6.11, we calculate c$. 9 For other applications of matrix methods to communication networks in Oceania, see Hage (1979b) and Hage and Harary (1991). 10 The procedure described here is due to Maria Hasse (1961).

Centrality 4251

197

= (0 * oo) + (2 * 4) + (oo * 4) + (oo * 3) + (oo * 0) + (1 * 3) =

4

Each term of the equation, shown in parentheses, gives the cost distance of a walk from v1 to v5, whose length is at most 2, and the "modified product" of these is the minimum cost distance among these walks. Thus, c^ gives the minimum cost among all such walks, and hence all paths from v-t to v^ and in general cff is the minimum cost among all paths from v{ to Vj whose length is at most n. The following theorem from Harary et al. (1965) summarizes these observations and states the general procedure for finding the cost distance from v{ to Vj in any cost network. THEOREM 6.2. Let C be the cost matrix of a network N. Let n be a positive integer such that the "modified powers" C[n] = C[n+1]. Then ctf is the cost distance from vi to Vj, and On] = F, the cost-distance matrix of N.

By carrying through the calculation prescribed by this theorem for the cost matrix C for Fig. 6.11, we find the cost-distance matrix of a network N as follows. Here we observe that n = 3 is the smallest positive integer such that C^ = O + 1 l 0 2 3 00 4 1 02]

2 0 3 1 00 3 4 4 13

1 0 2 4 7

3 2 0 3 6

4 4 3 0 3

3 7 6 3 0

C [3] = C [4] =

p

=

0 2 3 5 4 1

2 0 1 3 4 3

3 1 0 2 4 4

5 3 2 0 3 6

4 4 4 3 0 3

1 3 4 6 3 0

The median of a network is the node whose row sum in F is least. In Fig. 6.11 the median is v2. Theorem 6.2 provides a way of finding the distance matrix of a graph or digraph, which is often more economical than that given by Theorem 6.1. COROLLARY 6.2. Consider a graph G as a network in which every edge has value 1. The distance matrix of G is given by the cost-distance matrix F, as obtained by the method of Theorem 6.2.

In an analysis similar to Irwin's (1978) study of the rise of Mailu Island as a trade center in coastal Papua New Guinea, Davis establishes a

198

Island networks

correlation between the wealth of the island of Delos, as inferred from archaeological remains, and its central location in a trade network that evolved through four successive periods of Greek prehistory from 1600 to 700 B.C. In the Minoan period, from c. 1600 to c. 1400 B.C., Delos was neither wealthy nor centrally located: "Delos does not seem to have enjoyed a particularly central location in the Aegean world of Minoan times. In LM I [Late Minoan I] there is evidence for a major route of travel from Crete through the southern and western Cyclades, the so-called Western String, and from Crete through the Dodecanese to the western coast of Asia Minor, but strikingly little for cross-Aegean travel" (Davis 1982:23). In the Mycenaean period, from c. 1400 to c. 1100 B.C., the archaeological remains are rich, and "there is considerable reason to think that Delos belonged to a world which included the western coast of Asia Minor and the central as well as the eastern Cyclades" (1982:24). In the Dark Ages, from c. 1100 to 800 B.C., there are few archaeological finds and less evidence for cross-Aegean travel: "Both coasts of the Aegean, as well as the islands, were not, as seems to have been the case in Mycenaean times, part of the same sphere of interaction" (1982:24). In the Archaic period, at the end of the eighth century, Delos was prosperous, well known - celebrated in Homeric hymn - and the center of an Ionian amphictyony. It was also centrally located in a network of major city-states in Euboea, Attica, the Cyclades, and the western coast of Asia Minor. Davis assumes that ease of access in this network is "approximately equivalent to absolute distance between the major Ionian centers measured in a straight line, in as much as the inhabitants of each were subject to the constraints of the same winds and currents and in as much as nearly all communication within the Ionian world would have been by sea" (1982:28). Although Davis does not show the network or give any figures, he states that Delos ranks third after Mykonos and Tenos in relative accessibility, that is, median centrality. Fig. 6.12 shows our version of the network, based on Davis's distributional map of major city-states. Table 6.6 gives the relative centrality of nodes in this network and in a subnetwork consisting of the Cycladic Islands, calculated using the cost matrix of N. Delos has been known since ancient times as the "hub" of the Cyclades, even though it is not the geographical center of the group. Davis's hypothesis is that Delos emerged as an important regional center in the Archaic period because of its central location not in the Cyclades but in the larger Aegean network of city-states shown in Fig. 6.12. Table

TURKEY EPHES0S

50

Figure 6.12. The Archaic Ionian city-state network, based on J. L. Davis (1982).

100

200

Island networks

Table 6.6. Relative centrality of city-states in the Archaic Ionian network Median centrality in the Ionian network City-states 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29.

Chalkis Eretria Athens Keos Kythoros Seriphos Siphnos Andros Tenos Syros Mykonos Rheneia Delos Paros Naxos los Amorgos Phocaea Chios Erythrai Clazomenai Teos Lebedos Colophon Ephesos Samos Priene Myos Miletos

(km)

Rank

6,792 6,333 6,235 4,426 4,350 4,442 4,458 3,814 3,433 3,612 3,365 3,417 3,413 3,823 3,749 4,540 4,424 5,775 4,075 5,101 6,693 4,810 4,853 5,019 5,130 4,231 5,075 5,383 5,131

29 27 26 13 11 14 15 7 4 5 1 3 2 8 6 16 12 25 9 21 28 17 18 19 22 10 20 24 23

Median centrality in the Cycladic subnetwork (km)

Rank

1,143

13 10 8 9 11 6 1 5 2 3 4 7 12 14

998 950 952

1,035 787 683 780 686 706 761 865

1,112 1,348

Source: Based on J. L. Davis (1982).

6.6 shows that Delos is highly central in both networks. In the Ionian network Delos ranks second, slightly behind Mykonos, and in the Cycladic network Delos ranks third, just behind Syros and Rheneia. On the basis of present archaeological evidence it is not known whether Delos was a trade center and manufacturing specialist like Mailu Island in Melanesia (Irwin 1978) or a port of trade like Cozumel Island in Mexico (Rathje and Sabloff 1973-4), but it is certain that Delos was the panIonian cult center and quite likely a trade center as well. Davis makes two important observations on the significance of centrality in this network. The first concerns the relation between centrality and travel time. Given the relatively low speed of ancient ships, even

Centrality

201

modest differences in travel time would have posed a significant problem for more peripherally located cities: for a city such as Chalkis, the selection of Naxos, for example, over Mykonos as a Pan-Ionian center would involve a trip of five to nine hours longer. Over the round trip, this could easily amount to an extra day away from home. It seems clear that siting the festival at a still less favorable location such as Athens or Miletus would entail considerable hardships for many of the Ionian cities (1982:28). The second observation concerns the relation between centrality and environment. As shown in Table 6.6, centrality cannot be the only explanation for the rise of Delos, since the several top-ranking cities are separated by relatively minor distances. Delos's advantage over these other cities may have been its small size and natural poverty. "Unattractive to the city-states of the Cyclades, Delos could have offered a neutral ground outside the territory of any member of the amphictyony" (1982:27).

Self-centered networks There may be networks in which no island or islands have a locational advantage with respect to centrality, for example the ur-kula ring (Hage and Harary 1991) or the prehistoric Tonga-Fiji-Samoa trade network (Chapter 4), each of which consisted of a single cycle. The graphs of such networks are said to be "self-centered" or "self-median." To complete our presentation of centrality we give a general characterization of these graphs from Buckley (1979) and Sabidussi (1966). Consider two graphs G = (V, £) and G' = (V, £') both with n nodes. We label the nodes in V by the first ^-positive integers 1, 2,. . ., n. With this labeling of G fixed, then, as defined in Chapter 2, we say that G and G' are isomorphic graphs, provided it is possible to label the nodes in V as 1', 2 ' , . . . , n' in such a way that nodes i and / are adjacent in G if and only if nodes /' and /' are adjacent in G'. This means that whenever /, / are adjacent in G, we must have /',;' adjacent in G' and also that if / and / are not adjacent in G then /' and /' are not adjacent in G'. Now consider just one graph G = (V, £). We say that a is an automorphism of G if a maps V = {1, 2,. . ., n] onto itself in such a way that isomorphism is preserved. In other words, a is a relabeling of the same set V of nodes with either the same labels or with a different labeling 1', 2 ' , . . ., n'. Fig. 6.13 shows the four automorphisms of the graph K4 - e. The center of a graph G, C(G), as defined previously, consists of all

202 1

Island networks 2

1

2

3

4

Figure 6.13. The four automorphisms of the graph K4 - e.

K 3,3

Figure 6.14. Three node-symmetric graphs. nodes of G whose eccentricity is minimum. Writing V(G) for the node set, the graph G is self-centered if C(G) = V(G). A graph G is called node-symmetric if for any two nodes w, v there is an automorphism of G sending u onto v. The most mentioned examples are the cycles CM, the complete graphs Kp, and the complete bipartite graphs Kmn, as illustrated in Fig. 6.14 by the graphs C5, KA, and K33. The next result, due to Fred Buckley, is from Buckley and Harary (1990). THEOREM 6.3. Every node-symmetric graph is self-centered, but not conversely.

The smallest counterexample to the converse is the graph C5 + e in Fig. 6.15. Thus an island network may actually be self-centered even though it does not appear to be. A self-median graph G is one in which all the nodes have the same status s(u). The complete bigraph Kmw, for example, is self-median. If the southern Lau network actually consisted of three limestone islands exchanging goods with three volcanic islands (and not with each other), as Thompson implied, its underlying graph would be K3j3 and there would be no structural basis for a trade center. The graphs Cn and Kp are also self-median. Regular graphs are often self-median. Nonregular self-median graphs are rather rare. One example is given in Fig. 6.16 (from Sabidussi 1966).

Centrality

203

Figure 6.15. A self-centered graph that is not node-symmetric.

Figure 6.16. A nonregular self-median graph (from Sabidussi 1966).

The graphs Kp, Kmn, and Cn are self-centered as well as self-median. A graph may, however, be self-centered but not self-median - for example, the graph in Fig. 6.15. In most island networks certain islands will have a locational advantage with respect to centrality, but whether this can be turned to account depends on historical and material circumstances. Central location in the Gilbert and Tuamotu Islands, for example, had no apparent economic or political significance. In the Gilberts there was no trade to speak of, and hence there were no trade centers. In the Tuamotus, political power was based on dominating sets in the voyaging network.

Dominating sets

Veni, vidi, vici. Julius Caesar

Every network N, with underlying graph G, has one or more dominating sets. Historically, this concept originated with von Neumann in his pioneering work with Morgenstern (1944) on the theory of games. In game theory, a given game may have several strategies deciding which move to make in any given game situation. A strategy is said to dominate another one if the person using the first strategy defeats his opponent using the second one in a two-person game. This was formalized to domination in digraphs by Richardson (1953) and studied by Harary and Richardson (1959). Ore (1962) generalized the concept of domination in digraphs to graphs G. This is entirely analogous to the domination of the 64 squares of a conventional chessboard by Queens. This Queen domination problem was mentioned in Chapter 1. In particular, the placing of eight Queens on a chessboard so that no Queen threatens (dominates) any other Queen was completely solved by Euler in the eighteenth century. Ore defined a node v in G as dominating itself and all nodes adjacent to it, that is, v dominates its closed neighborhood N[v]. Domination in graphs is now the most active area of research in graph theory (Laskar and Walikar 1981; Hedetniemi and Laskar 1990). For our present purposes, every island network has some dominating set of islands. We now use the combinatorial model of domination in graphs to describe local political hierarchies in the Caroline Islands in Micronesia, alliance structures in the Tuamotu Islands in Polynesia, and pottery monopolies in two trade networks in Melanesia. 204

Dominating sets

3

205

4

Figure 7.1. A graph to illustrate dominating sets.

Local domination in the Caroline Islands Alkire (1970) distinguishes three levels of social organization in the western Caroline Islands: the intra-atoll level, in which islets or districts of islets were joined; the inter-island level, in which small clusters of islands were joined; and the overseas tribute system (the Yapese Empire), in which all islands were joined. The second level is illustrated by the Lamotrek, Elato, Satawal cluster. In this system, called the hu (lit. the "hook"), Elato and Satawal paid tribute to Lamotrek in return for the right to exploit neighboring uninhabited islands for marine resources. Although the hu is the only example Alkire gives of a local inter-island hierarchy, others are known or can be inferred. Collectively they constitute an "independent dominating set" in the graph of the western Carolines voyaging network. A dominating set S of nodes of a graph G has the property that every node not in S is adjacent with at least one node in S. A minimal dominating set is one in which no proper subset has this property. A minimum dominating set is a dominating set of nodes of smallest cardinality. In the graph of Fig. 7.1, the set {1, 6, 8, 9} is a dominating set and the set {1, 8, 9} is a minimal dominating set, in fact it is a minimum dominating set. If u and v are not adjacent in a graph G, then u and v are independent nodes of G. In an independent set no two nodes are adjacent. In Fig. 7.1 the set {1, 4, 7, 10} is an independent dominating set. THEOREM

1962).

7.1. Every graph has an independent dominating set (Ore

206

Island networks

Ulithi Fais Namonuito Faraulep Sorol ^ ^ ^ \ W o l e a i \

/ \ Ifaluk

Lamotrek

,

, _. r ulap

Elato

Eauripik Pulusuk

Figure 7.2. A graph of the western Carolines voyaging network (from Hage and Harary 1991). The domination number <x(G) is the smallest number of nodes in any dominating set, hence it is the cardinality of a minimum dominating set.

In Fig. 7.1, a(G) = 3. The independent domination number ct'(G) of a graph G is the smallest number of nodes in any independent dominating set. In Fig. 7.1, a(G) = 3 and a'(G) = 4. Fig. 7.2 shows a graph G of the voyaging network that joined the inhabited islands of the western Carolines from Ulithi to Namonuito (based on Hage and Harary 1991). Given the traditions of warfare, conquest, and hierarchy in the Carolines, we assume that all islands in this network were either dominated or dominating. Since a(G) and a'(G) = 4, we will look for three more dominating islands in addition to Lamotrek. To the east of Lamotrek, Puluwat dominated, through conquest and colonization, the neighboring islands of Pulusuk, Pulap, and Namonuito. Puluwat resettled Pulap and in one of its conquests wiped out the chiefly clans of Pulusuk and installed its own dynasty (Gladwin 1970). All three islands were tributaries of Puluwat (Damm and Sarfert 1935). The position of Lamotrek and Puluwat as independent nodes of the graph in Fig. 7.2 suggests that we should look for an independent dominating set of islands. With no two dominating islands adjacent, each could then be said to have its own structurally demarcated "sphere." Analogous to the Queens Problem in chess, dominating islands do not threaten each other. In the western part of the network, Ulithi dominated Fais and Sorol in an informal but fundamental way: Ulithi did not exact tribute from Fais and Sorol, but it did control their travel and trade. These two islands, alone among all others in the western Carolines, had, at an early date, lost their navigational skills and had to rely on Ulithi for communica-

Dominating sets

207

tion with all other islands (Lessa 1950). No explanation has been given for this loss, although mythological accounts clearly express Ulithi's dominance over Fais, especially in its communication with the high island of Yap (Damm and Sarfert 1935). The significance of such control is underscored by cases in which certain islands in the Carolines - Yap, Lamotrek, and Losap, for example - attempted to control the voyaging of their neighbors (Hage and Harary 1991). Given Lamotrek, Puluwat, and Ulithi, the other member or members of the independent dominating set must be Woleai or Ifaluk or Faraulep and Eauripik. Historical evidence favors the larger islands of Woleai and Ifaluk. Woleai evidently did not dominate its neighbors in the recent past but instead turned in on itself and developed an intra-atoll structure: "Woleai does not seem to have any formalized hook ties to its neighboring atolls of Eauripik, Ifaluk or Faraulep. However a hook system does exist within the atoll itself" (Alkire 1970:6). But Kotzebue (1821) mentions that Woleai once dominated a number of atolls in the Carolines. Although Ifaluk is not known to have dominated its neighbors in historical times, local legends, also reported from Faraulep and Lamotrek, describe Ifaluk's conquest and colonization of Woleai, Faraulep, and Eauripik, making it the "mother country" of neighboring atolls (Burrows and Spiro 1957). We suppose that either Woleai dominated Eauripik, Faraulep, and Ifaluk or that Ifaluk dominated Eauripik, Faraulep, and Woleai or, perhaps most likely, that Woleai and Ifaluk dominated at different times. As an independent dominating set of the graph of the western Carolines network we then have either {Lamotrek, Puluwat, Ulithi, Woleai} or {Lamotrek, Puluwat, Ulithi, Ifaluk}. In each set the dominating atolls of Lamotrek, Puluwat, and Ulithi played a special role in Carolinian social organization. Ulithi was the intermediary between Yap and the outer islands in the Yapese tribute system, Puluwat was the dominant trading community in the Carolines, and Lamotrek once dominated all the atolls in the group (Lessa 1962; Hage and Harary 1991). Lamotrek's and Puluwat's domination of their immediate neighbors can be seen as their local power base.

Alliance structures in the western Tuamotus In the western Tuamotus in East Polynesia (see Chapter 3 and Map 3.1) the island of 'Ana'a once dominated, through conquest, all other islands. The success of 'Ana'a is usually attributed to its superior resources and numbers (Lucett 1851; Corney 1913-19; Emory and Ottino 1967; Haddon and Hornell 1975; Alkire 1978), but it appears that its alliance with other islands must also be taken into account. The struc-

208

Island networks

ture of this alliance and, we conjecture, of earlier opposing alliances in the western Tuamotus took the form of a minimum dominating set of the inter-island voyaging network. In the Tuamotus, warfare was a pervasive feature of inter-atoll relations, celebrated in native traditions (Audran 1917, 1919; Emory and Ottino 1967) and emphasized in early European reports (Ellis 1829; Moerenhout 1837). Most wars were local, in the nature of raids motivated by severe and recurrent food shortages and by the pursuit of vengeance and glory. Individual islands or small clusters of islands were sometimes opposed as mutual enemies and at other times joined in shifting alliances. Among all Tuamotuans, the warriors of 'Ana'a were by reputation the most ferocious, referred to as paratas, the name of 'Tun des plus terrifiants requins mangeurs d'hommes" (Emory and Ottino 1967:29). The intensity and extent of their wars in the early nineteenth century are vividly described by Moerenhout: In general I will portray the inhabitants of the Dangerous Archipelago [i.e., the Tuamotus] by a double insignia: hardy navigators and redoubtable warriors; but those of 'Ana'a are distinguished among all by a spirit of pursuit of which there has never been a more striking example. They are almost constantly at war with their neighbors . . . They have been seen pursuing their attackers from island to island all the way to [Marokau and Ravahere], pillaging, razing by fire, massacring the men to devour them (because they are cannibals) and sparing young women and children to make slaves of them. In their blind and savage fury, as if they dreaded leaving any resources to the small number of victims who were able to escape them, they burned the coconut trees everywhere on the islands before departing, changing into frightful deserts these same islands which, before being invaded, had boasted of sizable populations and a suitable vegetation with which to nourish them. These devastating attacks went on from 1800 to 1815 and took on a fearful degree of barbarity, especially toward the end of the period and toward the people of the Palliser Islands [Kaukura, Arutua, and Apataki] and Tiooka [Takoroa], whom the 'Ana'a ns finished by expelling, and had the further audacity to pursue all the way to O-taiti [Tahiti] (Moerenhout 1837:370-1, our translation). Early traditions of 'Ana'a recount individual contests between famous warriors from 'Ana'a and the islands of Kaukura, Fangatau, and Aratika. Later traditions, of the seventeenth and eighteenth centuries, describe 'Ana'a's conquests of the central and western islands and its mili-

Dominating sets

209

tary campaigns, which ranged as far east as Vahitahi. In the west 'Ana'a conquered all the islands of the Mihiroa, Vahitu, and Tapuhoe-Tauaro districts. (See Fig. 3.3.) The bases of 'Ana'a's domination of other islands were demographic, material, and structural. 'Ana'a was the most populous of all islands in the Tuamotus, having an estimated 2,500 or more inhabitants in 1825 (Emory and Ottino 1967),1 compared to the several dozen to several hundred inhabitants of other islands. It was also the richest: Their pre-eminence arose mainly, no doubt, from the greater productiveness of their atoll, owing to its size and the nature of the soil on i t . . . They had also the best supply of timber, built larger and more seaworthy canoes than their neighbours, and ventured farther afield, not only to Tahiti (which they once designed to conquer, and did actually invade, but consented, after a parley, not to molest), but even to the windward atolls of Hao, Vairaatea, and Nukutavake (Corney 1915:117; quoted in Haddon and Hornell 1975). 'Ana'a also enjoyed a locational advantage in interregional trade. Lying on the western border of the archipelago, it was an intermediary in the trade with Tahiti, visited by islanders as far away as Takume 400 kilometers to the east (Lucett 1851; Caillot 1909; Dening 1963). The social basis of 'Ana'a's success in the nineteenth century was its permanent alliance through ties of marriage and kinship with the atolls of Takaroa and Takapoto: When the Anaa warriors conquered the people of Takaroa and Takapoto in the beginning of the nineteenth century, taking many of them to Anaa as captives, they did not annex the land as theirs by right of conquest. It still belonged to the kindred [(dti] whose maraes stood upon it, as it does to this day. But, through intermarriage with the families who had rightful title to the land, the conqueror's children came to inherit it (Emory 1947:7). Takaroa was a significant choice as ally. The island was relatively rich: "Cette ile [Takaroa] est apres Pile de Chaine ['Ana'a], la mieux boisee de tout l'Archipel. Elle est partout, garnie de cocotiers. On y trouve des cochons, de la volaille et des noix de coco" (Moerenhout 1837:202). It was also a traditional center of power, the "seat of a powerful tribe, the Vahitu or Ahitu, who dominated all of the islands in the 1 Emory estimates that as many as 1,000 of the 2,500 inhabitants of 'Ana'a in 1825 were captives taken in war.

210

Island networks

neighborhood" (Emory 1932:44). The most famous of all legendary Tuamotuan navigator-warriors, Moeva, "Chief of Vahitu," was born in Takaroa and was said to have conquered a great many islands, possibly including Rangiroa, Kaukura, Kauehi, Apataki, Niau, Fakarava, Makemo, and 'Ana'a. Takaroa also had, from the point of view of 'Ana'a, the advantage of a strategic location. Although 'Ana'ans carried out punitive raids - "if the injured party should dare to rise in their own defence, on the news reaching Ana, they collect an overwhelming force in their large double canoes, of which they have from fifty to sixty, and take fearful vengeance" (Lucett 1851:260) - they had few means at their disposal to maintain control over all the islands they conquered. Some of the more distant islands in the central region - Raroia, for example - eventually regained their independence from 'Ana'a (Danielsson 1956). A region of effective or enduring control is implicit in Emory and Ottino's conclusion that the traditions of 'Ana'a relate two major events in western Tuamotuan history: a war of reprisal against 'Ana'a, led by a warrior, Turihono, of the neighboring island of Fakarava, followed by 'Ana'a's wars against the western and far-western islands. In reconquering Rangiroa and Kaukura, 'Ana'a forced the inhabitants of these islands, as well as those of Arutua and Makatea, to flee to Tahiti, where they remained until 1820, when, under orders from Pomare II, they were allowed to return. The alliances and the outcome of such wars suggest that in general 'Ana'a, or any other island in the region (for example, Takaroa or Fakarava), could potentially dominate over the long term only the islands to which it was "adjacent." An empirical basis for determining adjacency in the graph of the Tuamotuan network is provided by the "100-mile overnight-voyaging rule" used by Marck (1986) in his study of the relation between dialects in Micronesia. This rule, derived from ethnographic accounts of Oceanic voyaging (Gladwin 1970; Lewis 1972), says that "100 miles is the approximate limit of the overnight voyage, i.e., if one leaves a certain island at dusk one day, landfall can reliably be made in daylight of the next day only if the target island is less than 100 miles away and the winds are favorable in direction and strength" (1986:253). This is a reasonable assumption for Tuamotuans. They were exceptional sailors and evidently made voyages of 100 miles or more quite routinely (Wilkes 1845). A graph G of overnight-voyaging connections between the islands of the Mihiroa, Vahitu, and Tapuhoe-Tauaro districts plus the island of 'Ana'a of Parata district is shown in Fig. 7.3. We have treated Takaroa and Takapoto as a single node in G, on the grounds that these two neighboring islands formed a single politically autonomous group, a

Dominating sets

211

Manihi

TakaroaTakapoto

Ahe

Kauehi

Fakarava

Faaite

'Ana'a

Figure 7.3. A graph of the overnight-voyaging network in the Vahitu, Tapuhoe-Tauro, and Parata districts of the western Tuamotus. "single nation," according to Danielsson (1956), and in fact a single community: "Taapouta [Takapoto] est si pres de Taaroa [Takaroa] qu'elle est continuellement visitee par les habitans de cette derniere. Je crois meme que les Indiens qu'on y trouve ne sont que des visiteurs de Taaroa" (Moerenhout 1837:205). In Fig. 7.3, 'Ana'a and Takaroa-Takapoto are collectively adjacent to all other islands and hence form a dominating set, in fact a minimum dominating set, of the graph. Caillot (1909:36) alludes to a number of alliances against 'Ana'a: "A different reprises les naturels de plusieurs iles se liguerent, dit-on, pour se venger des habitants d'Anaa." Emory and Ottino suggest that shifting alliances in the western Tuamotus might be revealed through a comparative study of island traditions. The model of dominating sets provides a basis for hypotheses concerning the composition of such structures. The

212

Island networks

next two theorems, from Ore (1962), show one line of approach for studying competing alliances that might at various times have dominated this network (or any similar network in the Tuamotus). 7.2. Every nontrivial connected graph G has a dominating set S such that the remaining setV - S of nodes is also a dominating set. THEOREM

THEOREM 7.3. If G is a connected graph, then for any minimal dominating set in G there exists another one disjoint from it.

'Ana'a and Takaroa-Takapoto are a dominating set, and by Theorem 7.2 all the remaining islands are also a dominating set, but one that might have been too large a set to function as an effective alliance. Let us assume that an effective alliance is a minimal alliance, like that between 'Ana'a and Takaroa-Takapoto. By Theorem 7.3, there must be another such minimal alliance disjoint from it. Inspection of the graph reveals a number of minimal alliances that are also minimum, for example, {Manihi, Fakarava}, {Manihi, Kauehi}, {Faaite, Aratika} and {Fakarava, Aratika}.2 Of these alliances {Aratika, Fakarava} looks very possible as an alliance against {'Ana'a, Takaroa-Takapoto}. Aratika was an early opponent of 'Ana'a, and Fakarava was a later opponent and once a member of an opposing alliance. Fakarava was, in addition, an island of great historical and social importance. Called "Hawaiki," the name of the Polynesian homeland, by Tuamotuans (and "Wittgenstein Island" by Europeans), Fakarava was the ancestral home of chiefs of many neighboring islands: "The highest ranking chiefs of the western half of the Tuamotus trace their lineage back to the ancestors of the Fakarava chiefs. Pomare I of Tahiti, on his father's side, was of extraction from this Fakarava stock" (Emory 1932:44). On historical, social, and structural grounds it would not be surprising to discover that Fakarava and Aratika had once formed an alliance that dominated the islands of the western Tuamotus.

Pottery monopolies in Melanesian trade networks In the graph of a communication network, the nodes in a dominating set can, collectively, reach all other nodes in a single step. In networks in which effective communication from a source is not limited to one step but extends to two steps, or three steps, and so forth, the appropriate graph theoretic model is an "w-step dominating set." This is a useful model for analyzing the distribution of monopolies in trade networks. 2 A method for enumerating the minimal dominating sets of a graph is given in Hammer and Rudeanu (1968).

Dominating sets

213

In a comparative study of pottery trading in Melanesia, Allen (1984) arrives at two propositions concerning specialist traders in general and pottery traders in particular: (1) "specialized middlemen traders . . . are likely to be located in marginal zones between agriculturally richer areas"; (2) "pottery provides an important item of subsistence exchange, and . . . capturing a monopoly over its manufacture and/or distribution permits specialized traders to occupy or transform agriculturally poor areas" (1984:435-6). As examples Allen cites the Mailu Islanders, off the southeastern coast of Papua New Guinea (Irwin 1974, 1978); the Western Motu of the Port Moresby region (Allen 1984); the Titan of Manus in the Admiralty archipelago (Mead 1930b, 1937); the Siassi Islanders of the Vitiaz Strait (Harding 1967); and the Amphlett Islanders of the Massim (Malinowski 1922). The first proposition, as Allen notes, "carries an inherent suggestion of centrality." Whereas Mailu Island and the Siassi Islands are centrally located in their networks, the Amphlett Islands are not. Their "apparently anomalous" position in the kula ring (Irwin 1983) suggests that the model of dominating sets would be more suitable than that of centrality for describing the location of pottery monopolists in larger trade networks (Fig. 7.4). There were actually two major pottery-producing communities in the kula ring - the Amphlett Islands and Tubetube. The Amphlettans were a small group of skilled potters and sailors inhabiting barren but strategically located islands. Malinowski (1922:46) describes them as "a numerically weak tribe, easily assailable from the sea, getting hardly enough to eat from their rocky islands; and yet through their unique skill in pottery, their great daring and efficiency as sailors, and their central position half way between Dobu and the Trobriands, they have succeeded in becoming in several respects the monopolists of this part of the world." They were not in fact self-supporting but exchanged pots for food with the Dobuan and Trobriand Islanders. "If we imagine a commercial diagram drawn on the map, we would first of all notice the export in pottery, radiating from the Amphletts as its source. In the inverse direction, flowing towards them, would be imports in food such as sago, pigs, coco-nut, betel-nut, taro and yams" (Malinowski 1922:286-7). Amphlettan pots were of superior quality and had a broad northsouth distribution: The strongest and best decorated pots in the Possession are made on the islands of the Amphlett Group whence they are traded in two directions, northwards to the Trobriands and southwards and eastwards to Milne Bay and the neighboring islands. These handsome pots do not reach Tubetube, at least I

214

Island networks Kiriwina

(TROBRIANDIS.) Kayleula Marshall Bennetts

Sinaketa

Woodlark Laughlan

Amphlett Is.

East Cape (MILNE BAY) Misima

East End Is.

Wari

Figure 7.4. The kula ring (after Irwin 1983).

saw none upon the island and as far as my experience goes they are very much less common in Milne Bay than pots made elsewhere (Seligmann 1910:531). Tubetube, like the Amphletts (and Mailu Island) lacked sufficient food resources to support its population and manufactured and traded pottery as a subsistence strategy: it is scarcely an exaggeration to say that everything in daily use, including food, was imported into Tubetube at one time or another. The exports from Tubetube are pots and shell ornaments; the pots made upon the island are traded over a wide area and a considerable number of the nose ornaments called wanepa are made locally, but with the exception of these two articles there are no local manufactures to exchange for the many articles imported into the island (Seligmann 1910:526).

Dominating sets 1

G:

215

Gz:

7 6 ^ \ _ ^ ^

7

4

5 Figure 7.5. The square G2 of a graph G.

Pots of Tubetube origin have been found at Vakuta and Sinaketa but not in the majority, and only a few sherds of Tubetube-Wari origin have been found elsewhere in the Trobriands (Lauer 1971). But Tubetube was the major supplier to Woodlark (Fortune 1932), probably to Dobu and Southwest Dobu (Macintyre 1983), and apparently the Marshall Bennetts (Munn 1986).3 In some cases the distribution of pottery from Tubetube and the Amphletts was practically complementary: for example, only Amphlettan pottery reached the northern Trobriands (Lauer 1971). In other cases it overlapped: for example, Amphlettan pottery was found at Milne Bay but not in the majority (Seligmann 1910).4 A precise statement of the distributional frequencies of pottery from these two sources awaits further study, but for the present we can say that the model of the pottery monopoly in the kula network is the "2-step dominating set" of a graph, a concept first introduced by Harary and Richardson (1959). The square G2 of a graph G has V(G2) = V(G) with w, v adjacent in 2 G whenever d(u, v) < 2 in G (Harary and Ross 1960). The powers G3, G4, . . . , Gn of G are defined similarly. The square of a graph is illustrated in Fig. 7.5. An n-step dominating set of nodes Sn of a graph Gn is a set in which every node not in Sn is adjacent to at least one node in Sn. A minimal nstep dominating set is one in which no proper subset has this property. A minimum n-step dominating set is a dominating set of nodes of smallest cardinality. 3 The attribution to the Amphletts of pots found in the Marshall Bennetts may be due to a misidentification on the part of Barton as reported by Seligmann (1910) and cited by Malinowski. 4 Paneati (in Misima) and Wari were evidently secondary pottery producers. Their exports and those of the Amphletts to non-kula communities (Tindale and Bartlett 1937; Lauer 1970) imply a larger Massim-wide network.

ON

I 4—'

c

2


Dominating sets

217 n

M

The n-step domination number of G, written a (G) = a(G ), is the minimum number of nodes in an n-step dominating set. In the graph of Fig. 7.5, a(G 2 )= 2. In the graph G of Fig. 7.4, the communities over which Tubetube and the Amphletts had a monopoly in the supply of pottery are one or at most two steps away, and the set of these communities includes all other nodes of the graph. There is no single node that can reach all other nodes in two steps or less. Hence the Amphletts and Tubetube are a minimum dominating set of G2 and a(G2) = 2. It is interesting to consider the graph of the Mailu network in Fig. 7.6 from this point of view. Irwin (1974) has shown that Mailu Island is the median of this graph. But one might say that what mattered for Mailu Island as a pottery monopolist was not so much its total distance, S^ ; , to all other communities as its maximum distance, max d^ to any other community. Inspection of the graph in Fig. 7.6 shows that the maximum distance is 2. Mailu Island is (uniquely) a minimum 2-step dominating set of G, and so a(G2) = 1.

8 Digraphs

It goes without saying that . . . merely formal studies can never be an end in themselves. But there is, on the other hand, always the danger that in historical or functional studies of kinship problems this formal aspect may be unduly neglected. Paul Kirchhoff, "Kinship Organization" There have been two major attempts to construct formal evolutionary models of kinship organization in Oceania: Murdock's (1949) derivation of Malayo-Polynesian societies from a Hawaiian prototype, and Marshall's (1984) derivation of Island Oceanic sibling terminologies from a distributional prototype. Murdock's model, known more generally as the "bilateral hypothesis," is part of a universal theory of social evolution representing the culmination of a lifetime of cross-cultural research on kinship organization, while Marshall's model is the most recent contribution to a theoretical discussion of sibling classification and social organization which began with the ethnographic researches of Codrington (1891) a century ago. Both models are controversial, primarily because of arguments from historical linguistics (Blust 1980, 1984; Bender 1984; Clark 1984). But there are problems of interpretation as well. Our purpose is to examine the graph theoretic foundation of these models. We will find that Murdock's model provides no valid reason for inferring that kinship organization in Proto-Malayo-Polynesian (PMP) society was Hawaiian in type. If anything, it was Iroquois or Nankanse, neither of which is a bilateral type of social organization. We will see that Marshall's model can be replaced by the one implicit in Milke's (1938) historical reconstruction of Proto-Oceanic (POC) sibling terms. In our discussion of Murdock and Marshall, we will refer to the evolutionary reconstruction of social stratification in Nuclear Micronesia that we outlined in Chapter 5, which contradicts the conclusions of both of these authors. 218

Digraphs

219

Figure 8.1. A digraph to illustrate the classification of nodes.

Basic definitions A directed graph or digraph D consists of a finite set V of nodes and a collection of ordered pairs of distinct nodes. Any such pair (w, v) is called an arc or directed edge and will be denoted briefly by uv. The arc uv goes from node u to node v and is incident with u and v. We also say that u is adjacent to v and v is adjacent from u. When arc x = uv, we write u = fx, the first node of x, and v - sx, its second node. In a digraph D, each node has both an outdegree and an indegree. The outdegree of node v, od(v), is the number of nodes adjacent from it, and its indegree id(v) is the number of nodes adjacent to it. A natural typology of the nodes of a digraph uses both outdegree and indegree. An isolate is a node whose outdegree and indegree are both 0. A transmitter is a node having positive outdegree and indegree 0. A receiver is a node with outdegree 0 and positive indegree. A carrier is a node whose outdegree and indegree are both 1. Any other node is called ordinary. In the digraph of Fig. 8.1, node 1 is a transmitter, 2 is a carrier, 4 is a receiver, 3 and 5 are ordinary nodes, and 6 is an isolate. A {directed) walk in a digraph D is an alternating sequence of nodes and arcs z/0, xl9 . . ., xm vn (not necessarily distinct) in which each arc x7is Vj _ \Vj. The length of such a walk is w, the number of arcs in it. A closed walk has the same first and last nodes, and a spanning walk contains all the nodes of D. A (directed) path is a walk in which all nodes are distinct; a (directed) cycle is a nontrivial closed walk with all nodes distinct (except the first and last). The directed cycle of length n is denoted by Cn. If there is a path from u to z/, then v is said to be reachable from w, and the distance d(u, v) from u to v is the length of any shortest such path. There may be more than one shortest path. In Fig. 8.2, the sequence of nodes 1, 2, 3, 4, 2, 5 is a directed walk; the sequence 1, 2, 5 is

220

Island networks 4

o 1 2 5 Figure 8.2. A digraph to illustrate walks.

a path; and 2, 3, 4, 2 is a cycle. There is no spanning walk. The distance from node 4 to node 3 is 2. Each walk in a digraph is directed from the first node v0 to the last vn. We also need a concept that ignores the property of direction and is analogous to a walk in a graph. A semiwalk is again an alternating sequence v0, X\, vx,. . ., xn, vn of nodes and arcs, but each arc xt may be either Vj_ iVi or ViVi_ !• Then a semipath is a semiwalk in which all nodes are distinct, and hence a path is a semipath with consistent direction. A strict semipath is not a path. In Fig. 8.2, nodes 1 and 6 are joined by a semiwalk (1, 2, 3, 4, 2, 5, 6) and by a semipath (1, 2, 5, 6). A semicycle is obtained from a semipath on adding one arc joining its endnodes (or by identifying its endnodes). Whereas a graph G is either connected or is not, a digraph D may be connected in three different ways. A digraph is strongly connected, or, more briefly, strong, if every two nodes are mutually reachable; it is unilaterally connected, or unilateral, if for any two nodes at least one is reachable from the other; and it is weakly connected, or weak, if every two nodes are joined by a semipath. Clearly, every strong digraph is unilateral, and every unilateral digraph is weak, but the converse statements are not true. A digraph is disconnected if it is not even weak. We note that the trivial digraph, consisting of exactly one node, is vacuously strong, since it does not contain two nodes. Thus there are four different connectedness categories Q of a digraph D that indicate its level of connectedness. Fig. 8.3 shows digraphs that are disconnected (category Co), weak but not unilateral (Q), unilateral but not strong (C2), and strong (C3). In contrast to the connected components of a graph, there are three different kinds of digraph components: strong, unilateral, and weak subgraphs that are maximal with respect to these properties. The following theorem from Harary et al. (1965) states the necessary and sufficient conditions for a digraph to satisfy each of the three kinds of connectedness.

Digraphs

221

o 4

(c 2 )

(c3)

4

4

Figure 8.3. Digraphs to illustrate connectedness categories.

THEOREM 8.1. A digraph is strong if and only if it has a closed spanning walk, it is unilateral if and only if it has a spanning walk, and it is weak if and only if it has a spanning semiwalk.

Directional duality in a digraph is based on the operation of taking the "converse." The converse digraph D' of D has the same set of nodes as D, and the arc uv is in D' if and only if arc vu is in D. The digraphs in Fig. 8.4 are converses of each other. THEOREM 8.2. The converse of the converse of a digraph D is D itself: Symbolically, D " = D (Harary et al. 1965).

The adjacency, distance, and reachability matrices of a digraph, written A(D), Dis(D), and R(D), are defined analogously to those of a graph (Chapter 6). We require one additional matrix. The universal matrix J is a matrix all of whose entries are 1.

222

Island networks 2 D:

y

\

D':

1 (a)

(b)

Figure 8.4. Illustrations of converse digraphs.

Murdock's maze: The bilateral hypothesis of Proto-Malayo-Polynesian social organization The historical reconstruction of Micronesian social organization that we proposed in Chapter 5 is radically different from the one given by Murdock (1948). Consistent with the principles set forth in Social Structure', Murdock (1949) assumes that all Micronesian and more generally all Malayo-Polynesian (MP) societies developed from a bilateral, "Hawaiian" type of social structure, characterized by Generation-Hawaiian kinship terminology, bilateral kindreds, bilocal extended families, a bilateral extension of incest taboos, and the absence of unilineal descent groups. By implication, this type is unstratified. In the case of Micronesia, Murdock assumes that the development of social stratification was contingent upon the transition from bilateral Hawaiian to matrilineal "Iroquois" and "Crow" types of social structure and thence on patrilineal modifications in residence, inheritance, and succession characteristic of petty feudal states. Thus, for example, whereas we view Trukese society as an egalitarian (tribal) derivative of a stratified Proto-Nuclear Micronesian (PNM) society, Murdock views it as the "germinal form" of all the stratified (feudal) societies in Micronesia. A number of Oceanists, most notably Goodenough, have accepted Murdock's reconstruction of Micronesian and more generally MP social organization as a virtual demonstration: That the Hawaiian type of kinship system was ancestral not only to the system now found in Truk but to those found throughout Micronesia has been demonstrated by Murdock (1948) (Goodenough 1951:95).

Digraphs

223

Despite the wide differences in the social systems which now exist among Malayo-Polynesian societies, Murdock (1948; 1949:228-31, 349-50) offers convincing evidence that they are derived from an original "Hawaiian" type of structure (Goodenough 1951:71). Goodenough proposed a major amendment to Murdock's Hawaiian type by adding as a basic feature cognatic landowning descent groups but assumed with Murdock that the development of more complex social systems depended on the emergence of unilineal descent groups. In archaeology, Bellwood (1978:113) entertains Austronesian (AN) (Murdock's MP) z/r-bilaterality as a working hypothesis, assuming the following sequence of events: 1. The settlement by 30,000 years ago of Island Southeast Asia, Australia, and western Melanesia by non-Austronesian (NAN) speakers with a "patrilineal form of social structure." 2. The presence in Island Southeast Asia 5,000 years ago of AN speakers with a "bilateral form of social structure." 3. An expansion of AN speakers into Oceania and the development of "unilineal and ambilineal forms of social structure" as a result of contact with NAN (Papuan) societies and/or population pressure on small islands. (We note that Goodenough uses a similar demographic argument to infer cognatic descent in PMP society. See Chapter 2.) The alternative to Murdock's bilateral hypothesis is Blust's (1980) "asymmetric connubium" reconstruction of early AN social organization. Blust's reconstruction is inspired by van Wouden's (1968[1935]) comparative study of clan organization and matrilateral cross-cousin marriage in eastern Indonesia. Blust assumes, as do many Dutch scholars, that matrilateral cross-cousin marriage was once widespread in Indonesia, and he agrees with Levi-Strauss (1969) that much of the Austronesian world should be viewed as an eastward extension of an ancient "Sino-Tibetan axis of generalized exchange." As evidence for the presence of generalized exchange (marriage with the MBD) and associated unilineal descent groups in Proto-Austronesian (PAN) or at least in PMP society, Blust adduces a number of linguistic reconstructions and statistical correlations. These include: (1) the reconstruction of a parallel-cross distinction in PMP sibling terminology,1 a feature that, as Murdock (1968) has shown, is negatively associated with bilateral kindreds 1 This is an extension of Milke's (1938) reconstruction of POC sibling terminology, described in the next section.

224

Island networks

and strongly associated with ambilineal, matrilineal, and double descent groups; (2) the reconstruction of a PMP term, *ma(n)tuqa, meaning "mother's brother/wife's father," which implies both unilineal descent and MBD marriage; and (3) the reconstruction of a PAN term *aya, meaning "father's sister," which, since it has no evident application to affines, that is, to mother-in-law, implies asymmetric rather than symmetric cross-cousin marriage. Further evidence is provided by "crosssibling substitution drifts" (Blust 1993). This refers to the process whereby terms for male/female and child + male/female, which originally applied to wife-giving and wife-taking groups, were later applied to cross-siblings, replacing the reflexes of PMP *naRa = B(w.s.) and *betaw = Z(m.s.). It is inferred that at the time of replacement the societies in question practiced matrilateral cross-cousin marriage. The process is called "drift" because it occurred independently in different languages, motivated by structural pressures in social organization. In MP societies that have these terms but that do not practice MBD marriage, the terms are interpreted as a reflection of an earlier such practice. In support of his reconstruction, Blust argues that the "bilateral hypothesis" is not supported by the linguistic evidence or by Murdock's own model. In order to explicate Murdock's model and Blust's interpretation of it, let us put the matter in graph theoretic terms. In Murdock's evolutionary model the social structure of every society can be classified by type, and all types can be derived from other types in specified sequences of transitions. On the basis of descent rules and kinship terminology, Murdock defines 11 primary types of social structure: Eskimo: Bilateral; Eskimo Hawaiian: Bilateral; Hawaiian Yuman: Bilateral; Iroquois Fox: Bilateral, Patrilineal; Crow, Omaha, Sudanese Guinea: Patrilineal; Eskimo, Hawaiian Dakota: Patrilineal; Iroquois Sudanese: Patrilineal; Sudanese Omaha: Patrilineal; Omaha Nankanse: Matrilineal, Double; Eskimo, Hawaiian Iroquois: Matrilineal, Double; Iroquois Crow: Matrilineal, Double; Crow, Omaha, Sudanese Of these 11 types, Eskimo and Hawaiian are stable bilateral structures, Dakota and Omaha are stable patrilineal structures, and Iroquois and Crow are stable matrilineal structures. The alternative of double descent in Iroquois and Crow indicates that matrilineal structures are less stable than patrilineal ones.

Digraphs

225

The 11 primary types are divided into 47 subtypes. The "Normal" subtypes are those from which the others are derived. The "stable" structures are more precisely defined as those in which the rules of residence are consistent with the rules of descent and the kinship terminology: Normal Eskimo is neolocal, Normal Hawaiian is bilocal, Normal Dakota and Omaha are patrilocal, and Normal Iroquois and Crow are matrilocal. The other subtypes are defined by their "deviant rules of residence, e.g., Patri-Eskimo, Matri-Hawaiian, Bi-Dakota, Neo-Omaha, and Avuncu-Crow or their atypical rules of descent, e.g., Patri-Fox and Duo Iroquois" (Murdock 1949:225). The "normal order of change" consists of one-step transitions beginning with a change in residence followed by changes in descent and kinship terminology. For example, a Normal Hawaiian society becomes Matri-Hawaiian with a shift to matrilocal residence, and thence Normal Nankanse with a shift to matrilineal descent, and either Iroquois or Crow with a change to an appropriate matrilineal kinship terminology. This would represent the sequence of events in the evolution of Marshallese and Trukese societies in Micronesia. Murdock regards his classification as "a maze in which a society can start at any given point and arrive at any other point whatsoever but only by a limited number of possible routes" (1949:221). By means of this maze one can trace out the prehistory of individual societies. To determine the ancestral form of societies belonging to the same linguistic stock it is only necessary to find the single subtype in the past toward which they all converge. According to Murdock, the 40 Malayo-Polynesian societies in his worldwide sample of 250 societies converge toward Normal Hawaiian providing "overwhelming support for the hypothesis that the original Malayo-Polynesian speech community had a social organization of Hawaiian type" (1949:350). Murdock regarded this particular result as the "most striking confirmation of the method." Blust, however, finds no significant support for this conclusion. To understand why, we need to distinguish between the long and short forms of Murdock's method. In the long form, a society is first classified by subtype. If this subtype can be derived from only one other subtype, then the latter is its antecedent form. If the subtype has alternative derivations, the "most probable" antecedent is determined by the presence of survivals in various secondary features of social structure. For example, a bilateral extension of incest taboos in a unilineal, unilocal structure indicates a bilateral antecedent, and Generation-Hawaiian terms for aunts and nieces indicates a Normal Hawaiian antecedent. This is presumably the reason why Trukese society, which is classified as Normal Crow (a stable structure) is derived from Normal Hawaiian. In essence one keeps working backward toward simpler forms. Thus it is not surprising that of the 13

226

Island networks

linguistic stocks about whose reconstruction Murdock feels confident, 6 have a Normal Hawaiian derivation. As Levi-Strauss (1963:307) observed, Murdock's method "proceeds by extracting common elements pertaining to each stage, in order to define a previous stage, and so on. Therefore, it is obvious that systems placed at the beginning can be only those which exhibit the more general features, while systems with special features must occupy a more recent rank. It is as though the origin of the modern horse were ascribed to the order of vertebrates instead of to Hipparion." This being the case, we cannot agree with Blust, who suggests that it would be "useful to reexamine the Austronesian evidence using Murdock's method of survivals to determine whether he was justified in claiming that the derivation of attested Austronesian speaking societies from an earlier Hawaiian structure is significantly simpler than derivations from other types of structures" (1980:225). In the short form of Murdock's method, which is the procedure followed by Blust, and earlier by de Lint and Cohen (1960), only the transitions between subtypes are considered in determining ancestral forms. Blust traces out the specific sequences of transitions for all 40 of the Malayo-Polynesian societies in Murdock's sample from Normal Hawaiian and Normal Iroquois and finds that the derivational simplicity of the former is not significantly greater than that of the latter. We can simplify Blust's procedure, check his result with another version of Murdock's classification, and give a general solution for solving a Murdockian maze by using a breadth-first search tree (BFST) as defined in Chapter 4. Murdock's maze is implicitly a digraph whose node set consists of the 47 subtypes of social structure and whose arcs consist of one-step transitions between them. Murdock gives two different synoptic presentations of his maze. In Table 73 of Social Structure he lists the transitions from each subtype to other subtypes, while in Appendix A he lists the antecedents of each subtype. For some unexplained reason, probably an oversight, these two mazes are not identical - the converse of the digraph implicit in Appendix A is not isomorphic to the digraph in Table 73. Blust uses Appendix A, but we will use Table 73, which may be regarded as the canonical version of Murdock's model, since it gives the most systematic account of transitions by residence, descent, and kinship terminology.2 This digraph is shown in Table 8.1 by listing its arcs. Murdock's assertion that a society can start at any given subtype in his maze and arrive at any other subtype means that its digraph is A third version at the bottom of Tables 61-8 of Social Structure, which lists derivations and permissible changes of each subtype, is not isomorphic to either of the digraphs in Table 73 or Appendix A.

Digraphs

227

Table 8.1. A digraph (list of arcs) ofMur dock's (1949) evolutionary model of social organization 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47.

NOR-ESK: BI-ESK: MAT-ESK: PAT-ESK: NOR-HAW: NEO-HAW: MAT-HAW: PAT-HAW: NOR-YUM: BI-YUM: NEO-YUM: MAT-YUM: NOR-FOX: BI-FOX: NEO-FOX: MAT-FOX: PAT-FOX: NOR-GUI: BI-GUI: NEO-GUI: NOR-DAK: BI-DAK: NEO-DAK: NOR-SUD: BI-SUD: NEO-SUD: NOR-OMA: BI-OMA: NEO-OMA: NOR-NAN: AVU-NAN: BI-NAN: NEO-NAN: PAT-NAN: DUO-NAN: NOR-IRO: AVU-IRO: BI-IRO: NEO-IRO: PAT-IRO: DUO-IRO: NOR-CRO: AVU-CRO: BI-CRO: NEO-CRO: PAT-CRO: DUO-CRO:

MAT-ESK, PAT-ESK NOR-ESK, MAT-ESK, PAT-ESK, NOR-HAW BI-ESK, PAT-ESK, NOR-NAN, MAT-YUM NOR-ESK, BI-ESK, NOR-GUI, NOR-YUM NEO-HAW, MAT-HAW, PAT-HAW MAT-HAW, PAT-HAW, NOR-ESK NOR-HAW, PAT-HAW, NOR-NAN, MAT-YUM NOR-HAW, NEO-HAW, NOR-GUI, NOR-YUM NOR-DAK NOR-YUM, MAT-YUM, NOR-HAW NOR-YUM, NOR-ESK NOR-IRO PAT-FOX, NOR-SUD, NOR-OMA NOR-FOX, MAT-FOX, NOR-HAW NOR-FOX, NOR-ESK NOR-CRO NOR-SUD, NOR-OMA NOR-DAK, NOR-SUD, NOR-OMA NOR-HAW NOR-ESK BI-DAK, NEO-DAK, NOR-SUD, NOR-OMA BI-YUM, BI-GUI NEO-YUM, NEO-GUI BI-SUD, NEO-SUD, NOR-OMA BI-FOX, BI-GUI NEO-FOX, NEO-GUI BI-OMA, NEO-OMA BI-FOX, BI-GUI NEO-FOX, NEO-GUI AVU-NAN, BI-NAN, NEO-NAN, PAT-NAN, NOR-IRO, NOR-CRO NEO-NAN, PAT-NAN, AVU-IRO, AVU-CRO PAT-NAN, NOR-HAW PAT-NAN, NOR-ESK PAT-ESK, PAT-HAW, NOR-GUI, DUO-NAN, PAT-IRO NOR-GUI, DUO-IRO AVU-IRO, BI-IRO, NEO-IRO, PAT-IRO, NOR-CRO NEO-IRO, PAT-IRO, AVU-CRO PAT-IRO, BI-YUM, BI-NAN PAT-IRO, NEO-YUM, NEO-NAN NOR-YUM, NOR-DAK, DUO-IRO NOR-DAK, DUO-CRO AVU-CRO, BI-CRO, NEO-CRO, PAT-CRO NEO-CRO, PAT-CRO PAT-CRO, BI-FOX, BI-NAN PAT-CRO, NEO-FOX, NEO-NAN NOR-FOX, PAT-FOX, DUO-CRO PAT-FOX, NOR-SUD, NOR-OMA, DUO-IRO

Note: "Rare or exceptional" changes marked by an asterisk in Murdock are omitted. Abbreviations: NOR = Normal; BI = Bi; MAT = Matri; PAT = Patri; NEO = Neo; AVU = Avuncu; DUO = Duo; Esk = Eskimo; HAW = Hawaiian; YUM = Yuman; FOX = Fox; GUI = Guinea; DAK = Dakota; SUD = Sudanese; OMA = Omaha; NAN = Nankanse; IRO = Iroquois; CRO = Crow. NOR-ESK = Normal Eskimo; BI=ESK = Bi-Eskimo; MATESK = Matri-Eskimo, and so on.

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Table 8.2. Murdoch's (1949) classification of Malay o-Polynesian societies PAT-ESK: NOR-HAW: NEO-HAW: PAT-HAW: NOR-YUM: MAT-YUM: BI-FOX: NOR-GUI: NOR-DAK: NOR-SUD: PAT-NAN: DUO-NAN: NOR-IRO: BI-IRO: PAT-IRO: DUO-IRO: NOR-CRO: AVU-CRO: DUO-CRO:

Balinese Eddystone, Ifugao, Maori, Ontong Javanese, Samoans Marquesans Futunans, Hawaiians, Mangarevans, Tongans, Ulawans Eromangans Mentaweians Tokelau Tikopia Epi, Fijians, Tanala, Tannese Batak, Mailu Tetekantzi Pukapukans Arosi, Dobuans, Lesu, Marshallese, Minangkabau, Nauruans Kurtatchi Getmatta, Tismulun Wogeo Trukese Mota, Trobrianders Manus, Pentecost, Ranon

strongly connected. This assertion is true, as we will see, for the digraph D in Table 73 but not for the one in Appendix A of Social Structure, which contains three subtypes - Patri-Fox, Neo-Guinea, and NeoNankanse - which, once they are reached, cannot reach any other subtypes, that is, three nodes that are receivers of D. This is another reason for using Table 73 rather than Appendix A. Murdock's classification of the 40 Malayo-Polynesian societies in his sample is given in Table 8.2. In the digraph of Murdock's maze the "point of convergence" - or better, divergence - of a set of societies in the same linguistic stock or other such group is the node that can reach them all in the least total distance. To find this point we treat each node of D as the root of a BFST of D. In effect we regard each node of D as a potential ancestor. The following algorithm gives the method. ALGORITHM

8.1. Construction of a BFST of a digraph

Given. A strongly connected digraph D. Wanted. A BFST of D to obtain the distance from one node (the root) to all other nodes of D. Step 1. Choose a node u0 of D as the root of a BFST of D. Step 2. Go through the set of remaining nodes and join u0 to all the nodes u1 adjacent from u0. Step 3. Go through the set of remaining nodes and join u1 to all nodes u2 adjacent from uu using each u2-node exactly once.

Digraphs

229

BIYUM NEO YUM BIFOX

MAT FOX

NEO FOX NOR FOX PAT FOX DUO CRO

BIGUI NEO GUI

Figure 8.5. A BFST of Murdock's (1949) maze with Normal Hawaiian as the root.

Step 4. If all the nodes of D have been visited, stop. Otherwise continue the process in step 3. To illustrate with the digraph in Table 8.1, we take Normal Hawaiian as the root u0 of a BFST. We join Normal Hawaiian to the nodes to which it is adjacent, namely, Neo-, Matri-, and Patri-Hawaiian. We then join Neo-Hawaiian to Normal Eskimo, and so on. BFSTs with Normal Hawaiian and Normal Iroquois are shown in Figs. 8.5 and 8.6. We note first of all that the root nodes of the BFSTs in Figs. 8.5 and 8.6 can reach every other node in D. The same is true of every node in

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Island networks

- NOR NAN — AVU NAN

PATESK NOR GUI DUO NAN

- BI CRO

-

• NEOCRO • PAT CRO =

• BIFOX . NEOFOX - NOR FOX PAT FOX

Figure 8.6. A BFST of Murdock's (1949) maze with Normal Iroquois as the root. this digraph when it serves as the root of a BFST. In effect the distance from the root of a BFST to every other node gives its row entries in the distance matrix of a digraph (or graph). The distance matrix of the digraph in Murdock's Table 73 (not shown) has only finite entries. Hence its reachability matrix is universal, and it is therefore strongly connected, just as Murdock implies. This is established by the following result from Harary et al. (1965). THEOREM 8.3. A digraph D is strong if and only if its reachability matrix is universal, that is, R = J.

The total distance from Normal Hawaiian to all Malayo-Polynesian societies is 106, but from Normal Iroquois it is only 95. This compares with distances of 114 and 117 that Blust found using Appendix A of Social Structure. If we treat Table 73 as the authoritative version of Murdock's digraph, we should conclude that Normal Iroquois rather than

Digraphs

231

Normal Hawaiian was the ancestral form of MP society. This is an interesting result in view of Blust's (1980:22) opinion that "the linguistic evidence suggests that early Austronesian society can probably best be characterized in Murdock's typology as Iroquois." Consistent with this interpretation, the distance from Normal Nankanse, an "incipient and transient type of matrilineal organization," which leads directly to Normal Iroquois but not to Normal Hawaiian, is only 88, making it an even likelier ancestral form. We conclude that neither the long nor the short form of Murdock's method of historical reconstruction provides any support for the Hawaiian (bilateral) derivation of Malayo-Polynesian social organization.3 There are three Nuclear Micronesian societies in Murdock's sample: Marshallese, Nauruan, and Trukese. The first two are Normal Iroquois, and the third is Normal Crow. Using Murdock's short method, Normal Iroquois must be the ancestral form. We conclude that there is no reason to assume a Hawaiian (bilateral) origin of Micronesian society and therefore no objection on this score to our reconstruction of Proto-Nuclear Micronesian (PNM) society in Chapter 5. As noted in Chapter 1, the languages of Nuclear Micronesia exhibit greater genetic diversity than those of Polynesia, suggesting an earlier time of settlement. The Iroquoian derivation of PNM society would thus be consistent with the Iroquoian derivation of PMP society. In our view, Murdock's typology has limited relevance to the study of AN social organization. It allots no place to "preferential marriage customs," whose effects on social structure are considered to be "distinctly secondary" (Murdock 1949:222), and, as Service (1985) has emphasized, it ignores the difference between ranked and egalitarian descent groups. As Blust and Friedman have argued in their connubium and prestige-good system models, and as we have shown in Chapters 4 and 5, asymmetric marriage alliance and kinship rank are essential features of many Oceanic societies.

Sibling classification and culture history in Island Oceania Marshall (1984) has proposed a formal evolutionary model of sibling classification in Island Oceania based solely on geographical, statistical, and structural considerations. The model, which derives from the cross3 If one accepts the result of Murdock's short method and then insists on using the long method to work farther back, one would have to concede that all MP societies were Iroquois or Nankanse at one stage in their history.

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Island networks

cultural research of Murdock (1968) and the semantic analysis of Epling, Kirk, and Boyd (1973), is controversial because it classifies together terminologies from genetically unrelated AN and NAN languages and controverts the widely acceped linguistic reconstruction of POC sibling terminology of Milke (1938). It also disregards functional determinants that might account for differences between sibling terminologies and relies mainly on diffusion as an agent of change. In order to evaluate Marshall's model and present the alternative implicit in Milke's reconstruction, it will be helpful to give a joint historical and structural account of anthropological and linguistic research on Oceanic sibling terms. The analysis of sibling classification has a long and interesting history in both Oceania and kinship studies, involving problems of comparison, interpretation, notation, and reconstruction. In his classic work The Melanesians, Codrington (1891) identified two unusual features of sibling terms found in many of these societies: The terms equivalent to brother and sister are used on a different principle from that with which we are familiar, and according to which the sex of the person referred to determines the use of the word. In Melanesia, as elsewhere, one word describes the relationship of persons of the same sex, and the other word describes the relationship of persons of different sexes. Men are tasiu to men, and women tasiu to women; men are tutuai to women, and women tutuai to men. There is a further difference, the sex being the same, the elder man or woman is tugui to the younger, the younger man or woman is tasiu to the elder; but tasiu is the prevailing use (1891:36). These two features, relative sex (referred to as "parity" in modern terminology, with "parallel" for same sex and "cross" for opposite sex) and relative age (seniority) were regarded by Rivers (1914a) as distinctive features of the "classificatory" system of kinship in Melanesia. Rivers explained parity as a consequence of clan exogamy and noted the restriction of seniority to parallel siblings. Most anthropologists, from Firth (1936) to Murdock (1968), emphasize some social or psychological feature of cross-sex relations or some rule of descent as determinants of the parity relation. Virtually all Oceanists, except for Panoff (1965), interpret the seniority distinction as an expression of social rank.4

In a comparative study of three Polynesian kinship terminologies - Samoan, Futunan, and Tahitian - Panoff treats seniority, quite idiosyncratically, as an expression not of rank but of the "educational relationship" between senior and junior siblings. He also muddies the analytical waters by treating residence as a distinctive feature of these kinship terminologies.

Digraphs

233

The number of possible sibling terminologies is large, and their description in ordinary English cumbersome. In an elegant anthropological study in applied combinatorics, Nerlove and Romney (1967) counted the number of logically possible types of sibling terminologies and generated on purely deductive grounds a subset of "ideal" - that is, empirically probable - types. Given the three binary dimensions of parity (parallel vs. cross), seniority (elder vs. younger) and sex (male vs. female), there are eight different sibling kin types: el

el el

el yg yg yg yg

Br Br Si Si Br Br Si Si

One sibling term will partition the eight kin types in 1 way (the limiting case), two sibling terms in 127 ways, three sibling terms in 966 ways, and so forth. In general, the problem of counting the number of ways in which the eight kin types can be partitioned "may be stated as the number of ways of separating m different objects (the eight kin types) into k non-empty batches (the number of kin types, i.e. one to eight), or the number of ways of separating a set of m things into k nonempty sets (Niven 1965:112-13)" (Nerlove and Romney 1967:180). The number of logically possible sibling terminologies is an astounding 4,140. In Nerlove and Romney's classification of sibling terminologies, a primary distinction partitions the eight kin types into two subsets of four; a secondary distinction partitions only one subset of a primary partition; and a tertiary distinction partitions a secondary distinction, or one of the subsets produced by the intersection of two primary distinctions. On the basis of cognitive economy and universals in marking, Nerlove and Romney predicted that only 12 types of sibling terminologies would actually occur. Cognitive economy refers primarily to the avoidance of disjunctive categories, that is, categories which are defined by different combinations rather than by the joint presence of attributes (Bruner, Goodnow, and Austin 1956). In general, disjunctive categories contain three, five, six, or seven of the eight kin types. An example of a disjunctive category would be one that classifies together x el Br, || el Br, x el Si, x yg Br, || yg Br, and x yg Si. The vast majority of the 4,140 logically possible sibling terminologies contain disjunctive categories. Cognitive economy also refers to the avoidance of using the two relational

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Island networks

attributes of parity and seniority as primary distinctions, evidently on the grounds that relational attributes are more difficult to learn than criterial attributes such as sex.5 Nerlove and Romney assumed, on the basis of Greenberg's (1965) study of language universals, that unmarked categories - male as opposed to female and elder as opposed to younger - are more subject to differentiation than marked categories. Thus if (1) sex is secondary to seniority, it will partition the elder category; (2) if age is secondary to sex, it will partition the male category; (3) if age is secondary to parity, it will partition the parallel category. The third prediction implies that parallel, as opposed to cross, is also an unmarked category. Marking rules, as we will see, play an important role in interpreting the evolution of sibling terminologies. To represent sibling terminologies, Nerlove and Romney used box diagrams partitioned to show distinctions of sex, seniority, and parity. These diagrams, which were subsequently refined by Epling, Kirk, and Boyd and adopted by Marshall, are structurally equivalent to nested-set representations of twin binary trees as defined in Chapter 2. For convenience of exposition, we will use graphical rather than nested-set models. Nerlove and Romney's 12 ideal types of sibling terminologies are shown in Fig. 8.7. The root of each tree is labeled "S" for sibling. A terminology with no distinctions, and therefore having a single term, consists of this single node. In graph theory such a tree is called a degenerate tree, an evolutionarily appropriate designation for the corresponding type of sibling terminology in Oceania. The other nodes are labeled "m" and "f" for male and female, "e" and "y" for elder and younger, and "||" and "x" for parallel and cross. The individual endnodes of each tree represent the categories of a sibling terminology. Type 11, for example, has terms for "cross-sibling," "elder parallel sibling," and "younger parallel sibling" and represents the Melanesian type identified by Codrington. A disjunctive category, such as the example given earlier, can be defined as a category that cannot be represented by a single endnode of a twin binary tree. In a sample of 245 sibling terminologies, Nerlove and Romney found that 214 cases correspond to the types in Fig. 8.7. The exceptions consist of 21 "derivative" types, that is, those in which a tertiary distinction is present in addition to the primary and secondary distinctions defining an ideal type; 5 "reversals," that is, cases in which a marking assumption is reversed; 1 unique case; and 4 disjunctive terminologies. All but the rarest cases in Marshall's Island Oceanic sample of sibling terminologies are included in Fig. 8.7. Nerlove and Romney also state that parity will not occur as a primary distinction in a sibling terminology, but this is contradicted by types 7-12 in their table, as shown in Fig. 8.7.

Digraphs Typel

235 Type 4

Type 2

S

S

Type 5

Type 6

Type 7

Type 8

S

S

S

S

y

m

Figure 8.7. Nerlove and Romney's (1967) ideal types of sibling terminologies. In searching for functional determinants of sibling terminologies, Nerlove and Romney found a correlation between terminologies in which parity is a primary distinction (types 7-12 in Fig. 8.7) and a psychological syndrome of special concern with cross-sex relations and sex identity whose antecedent is a long postpartum sex taboo. The next step in the analysis of sibling terms was taken by Murdock (1968), whose work was to provide important cross-cultural evidence for Blust's model of early AN social organization and distributional evidence for Marshall's model of Island Oceanic sibling classification. Inspired by the kinship classifications of Lowie (1928) and Kirchhoff (1932) for avuncular and nepotic terms, and Spier (1925) and himself (Murdock 1949) for cousin terms, Murdock, after analyzing a worldwide sample of 800 societies, proposed a typology of seven basic patterns of sibling terms. Six of these patterns correspond to ideal types in Nerlove and Romney's typology. Murdock's Kordofanian or Undifferentiated Sibling type, Yoruba or Relative Age type, Algonkian or Skewed

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Island networks

Age type, Dravidian or Age-Sex type, and European or Brother-Sister type correspond to Nerlove and Romney's types 1, 4, 5, 6, and 2 respectively. Murdock's Melanesian or Relative Sex type includes four subtypes that correspond to Nerlove and Romney's types 7 through 10. All four of these subtypes are based on the parity distinction. Curiously, Murdock chose to disregard the seniority ("partial age") distinction emphasized by Codrington and Rivers and treated Nerlove and Romney's types 11 and 12 as "minor variants" of the Melanesian type. This may be explicable in terms of Murdock's disregard of the distinction between ranked and unranked kinship groups that is so vital to understanding the social organization of many Oceanic societies.6 Murdock's seventh type, the Siouan or Complexly Differentiated type, uses all three distinctions of sex, seniority, and parity and contains at least six terms. It would be included in Nerlove and Romney's derivative types. Their type 3 is treated as a minor variant of the Yoruba type. Murdock found that the "distribution of sibling terminologies follows very closely the boundaries of known linguistic divisions, especially language families and subfamilies" (1968:5). In Oceania these boundaries correspond to the traditional division into Polynesia, Micronesia, Melanesia, and Indonesia. He concluded from his survey that sibling terms are remarkably stable, "with a tendency to persist often for thousands of years among groups of people who speak related languages" (1968:5). Under the heading of functional determinants of sibling terminologies, Murdock pointed to kinship variables, in particular rules of descent. Matrilineal, double, and ambilineal descent, for example, are "conducive" to Melanesian relative sex terminologies whereas bilateral kindreds are not, as we mentioned in our discussion of Blust's reconstruction of early AN social organization. The first comparative study of sibling terms in Polynesia was by Firth (1936, 1970), who defined a taxonomy, mapped its distribution, and raised the question of transitions between types. Firth identified four different types of sibling terminologies, all of which are included in the Nerlove and Romney typology. In order of increasing complexity, they are: (1) the Kapinga type, which has a single term for "sibling"; (2) the Tikopia type, which distinguishes parallel from cross-siblings; (3) the Tokelau type, which distinguishes male from female cross-siblings; and As Service (1985:121-2) has observed in his historical analysis of the "clan-gens problem," "G. P. Murdock, in his comprehensive work Social Structure (1949), shows appreciation of the various forms social groups can take but nowhere does he show interest in the difference between ranked and egalitarian types, even though his discussion of consanguineal groups is a thorough mixture of references to societies of both types. This is not said to castigate Murdock, but to call attention to those times - and Murdock was a leader in those times."

Digraphs

237

Tikopia

Kapinga

S

S o tuahin(a) taina

kave m tuagane

Nukuoro

Futuna Vaitupu Ontong Java

Pukapuka

yo o m

of

teina tungane tuahine

Hawaii, Marquesas, Tahiti, Tuamotu, Tubuai, Niue, Rarotonga, Manihiki, Mangaia, Tongareva, Tonga, Samoa, Uvea, Rennell, Bellona

Figure 8.8. Polynesian sibling terminologies identified by Firth (1970). (4) the Maori type, which distinguishes elder from younger parallel siblings and male from female cross-siblings. Firth's diagrammatic representation of these types can be greatly simplified by using the twin binary tree notation in Fig. 8.8. Below each type are listed the other Polynesian societies where it is found. As is evident from Fig. 8.8, the Maori type is by far the most common in Polynesia. It corresponds to the "Core Polynesian" type in Marshall's classification and the wr-type in Milke's reconstruction of POC sibling terminology. The restriction of seniority to parallel siblings in the Maori type is consistent with Nerlove and Romney's assumptions regarding marking (there are no terminologies in Fig. 8.7 in which seniority is restricted to cross-siblings), and it may also be consistent with social constraints that make it practicable as well as thinkable. In his monograph on Manihiki and Rakahanga, P. H. Buck (Te Rangi Hiroa) (1932) conjectured that this restriction eliminates challenges to male succession to rank: The restriction of tuakana and teina to members of the same sex is to prevent their use between opposite sexes. A logical reason for this usage would be to prevent some danger. The danger, as I see it, was to the male succession to rank and title through seniority of birth. No danger could arise from the use of terms among sisters to denote their relative positions in the female sphere of activity. If, however, a sister were termed tuakana to a younger brother, her seniority to him in the family would be admitted, regardless of sex. A first-born female would be tuakana to the rest of the family, and her claims would be hard to combat. It seems plausible, therefore, that those who guided the evolution of social structure provided against such a contingency (Buck 1932:34).

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Firth interpreted the distinctions in Polynesian sibling terminologies as reflections of social institutions: parity reflects the brother-sister tapu and the "place of male and female lines in the transmission of group interests" (1936:280), and seniority reflects the emphasis on rank (primogeniture). In accounting for the simpler terminologies found in the smaller, more remote, less populous societies, Firth proposed that "the smaller the community in numbers and in geographical circumscription, the more face-to-face contact is likely to obtain and the less need for more complex forms of social identification" (1970:275). Firth also posed the really interesting evolutionary question: Granted the probable effect of environmental and demographic factors in the situation, a number of puzzling questions remain. There is the formal evolutionary problem: assuming all these Polynesian systems are variants on a common theme, are those of Tikopia and its congeners to be regarded as denuded versions of a richer prototype; or are they a basic version, and those of the Maori and analagous societies to be seen as products of cultural efflorescence; or are they all independent developments from an early simple base - each a separate response to diversification of environmental and differential growth of population and fission of social groups? (Firth 1970:275). Although Firth thought that any proposed general solution would be "debatable and speculative," the problem was addressed by Epling, Kirk, and Boyd (1973), solved by Clark (1975), generalized by Marshall (1984), and generalized and solved (before it had even been posed) by Milke (1938). In developing a solution to Firth's problem, Epling, Kirk, and Boyd constructed a lattice model of 146 "evolvable" sibling terminologies. The model shows how conjunctive sibling terminologies evolve through successive binary partition, that is, through the successive addition or deletion of the three binary dimensions of sex, seniority, and parity. Expanding Firth's sample of Polynesian societies from 16 to 23, the researchers found five types of sibling terminologies, which they arranged in an "upper-semilattice" as shown in Fig. 8.9 (using twin binary trees rather than their box diagrams). In Fig. 8.9, types 1, 2, 3, and 4 correspond to Firth's Kapinga, Tikopia, Tokelau, and Maori types respectively. Type 7 represents Epling, Kirk, and Boyd's Samoan type, which is different from the classification of Firth and everyone else.7 Types 5 and 6 are not found any7 With the exception of their Samoan type, their classification of Polynesian sibling terminologies agrees with Firth's classification of the same languages.

Digraphs

239 (i)

s

(2)

y

m

f

Figure 8.9. A graph of Epling, Kirk, and Boyd's (1973) "upper-semilattice of [Polynesian] sibling terminologies showing the two reconstructed evolutionary chains."

where in Polynesia, even though they are (theoretically) necessary precursors to type 7. Epling and his colleagues argued that type 5 is reconstructable for early nineteenth-century Marquesan, and they predicted that type 6 would be reconstructable for some society in Polynesia. In interpreting the semilattice in Fig. 8.9, they defined a general trend of increasing terminological "refinement" as well as subsequent "loss," or "denudation," to use Firth's terms, in the Polynesian outliers of Kapinga, Nukuoro, and Tikopia. According to them, 22 of the 23 Polynesian sibling terminologies studied correspond to types 1, 2, 3, or 4. The addition of the Samoan case and its hypothetical precursors results in a semilattice showing "two distinct lines or chains of evolution" (1973:1612-13).

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Island networks

Epling, Kirk, and Boyd accounted for the general trend from simple to complex terminologies in terms of Sahlins's (1958) evolutionary model of social stratification in Polynesia.8 They proposed that "the amount of information H, of a sibling terminology, is positively correlated with the degree of social differentiation and degree of (emphasis on) social ranking" (1973:1615). In Sahlins's sample of 14 Polynesian societies, Epling and his fellow researchers found that complex and simple sibling terminologies are associated, respectively, with societies exhibiting "high" and "low" degrees of social stratification. There are three objections to their model. 1. The classification of Samoan terminology is evidently incorrect. As Firth (1970) has shown in his reading of Mead (1930a), the term they identify for "younger sibling" is in fact merely a "domestic sibling term" based on residence and supplementary to referential terms. Samoan terminology corresponds to either the Maori (Firth 1970) or the Tokelau (Clark 1975) type. 2. The reconstruction of early nineteenth-century Marquesan terminology is suspect. As Clark (1975) has shown, it is based on isolated reports (Des Vergnes in Williamson 1924) at variance with early linguistic and later anthropological evidence (Crook 1799; Handy 1923). Further, as Clark has pointed out, the Des Vergnes system actually has four terms, based on seniority and parity. Even if it could be documented, it would not support Epling, Kirk, and Boyd's evolutionary model of Polynesian sibling terminology. Neither their Samoan type nor its immediate precursor occurs in Nerlove and Romney's typology, which suggests that these terminologies, while not impossible, are improbable. 3. The inferred evolutionary trend toward terminological complexity (refinement) is contradicted by the linguistic evidence, as we will next see. In his comment on their work, Clark (1975) essentially solved the problem of the evolution of Polynesian sibling terminologies. Applying the traditional methods of historical linguistics to 30 Polynesian languages, Clark reconstructed four sibling terms for Proto-Polynesian (PPN): 1. 2. 3. 4.

*tuakana *tahina or *tehina *tuangafane *tuafafine

"elder sibling, same sex" "younger sibling, same sex" "woman's brother" "man's sister"

This terminology corresponds to the Maori pattern of Firth and to Epling, Kirk, and Boyd's type 4.9 Clark argued that "all developments in 8 See n. 9, Chapter 4. 9 With the exception of Samoan, Clark's classification agrees with Firth's classification of the same languages.

Digraphs

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the attested languages can be accounted for in terms of two processes: (1) replacement of terms by forms other than the original four; (2) elimination of semantic distinctions" (1975:86). In Clark's model of successive binary partition, the dimensions of seniority, sex, and parity are eliminated in that order, just the converse of the evolutionary sequence implied in the right-hand side of Epling, Kirk, and Boyd's model in Fig. 8.9. Clark also pointed out an interesting pattern of "semantic survival" in which the younger, male, and parallel terms replaced the elder, female, and cross terms: there appears to be a fairly consistent pattern of survival, when one term extends to a larger category and ousts another. Thus *tahina replaces *tuakana; * tuanga'ane replaces *tuafafine; and *tahina replaces ^tuanga'ane, so that both Nukuoro and Easter Island which have (independently) reduced to one term systems, show reflexes of *tahina for this single term. An explanation for this might be sought in terms of semantic markedness (1975:86). If replacement is treated as the diachronic equivalent of neutralization that is, the leveling of oppositions - then the unmarked categories in Polynesian sibling classification are "younger," "male," and "parallel" (R. Blust, personal communication). The second and third replacements are consistent with the marking assumptions of Nerlove and Romney, but the first is not. A digraph model of Clark's reconstruction, showing this pattern of linguistic change, is given in Fig. 8.10. Clark's reconstruction seemed to solve the puzzle of Polynesian sibling terminologies. Nine years later, however, Marshall (1984), in an ethnological tour de force, developed an evolutionary model of sibling classification in Island Oceania that inverts Clark's Polynesian sequence as well as Milke's reconstruction of POC sibling terms. Marshall's model is based on Murdock's demonstration that the distribution of sibling terminologies coincides with traditionally recognized linguistic divisions in this case the traditional division of Oceania into Melanesia, Micronesia, and Polynesia - and on Epling, Kirk, and Boyd's assumption that sibling terminologies evolve through successive binary partition including the addition as well as the deletion of semantic features. Marshall ignores genetic and lexical relationships and classifies all terminologies, AN and NAN, solely on the basis of their semantic structure. He also rejects, in a very un-Murdockian way, the functionalist explanations offered by Firth and by Epling, Kirk, and Boyd by adducing counterexamples in which simple terminologies are found in populous or stratified societies and conversely. Essentially, Marshall relies on the facts of geographical distribution and relative frequency and on concepts from

242

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e

yO

O

Om

Of

*tuakana *tahina *tuanga 'ne *tuafafine

V

s ex

II cT

N) x

*tahina

/ \ mO Of *tuanga 'ane *tuafafine

*tahina

*tuanga 'ane

Figure 8.10. A digraph of Clark's (1975) model of the evolution of Polynesian sibling terminology. graph theory in the construction of his model. The argument unfolds in several steps. In a sample of 241 languages in Island Oceania, Marshall discovered 11 different types of sibling terminologies. His first step was to put these types in a graph of evolvable sequences, as shown in Fig. 8.11 (in which we again use twin binary trees rather than box diagrams). This graph shows symmetric changes between all pairs of adjacent terminologies. It eliminates as unfounded Epling, Kirk, and Boyd's Samoan type and its unattested precursors. Their remaining types of Polynesian sibling terminologies correspond to types 1, 3, 5, and 10 in Marshall's graph. All but three types in his graph - 7, 9, and 11 - are included in Nerlove and Romney's typology of ideal types, and these three types are rare, represented by two, five, and two cases respectively. They are also eliminated in Marshall's final analysis.

Digraphs

mo

f o

243

mo

fb

e

o

yo

e

Figure 8.11. A graph of Marshall's (1984) "chain of structural patterns of sibling classification in Island Oceania."

In his survey Marshall found that "each major geographical region of Island Oceania is characterized by a dominant pattern of sibling classification in terms of both frequency of occurrence and geographical spread" (1984:604). Type 3 is dominant in Micronesia and type 10 in Polynesia. Type 6, although it is only slightly more frequent than type 3

244

Island networks Core Micronesia S

II Core Western Polynesia/Eastern Melanesia

Core Melanesia

Figure 8.12. Marshall's (1984) "developmental core of structural patterns of sibling classification in Island Oceania."

in Melanesia, is restricted to this area and is therefore identified as the "Melanesian" type. Type 5 is concentrated in the intersection of eastern Melanesia and western Polynesia. These four types, graphed in Fig. 8.12, are said to constitute the "developmental core of structural patterns of sibling classification in Island Oceania" (1984:604). According to Marshall, the developmental core, or "core tetrahedron," can be distinguished on structural as well as statistical grounds when the detailed graph in Fig. 8.11 is simplified to the graph in Fig. 8.13: When all the types [of sibling classification] and their interconnections are redrawn as a directed graph [Fig. 8.13] the critical role played by this "core tetrahedron" in sustaining the connectivity of the digraph can be clearly seen. The digraph presentation also eliminates the tendency, on viewing [Fig. 8.11]

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245

Figure 8.13. Marshall's (1984) graph of "relationships among structural patterns of sibling classification in Island Oceania."

for example, to assume that more complex forms always developed from simpler ones. Finally, it must be noted that these four core types account for 80% of the cases in the sample (1984:604). Fig. 8.13 is an undirected (not directed) graph of purely hypothetical two-way changes between the 11 types of sibling terminologies in Island Oceania. In this graph the core relationships between types 3, 5, 6, and 10 are emphasized by showing darker edges. The next step in Marshall's analysis was to construct a more restricted digraph of probable genetic relationships between sibling terminologies as inferred from the facts of numerical frequency and geographical distribution. Marshall assumes that there are two-way changes between the most common types, that is, types 3, 5, 6, and 10 in his developmental core, with one-way changes leading from the core to the least common, geographically localized types, as shown in Fig. 8.14. In Fig. 8.14, type 2 is derived from type 1 rather than type 7, because type 7 is restricted to two societies in Melanesia whereas types 1 and 2 are both common to Micronesia. Type 7 is derived from type 6 rather than types 2 or 3, because types 6 and 7 belong to the same linguistic group, Siassi (Bariari) in New Britain. Type 8 is derived from either types 4 or 5 rather than type 11, because types 4 and 5 are in proximity to the two local groups of type 8 societies. Type 9 presents a minor problem, since it is found in four Papuan societies in the Solomon Islands but also in Palau in Micronesia. Type 9 is derived from types 4 or 6, which are found in the Solomons and in nearby islands in Melanesia. The Palauan type 9 is regarded as an intrusion from outside Island Oceania, that is, from Island Southeast Asia. Type 11 is found only in

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Island networks

Figure 8.14. Marshall's (1984) digraph of "probable genetic relationships among the eleven structural patterns of sibling classification in Island Oceania." two closely related societies in Malekula in Vanuatu. It is derived from type 10, which is also found in Malekula, whereas types 8 and 9 are not. Marshall then argued, on both distributional and structural grounds, that type 3, a two-term system based on the parity distinction, is the ancestral form of all sibling terminologies in Island Oceania. This hypothesis is offered as an alternative to the linguistic reconstruction of type 10 for POC sibling terms (Milke 1938). The distributional argument is that type 3 is widespread in Near Oceania, while type 10 is concentrated in Remote Oceania, being found in all Polynesian subfamilies. Marshall supposes that type 10 originated in western Polynesia from either type 5 or type 6 - the core types in western Polynesia-eastern Melanesia and Melanesia - and that its presence in Melanesia is due to diffusion and back-migrations. This hypothesis is consistent with the assumption that Island Oceania was settled from west to east, and with Murdock's conclusion regarding the stability of sibling terms. The structural argument is an economic one based on the concept of distance in a digraph: it is easier to derive all the types of sibling terminologies in Island Oceania from type 3 than it is from type 10 (or any other type), because the number of steps from type 3 to all other types is least. The final step in Marshall's analysis was to drop all the receivers in

Digraphs Polynesia

247 Micronesia

Melanesia

Figure 8.15. Marshall's (1984) digraphs of "probable developmental sequences of major structural patterns of sibling classification in Polynesia, Micronesia, and Melanesia."

the digraph of Fig. 8.14 that represent rare types of sibling terminologies and define probable developmental sequences for the major patterns of sibling classification in Polynesia, Micronesia, and Melanesia, as shown in the asymmetric digraphs in Fig. 8.15. Type 3 is the origin (unique source) of sibling terminologies in all three areas, whereas type 10 is a later development in Polynesia and Melanesia. There are serious linguistic, anthropological, and structural objections to Marshall's model. Linguists (Bender 1984; Blust 1984; Clark 1984) see no reason to reject Milke's reconstruction of POC sibling terms as type 10, and they insist that lexical relationships cannot be disregarded and that evolutionary models must deal with genetically related languages. The case for a four term (Type 10) Pro to-Oceanic system is based on the fact that languages from such widely separated areas as New Britain, Vanuatu, and eastern Polynesia agree not simply in having systems of Type 10 but in having the same cognate forms for each term, forms reflecting Proto-Oceanic *tuqaka, ||e, *tansi9 ||y, * Qmaqane^ xm, and *papine xf. If Type 10 had evolved independently in these areas from some simpler system, it is extremely improbable that all would have chosen just the same terms for the new categories (Clark 1984:628). Clark also notes two difficulties with Marshall's diffusionist explanation for the presence of type 10 terminologies in Melanesia: (1) it is difficult to distinguish Polynesian innovations that have supposedly diffused to Melanesia from POC retentions; (2) reflexes of POC sibling terms are not confined to areas where type 10 terminologies occur but are found in all areas of Melanesia and Micronesia.

248

Island networks

Anthropologists and archaeologists who are otherwise impressed by Marshall's model are critical of its formalism, that is, its lack of any explanatory mechanisms to account for the emergence of the different types of sibling terminologies in Island Oceania (Alkire 1984; Smith 1984; Kirk and Boyd 1984) and its independence from linguistics and archaeology (Chowning 1984; Kirch 1984b). As Bellwood has observed, The problem with basically synchronic (or "ethnographic present") data of this kind is that they can only be used to support or reject the historical reconstructions of archaeologists and linguists and do not provide convincing data sources for independent reconstructions in themselves. The classificatory and distributional data presented [by Marshall] could no doubt be interpreted in several conflicting ways, with both historical and functional biases, if taken in isolation (Bellwood 1984:625). There are two problems with Marshall's graph theoretic analysis. First, contrary to Marshall's assertion, the developmental core or tetrahedron consisting of types 3, 5, 6, and 10 does not play any special role in sustaining the connectivity of the graph in Fig. 8.13. The subgraph induced by the four nodes 3, 5, 6, and 10, shown by the darker lines in the figure, is a quadrilateral whose removal does disconnect the graph. But so does any cycle that contains the nodes 3 and 6 or the nodes 3 and 7, for example 3, 4, 9, 6, 7, 3 or the triangle 3, 6, 7 or the quadrilateral 3, 4, 9, 6. There is nothing special about this particular quadrilateral in this graph. Secondly, the argument for the derivational simplicity of type 3 in the digraph of probable genetic relationships among the 11 types of sibling terminologies in Island Oceania in Fig. 8.14 is tenuous. The distance matrix of this digraph is 1 2 3 4 Dis(D)= 5 6 7 8 9 10 11

1 2 3 4 5 6 7 8 0 1 00 00 00 00 00 00 0 00 00 00 00 00 1 2 0 1 1 1 2 00 00 00 0 00 00 00 2 3 1 2 0 2 3 2 3 1 2 2 0 1 00 00 00 00 00 00 0 00

00

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2 2 2 3 1 1 00 00 1 3 1 2 3 1 1 2 00

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0 00 00 3 1 1 2 2 2 0 1 00 00 00 00 00 00 00 0 00

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Digraphs

249

The derivational simplicity of each type of sibling terminology is its total distance to all other types, which is given by its row sum in Dis(D). The row sum of type 3 is least, with S^y = 17, but the differences between type 3 and types 5, 6, and 10 are not great. Given Marshall's assumptions concerning the evolution of sibling terminologies, this suggests that any of these four types might be the ancestral form. In response to his linguistic critics Marshall concedes this possibility: I am well aware that this hypothesis contradicts the reigning conventional wisdom, which is based entirely on the hypothetical reconstruction of Proto-Oceanic sibling terms. Bender, Blust, Clark, and Kirch prefer to stay with the conventional wisdom, and they advocate a prototypic Type 10 pattern. Interestingly, Chowning hypothesizes a prototypic Type 6 structure on somewhat similar grounds. The data and the model presented in my paper admit of several possible alternative interpretations, and the commentators are clearly aware of this. The formal model strongly suggests that there are only four likely possibilities as competing hypotheses, namely, the four structural patterns contained in the core tetrahedron (Types 3, 5, 6, 10; [Fig. 8.12]. Conceivably, any one of these four patterns may have generated the entire system. Further research in Pacific archaeology and linguistics eventually may permit us to select among these possibilities with greater confidence than at present (Marshall 1984:633). If type 10 is conceded as a potential source of sibling terminologies in Island Oceania, then we should consider Milke's reconstruction of sibling terms in the Oceanic (OC) languages, a group that includes most of those in Marshall's sample. Milke's paper, which is well known to linguists but inexplicably ignored by many anthropologists, in particular by Firth and by Epling, Kirk, and Boyd, provides a basis for the "full integration of anthropological and linguistic methodologies" recommended by Bender (1984) in his comment on Marshall. In a paper entitled "Die Benennungen der Geschwister in den austronesischen Sprachen Ozeaniens," Milke (1938) reconstructed POC sibling terminology and gave a brief account of its probable social background and subsequent historical development. In Milke's reconstruction, based on a sample of 125 AN languages, Ur-Melanesian (UMN), that is, POC sibling terminology, consisted of four terms based on the distinctions of parity, seniority, and sex: Das Urmelanesische System unterschied vier Geschwisterbeziehungen, namlich: alteres Gleichgeschwister, jiingeres

250

Island networks Gleichgeschwister, mannliches Kreuzgeschwister, weibliches Kreuzgeschwister (Milke 1938:61).10

The terms as written by Milke are: tuwa-aka (e, ||), t-ad'i (y, ||), mane (xm), and vavine (xf). Evolutionary paths leading from this prototype consist mainly of simplifications resulting from the loss of the seniority distinction or the loss of the sex distinction, or both: Eine Vereinfachung des Systemes durch Zusammenlegung der beiden Gleichgeschwister- oder der beiden Kreuzgeschwisterterme, oder beider Paare, so dass nur noch zwei Beziehungen unterschieden werden, namlich Gleichgeschwister und Kreuzgeschwister (1938:61 ). n Milke mentions only a single case of differentiation, resulting from the addition of the sex distinction to parallel siblings: anderseits eine Vermehrung der unterschiedenen Beziehungen, indem die Gleichgeschwisterterme nach dem Gechlecht des Geschwisters differenziert werden, zumeist durch beigefiigte Pra-oderSuffixe(1938:61).12 A third change, found especially among coastal populations of New Guinea, resulted from the loss of the parity distinction - "das eigentliche Ruckgrat der urmelanesischen Geschwisterterminologie" - and the use of seniority or sex distinctions or both: Das dadurch entstandene System weist entweder vier Terme auf (alterer Bruder; j lingerer Bruder; altere Schwester; jiingere Schwester) oder zwei Terme (Bruder; Schwester bzw. alteres Geschwister; jiingeres Geschwister). Diese Systeme bringen also entweder das absolute Geschlecht (statt, wie das UMN-System, das relative Geschlecht) und das relative Alter zum Ausdruck oder nur eine dieser beiden Kategorien (1938:61).13 10 Translation: "The Proto-Melanesian system distinguished four sibling relations, namely: elder parallel-sibling, younger parallel-sibling, male cross-sibling, female cross-sibling." 11 Translation: "A simplification of the system through a merging of both parallel-sibling or both cross-sibling terms, or both pairs, so that only two relations are distinguished, namely parallel-sibling and cross-sibling." 12 Translation: "On the other hand, an increase of distinguished relations, by differentiating the parallel-sibling terms according to the sex of the sibling, mostly by added prefixes or suffixes." 13 Translation: "The resulting system shows either four terms (elder brother; younger brother; elder sister; younger sister) or two terms (brother; sister; or elder sibling; younger sibling, respectively). These systems express either absolute sex (instead of relative sex as in the Proto-Melanesian system), and relative age, or only one of these two categories."

251

5

6

7

4

5 6

Figure 8.16. Two depictions of the same rooted tree.

The structural model implicit in Milke's reconstruction is an "uppersemilattice." In order to define a semilattice it is best to explain the concept of a mathematical "lattice" first and then reduce the two essential axioms by one, thus getting roughly half a lattice, or a "semilattice." Mathematically a lattice may be defined in a formal axiomatic manner as a partially ordered set of elements (nodes) in which every two nodes have a least upper bound (LUB) and a greatest lower bound (GLB). Using the standard terminology of Birkhoff (1967), we may say that a semilattice is a "partially ordered set" of nodes in which every two nodes have an LUB. We should point out that the presence of a GLB is deliberately excluded from this definition. It therefore follows that every lattice is a semilattice but not vice versa. In this sense, a semilattice is a more general mathematical structure than a lattice. Every rooted tree (Fig. 8.16) is likewise a semilattice, but the converse is not true. We must now explain each ingredient in the definition of a lattice. Recall that a (binary) relation R is a set of ordered pairs («, v) of elements from some set V, and that the elements of V are called nodes. When («, v) is in R, we also write uRv. A partially ordered set consists of a set V of nodes and a binary relation R that satisfy the following three properties: 1. The relation is irreflexive: no node is in relation R to itself. 2. The relation is asymmetric: if uRv, then v is not in the relation R to u. 3. The relation is transitive: for every three distinct nodes w, v, w, if uRv and vRw, then uRw. (For a detailed explanation of these three properties of relations, with illustrations, see Harary et al. 1965, Chapter 1, and Hage and Harary 1991, Chapter 8.)

252

Island networks

Figure 8.17. A directed path.

In these terms a digraph, D, is simply defined as a finite irreflexive relation. Thus D may or may not be symmetric and also is not necessarily transitive. An oriented graph is an asymmetric digraph; it has no symmetric pairs of arcs joining the same two nodes. The discussion of the other two elements, namely the LUB and GLB, will be facilitated by referring to another structural concept. A directed path from one node / to another node / of a digraph is illustrated in Fig. 8.17. In this case we say that there is a sequence of directed arcs, and that node / is reachable from node i. Note that all nodes of a path are taken as distinct. Given a lattice L = (V, R) with node set V and relation R, we write D(L) for its digraph. Thus arc uv is in digraph D(L) whenever uRv is in L. When both arcs uv and vw are in D(L), v is between u and w, written (uvw). A node u covers node w if there is no node v for which (uvw). The covering digraph C{L) is that subdigraph of D(L) in which only those arcs xy appear when x covers y. It is customary, for simplicity, not to draw all the arcs in a digraph of a partial order but only those in its covering digraph. It is also a convention to omit the directional arrow in a covering arc uv by showing in the diagram node u above v joined by an edge, but we will retain the arrow for clarity. In the mathematical literature of lattice theory this representation of a lattice (without the arrows) is generally called its "Hasse diagram," after the German mathematician who introduced it. It is instructive to compare the three types of structure: lattice, semilattice, and oriented graph, all of which are special classes of digraphs, and this is done in Fig. 8.18. It can be seen that the LUB of nodes u and v in Fig. 8.18a is node w because a directed path exists from wtou and from w to i/, making it an upper bound. Further, every other node that can reach both u and v can also reach w, making it a least upper bound. Similarly, in the tree of Fig. 8.16 the LUB of nodes 4 and 5 is node 2, because this node can reach both 4 and 5, and every other node (such as node 1) that can reach 4 and 5 can also reach 2. It is therefore evident that the LUB of nodes 3 and 4 is 1.

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253

(b)

(c)

Figure 8.18. (a) Lattice, (b) semilattice, and (c) oriented graph.

On the other hand, the GLB of the two nodes u and v is defined analogously by directional duality, that is, the consideration resulting from reversing all the orientations on the arcs. For example, node x of Fig. 8.18a is the GLB of u and t>, but in Fig. 8.18b the nodes u and v do not have a GLB. The lack of a GLB in this latter instance makes it an uppersemilattice, since every pair of nodes do have an LUB. If this graph were inverted, it would then become a lower-semilattice. The oriented graph of Fig. 8.18c is a directed cycle in which the nodes u and v have neither an LUB nor a GLB. Hence it is neither a lattice nor a semilattice. One real-life example of this is to be found in the gridiron street plan, often called the "Manhattan Plan," common to many cities. By and large, a network of intersecting roads forms a graphical structure more general than a semilattice. Fig. 8.19 shows the upper semilattice implicit in Milke's historical reconstruction of POC sibling terminologies, with each type labeled as in Marshall's classification. We have added type 1, which is given in Milke's tables of sibling terminologies. Although Milke's sample is smaller than Marshall's and restricted to the AN languages of Oceania, it contains all of the more common types found by Marshall with the exception of type 4. It also includes two types, marked by asterisks, which were not found by Marshall, perhaps because he excluded the AN languages of coastal New Guinea from his sample. Several observations may be made on the digraph in Fig. 8.19. 1. The semilattice in Fig. 8.19 shows that sibling terminologies evolve through sequences of binary steps. Unlike Epling, Kirk, and Boyd's model, however, the changes are strictly asymmetric and, unlike Marshall's models in Fig. 8.15 (which are semilattices), all changes, with one excep-

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Figure 8.19. A semilattice of the evolution of Proto-Oceanic (POC) sibling terminologies, based on Milke (1938).

tion, consist of binary deletions. The significance of this trend was emphasized by Blust in his comment on Marshall: Marshall's error is to treat all types of structural changes as equally "costly" in terms of the simplicity metric of derivational parsimony. But changes that increase distinctions are less likely to occur independently than those that delete them particularly if the added distinction is correlated with cognate linguistic forms (Blust 1984:626).

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2. The core tetrahedron identified by Marshall, consisting of types 3, 5, 6, and 10, appears in Fig. 8.19, but with type 10 rather than type 3 as the source it is the converse of the one implied by Marshall and shown in his Melanesian digraph in Fig. 8.15. The tetrahedron in Milke's semilattice is the basis of Kirch's proposed alternative to Marshall's model: Is it not conceivable . . . that the early Lapita colonizers of island Melanesia and western Polynesia possessed a Type 10 structural pattern, given the working assumption that the Lapita people were speakers of Proto-Oceanic? The dominance of Type 10 throughout Polynesia would then reflect a retention of this early, ancestral system, while the Types 3, 5, and 6 structures that dominate eastern Melanesia and Micronesia would represent transformations from the Type 10 ancestral structure. This hypothesis would also account for the presence of certain scattered Type 10 systems along the original "Lapita network" (Kirch 1984b:629). 3. The evolutionary sequence of Polynesian sibling terminologies reconstructed by Clark (1975) in Fig. 8.10 (which is a semilattice) is contained in Fig. 8.19 as the directed path 10, 5, 3, 1. 4. By adding the dashed arc from type 6 to type 2, the semilattice in Fig. 8.19 can be shown to contain the evolutionary lattice of sibling terminologies in Nuclear Micronesia, as we will now explain. In the conclusion of his paper Milke interpreted the parity and seniority distinctions in POC sibling terminology as expressions of brother-sister avoidance and social rank: Die UMN Geschwister-Terminologie beruht auf der Trennung der Gleichgeschwister von den Kreuzgeschwistern und auf der Differenzierung der Gleichgeschwister-Terme nach dem relativen Alter. Dieses System kann als adaquater Ausdruck in Ozeanien weitverbreiteter Einrichtungen betrachtet werden, namlich der Meidung zwischen den Kreuzgeschwistern und des Vorranges der alteren Gleichgeschwister gegeniiber den jiingeren. Ein solcher Vorrang aussert sich in der Regelung der Erb-, Amts- und Rangfolge und verleiht dem alteren Bruder gegeniiber dem jungeren eine vaterahnliche Autoritat (1938:65).14 14 Translation: "The Proto-Melanesian sibling terminology rests on the separation of parallel-siblings from cross-siblings, and on the differentiation of the parallel-sibling terms according to relative age. This system can be regarded as an adequate expression of institutions that are widespread in Oceania, namely, avoidance between crosssiblings and the priority of the elder parallel sibling over the younger parallel-sibling.

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He also suggested that the presence of the sex distinction between the cross terms could be interpreted as a sign of the asymmetry in the brother-sister avoidance relation. Whether or not parity, which occurs in the great majority of Oceanic sibling terminologies, is interpreted as an expression of brother-sister avoidance, a concern with cross-sex identity (Nerlove and Romney 1967), a reflection of the "place of male and female lines in the transmission of group interests" (Firth 1936), or a correlate of matrilineal or ambilineal descent (Murdock 1968), it seems certain that seniority is an expression of rank and that it persists where primogeniture is emphasized. In a society where lineage rank and succession to the chieftainship are based on seniority of birth, the terminological distinction between elder and younger siblings is important as Polynesianists have often observed. (It may be no accident that the Polynesian languages which most strongly maintain the terminological distinction are generally spoken in those societies in which primogeniture is of particular importance in the transmission of rank, e.g., Tonga, Hawaii, and New Zealand, but not Samoa, Tokelau, or Nukuoro) (Pawley 1980:237). This correlation suggests an alternative to Marshall's Micronesian sequence. In Chapter 5 we conjectured, on the basis of linguistic evidence, that Marshallese and Tongan societies are genetically related variants deriving from a stratified POC society based on the conical clan and asymmetric marriage alliance. We proposed, contra Murdock (1948, 1949), that societies in Nuclear Micronesia should be viewed as devolutionary developments from a Marshallese prototype in which rank based on primogeniture is modified by competition and achieved status as in Pohnpei, eventually giving way to age-based hierarchies as in Truk, and in which elementary structures of marriage alliance are replaced by semicomplex and complex structures as in Truk and the Gilberts. Since Marshallese society is closely related to Tongan society, structurally and genetically, it would not be surprising to find that it once had a similar sibling terminology. In Marshall's classification, however, Tongan sibling terminology is a type 10, whereas Marshallese is a type 2 based on the single dimension of seniority. In Marshall's model, Nuclear Micronesian Such a priority is expressed in the regulation of succession, authority, and rank, and lends to the elder brother, in relation to the younger, a fatherly authority." Milke saw no reason to insist that subsequent developments in POC social institutions and sibling terminology would be in perfect correspondence.

Digraphs

257 Gilberts, Banaba, Pingelap Mortlocks, Truk, central and western Carolines

Pohnpei, Mokil, Ngatik, Nauru

o (T)

/*\. -/^ ® ^ s ~

Truk, Murilo, Namonuito

Marshalls, Kosrae (?)

Figure 8.20. Marshall's (1984) model of the evolution of Nuclear Micronesian (NM) sibling terminologies.

(NM) sibling terminology began with the single dimension of parity, from which there flowed two evolutionary paths: in one path parity was eliminated and followed by the emergence of seniority, and in the other the sex distinction was added to the cross category, as shown in Fig. 8.20. It appears, however, that Marshallese sibling terminology was originally more complex than it is now. All sources are consistent in giving two terms for Marshallese: jeo (elder sibling) and jato (younger sibling). There are however two "supplementary" terms: inb (man's sister) and mano (woman's brother), given by Spoehr (1949a) and also by Kramer and Nevermann (1938), Erdland (1914), and Mason (1954) and cited by Milke, who classifies Marshallese as a four-term system. According to Mason, Marshallese practice "favors" jeo and jato for parallel siblings and inb and mano for cross-siblings. The terms inb and mano appear to be reflexes of POC *papine (xf) and *rjmagane (xm). Marshallese sibling terminology was originally a type 10, like POC and Ton-

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Marshal Is

Truk, Murilo, Namonuito

Marshalls

Gilberts, Banaba, Pingelap Mortlocks, Truk, central and western Carolines

Marshalls, Kosrae (?)

Pohnpei, Mokil, Ngatik, Nauru

Figure 8.21. A lattice model of the evolution of Nuclear Micronesian (NM) sibling terminologies. gan terminology.15 This suggests the developmental sequence of NM sibling terminologies shown in the lattice of Fig. 8.21. In Fig. 8.21 there are several evolutionary paths flowing from a Proto-Nuclear Micronesian (PNM) type 10 terminology. In one path, which began with the stratified (markedly primogenitural) societies of the Marshall Islands, the sex distinction was lost first, then the parity dis15

Marshallese is one of Marshall's counterexamples to the functional interpretations of Firth and of Epling, Kirk, and Boyd - a demographically large society with a simple sibling terminology.

259

m mwddni

II cr pwii

f feefinej

>D x mwddni

Figure 8.22. Lexical relationships in the evolution of Trukese sibling terminology.

tinction, and finally the seniority distinction, ending in a terminology with a single term. Our hypothesis is that the terminologies of the decreasingly stratified and strongly acculturated societies of Pohnpei and Kosrae followed this route. In a second path, which includes the more egalitarian or weakly stratified societies of Nuclear Micronesia, the seniority distinction was lost first, followed by the sex distinction and then the parity distinction, also ending in a terminology with a single term. In a third path, a type 3 terminology was reached from either type 6 or type 5, perhaps the former for the Gilberts and the latter for Truk and the Carolines. There are two different sibling terminologies for societies in the Truk Lagoon: type 5 and type 3. The displacement of the female cross term by the male cross term, hypothesized in Fig. 8.22, is analogous to the process described by Clark (1975) in Fig. 8.10 for Polynesia. A consideration of the meaning of these terms provides strong support for the genetic model and the devolutionary hypothesis: Trukese mwaani- ("male sibling of opposite sex") is cognate with the second element of Tongan tua-nga'ane^ with the second element of Lakalai hata-male, and with Bwaidogan mogane-, all meaning "male" as adjective. Proto-Oceanic kinship terms used expressions meaning "my man" for male cross-sex sibling and "my woman" for female cross-sex siblings. Where one term was dropped and one survived, it was regularly the term for "my man," as in Gilbertese mwaane-, showing that

260

Island networks this was the unmarked term, whereas "my woman" was the marked term of the pair. The meanings of the two terms, however, make sense only if the two-way distinction was made originally with reduction coming later. It makes no sense to think of something meaning "my man" to have originally stood alone for both male and female cross-sex siblings with the term "my woman" having been added later (W. H. Goodenough, personal communication).

Clearly, as Goodenough, Blust, and Clark have shown, it is not possible to study the evolution of AN kinship terminologies without consideration of cognate forms and their derivational meanings. The lattice digraph of NM sibling terminologies in Fig. 8.21 (with type 1 as the GLB) is a subdigraph of the semilattice implicit in Milke in Fig. 8.19. We conclude this discussion on a historical note. In his paper on sibling terminologies Murdock outlined a colorful four-stage history of kinship studies. It begins with The Founder, Lewis Henry Morgan, whose Systems of Consanguinity and Affinity of the Human Family, published in 1870, created the field of kinship analysis. Morgan, as Levi-Strauss (1963:300) has written, identified the basic features of kinship systems that have made them a subject of enduring interest: "permanency, systematic character and continuity of changes." Morgan was followed by The Early Giants, who adopted, revised, and extended his ideas. Among their other contributions, Kroeber (1909) elucidated the underlying principles of kinship classification, Rivers (1910b, 1914b) invented the genealogical method and demonstrated the correlation between kinship terminology and forms of crosscousin marriage, Radcliffe-Brown (1930-1) discovered the major types and associated nomenclatures of Australian marriage systems, and Lowie (1920, 1928) critically evaluated and organized the voluminous material on kinship and social organization. Next came The Later Masters, who "produced wholly satisfying descriptive accounts and interpretations of social systems." They include Firth (1936), Fortes (1945, 1949), and Eggan (1950), who wrote classic monographs on Polynesian, African, and North American Indian societies, and Levi-Strauss (1949), who wrote a brilliant and far-reaching analysis of elementary marriage systems. Finally, The Modern Innovators - Goodenough (1956, 1964), Lounsbury (1956, 1964), and Romney and D'Andrade (1964), among many others - advanced the analysis of kinship terminology through the application of models from structural linguistics and cognitive psycholo-

Digraphs

261

gy.16 The contributions of these Giants, Masters, and Innovators pervade Chapters 4, 5, and 8 of this book. Murdock viewed his own work as a continuation of the main tradition running through these four stages. But he also pointed to two other sources of inspiration. The first was Nerlove and Romney's theoretical analysis of sibling classification, which provided the basis for Murdock's own study. The second was historical linguistics, with its reconstructions of proto-kinship systems, as exemplified by the work of Hoijer (1956) on Athapaskan, Elmendorf (1961) on Salishan, and Hockett (1964) on Algonquian. Although Murdock's research was basically inductive and was inspired mainly by parallels between social and linguistic processes,17 he had the right idea about future developments in kinship analysis. In the case of Austronesian and Oceania studies, much of the progress since Murdock has been due to applications of formal models and the comparative method of historical linguistics. 16 The fourth stage is also the beginning of a decline in which "certain self-styled 'social anthropologists' today no longer report kinship terms in their monographs or do so half-heartedly or incompletely - a tendency that would have profoundly shocked the Early Giants and Later Masters" (1968:2). One wonders how Murdock would have characterized the present period, when either the reality or the variety of kinship systems is denied - The Late Dwarfs, perhaps? 17 The parallels include "limitation in the possibilities of change, a strain toward consistency, shifts from one to another relatively stable equilibrium, compensatory internal readjustments, resistance to any influence from diffusion that is not in accord with the drift and noteworthy lack of correlation with accompanying cultural norms in technology, economy, property or government" (Murdock 1949:199).

Conclusion

New points of view usually produce new observations. Oystein Ore, Graphs and Their Uses

In a symposium entitled "Man's Place in the Island Ecosystem," held at the Tenth Pacific Science Congress in 1961, Vayda and Rappaport advocated research on ecosystems to elucidate the social and political organization of island populations: An attempt has been made to view human populations as neither more nor less than populations of a generalized and flexible species, for in the most fundamental respects man hardly differs from other animals. His populations participate in ecosystems, as do the populations of other species; they occupy particular positions in food webs as do others; and they are limited by factors little different from those that limit others . . . In the detail of man's commitment to, and participation in, the biotic communities in which he has his being there is much to illuminate his social and political organization (Rappaport 1963:168-9). In spite of a certain enthusiasm for ecological studies at that time, due in large part to Goodenough's (1951) paper on land tenure and cognatic descent in PMP society and Sahlins's (1958) adaptive radiation model of social stratification in Polynesia, H. E. Maude was skeptical that this proposal would produce results even for simple atoll societies: Vayda and Rappaport hint at the rewarding possibilities of studying the effect of the ecosystem on the social organization of the human population. One hopes that they will attempt this at least for the limited and demanding environment of the low islands, where one would expect similar patterns to develop in the face of a very similar environment; but where in my limited experience, no such thing happens (Maude 1963b:173-4). 262

Conclusion

263

By now it has become apparent to many Oceanists that very little can be inferred about an island society without taking into account its social, historical, and linguistic connections to other island societies. Our purpose has been to provide a set of useful and interesting graph theoretic models for studying these connections. We began with a general theorem on trees, the simplest of all graphs and of all network models. We used rooted trees in Chapter 2 to describe the control of communication in the Yapese networks in western Micronesia, in-trees to describe the flexibility of cognatic kinship ties in atoll networks, and twin binary trees to give a single characterization of a primordial AN classification system that has been independently discovered in a number of Oceanic and Indonesian societies. The twin concepts of a spanning tree and the cycle rank of a graph provided a measure of network connectedness useful for studying linguistic diversity in trade networks and elite monopolization of exchange in prestige-good systems. In Chapter 3 we introduced the minimum spanning tree (MST) network model from applied combinatorics. We presented three different MST algorithms and an application of each: (1) Kruskal's algorithm and the concept of clustering in an MST served to describe the partitioning of the Tuamotu atolls in East Polynesia into dialect groups and marriage isolates; (2) Boruvka's algorithm was employed to simulate the evolution of overseas chiefdoms in eastern Fiji; (3) For reasons of computational efficiency and mathematical explicitness, Prim's algorithm was exploited rather than the standard method of close-proximity analysis in archaeology. From computer science we borrowed the model of a search tree to study kinship structures in Chapters 4 and 5. Depth-first search trees (DFSTs) enabled us to give a clear and general characterization of rank in the conical clan. This descent group constitutes the basic framework of many Polynesian societies and probably PPN and POC societies as well. It was then easy to recognize the essential similarity of this model to Mason's numerical description of the conical clan in the Marshall Islands in Micronesia. This formal insight, together with evidence from historical linguistics, provided the basis for the reconstruction of PNM society and for a comparative analysis of social organization in Nuclear Micronesia. We also used breadth-first search trees (BFSTs) to evaluate Murdock's bilateral hypothesis of MP origins, and we described the role of BFSTs and DFSTs as cognitive models in the organization of genealogical knowledge. In order to define the variety of ways in which islands may be advantageously located in voyaging and trade networks, we utilized, in Chap-

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Island networks

ter 6, six different concepts of centrality. In the graph G of a trade network, degree centrality was used to index a community's direct trading activity, median (closeness) centrality its access to resources in other communities, and betweenness centrality its position as an intermediary. All three concepts were suitable for describing trade centers in southern Lau. Betweenness characterized the crossroads position of Lakemba and its success over competing power centers in the Greater Lauan trade network in eastern Fiji. The traditional unmarked concept of centrality accounted for the location of political and mythological centers in the Marshall Islands. Some trade networks are rooted, in the sense that the relative economic success of each community depends on its intermediate position with respect to a source community. This was the case in the canoe-purchasing networks of Torres Strait, and we therefore introduced the concept of betweenness in a rooted graph to study the relation between trading success, subsistence techniques, and social stratification. To accommodate situations in which physical distance must be considered, we gave a matrix method for finding the median of a network N, using an example from classical archaeology. Finally, we characterized the limiting case of networks whose graphs are self-centered. Where networks are dominated by combinations of island communities, the appropriate structural model is the decomposition of a graph into its dominating sets. In Chapter 7 we developed several variations of domination, including independent dominating sets, to describe locallevel structures in the western Carolines, minimum dominating sets to elucidate alliance structures in the Tuamotus, and n-step dominating sets to account for the distribution of pottery monopolies in two Melanesian trade networks. Evolutionary models of kinship organization consisting of step-bystep transitions between structural forms are implicitly, if not explicitly, based on digraphs. An examination of the digraphic basis of Murdock's Hawaiian derivation of MP society and Marshall's distributional derivation of Island Oceanic sibling terminologies led in Chapter 8 to a rejection of both models. In Murdock's own terms, kinship structure in PMP society was shown to be of the Iroquois or Nankanse type. Murdock's bilateral hypothesis is therefore not an alternative to Blust's asymmetric connubium reconstruction of early AN society. In Marshall's model there are actually four different possibilities for the ancestral form of Island Oceanic sibling terminology, one of which is identical to Milke's reconstruction of POC sibling terms. The evolutionary digraph implicit in Milke's reconstruction was shown to be a semilattice whose structure is consistent with Clark's reconstruction of PPN sibling terms and our own reconstruction of NM sibling terms and PNM society. In reviewing these results we would like to consider in general terms

Conclusion

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the value of a graph theoretic perspective for network analysis in anthropology. One advantage lies in the relation of graph theory to several other fields of mathematics, including geometry, combinatorics, the logical theory of relations, matrix algebra, recreational mathematics, and computer science. Informally, a graph consists of a set of nodes, some pairs of which are joined by edges. Thus graph theory is a kind of abstract geometry (Harary 1989). The geometrical and pictorial aspects of graph theory permit an intuitive understanding of subtle mathematical concepts such as connectivity, symmetry, and isomorphism. Much can be gained in the way of clarity and insight by drawing the graph of an empirical structure. The hierarchical structure of the conical clan, which has proven so elusive in descent theory, stands clearly revealed once it is drawn as a DFST as in Chapters 4 and 5. Similarly, the ambiguities surrounding the concept of recursive dualism disappear completely when this structure is represented as a twin binary tree, as was done in Chapter 2. Graphs are basic tools of combinatorics, serving as models for patterns, designs, and arrangements of various kinds. Although a concern with typologies is as old as anthropology itself, very few studies have risen to the level of being explicitly combinatorial in nature. There are three basic problems of combinatorics. The existence problem asks: Is there a structure of a certain type? The enumeration problem asks: How many such structures are there? The optimization problem asks: Which of these structures is best, according to some criterion? (Roberts 1984). The existence problem occurred where theorems were given specifying the conditions for various types of graphs, for example, trees and strongly connected digraphs. The optimization problem arose in Chapter 3 in applications of the MSTP to network structures. Nerlove and Romney's study of sibling terminologies, which served as the framework for the second half of Chapter 8, illustrates both the enumeration and the optimization problems. They first determined the number of logically possible sibling terminologies and then predicted which of these types were empirically likely, given the constraints of cognitive economy and universals in marking rules. This deductive approach to kinship organization contrasts with more familiar inductive methods that may miss actual types and that overlook the theoretical significance of logically possible but empirically rare or unknown types. It is important to understand why disjunctive sibling terminologies do not occur and why crosssiblings are not classified by seniority. In logic, a graph can be defined as a finite irreflexive relation. Graph theory is in fact coextensive with the theory of relations. From a graph theoretic point of view, many of the empirical structures studied in anthropology can be described in terms of properties such as their transi-

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Island networks

tivity, symmetry, and completeness. They can also be analyzed in terms of higher-order structures. Although their study was purely typological, Epling, Kirk, and Boyd made an important contribution to kinship theory when they proposed that the evolution of sibling terminologies could be modeled as a semilattice in which changes occur in sequences of binary steps. This was the basis of our genetic analysis of POC terminologies in Chapter 8.1 Algebraically, a graph can be characterized as a square, symmetric binary matrix (of zeros and ones) with only zeros on the main diagonal. In Chapters 6 and 8 we used simple techniques of matrix algebra to develop quantitative properties of networks. But the value of a matrix representation is not just computational. If a network is conceived from the outset in matrix form, then every entry must be recorded as zero or one, showing the presence or absence of a connection between every pair of locations, individuals, groups, events, or categories. The result is a complete description of a network, which permits and encourages an analysis of structural properties such as connectedness and centrality. Graph theory has an important historical relation to recreational mathematics. Many of its concepts originated in puzzles such as the Konigsberg Bridge Problem, Hamilton's Around the World game, round-robin tournaments, and map-coloring problems. In Chapter 4 it was helpful to conceive of a DFST as a maze-solving problem, while in Chapter 7 the application of dominating sets to networks was suggested by board-game problems. In our two previous books we used the concept of coloring in graphs to analyze the partitioning of social networks. A number of papers have now appeared on this subject, for example, Everett and Borgatti (1993), Borgatti and Everett (1994), Freeman and Duquenne (1993). Finally, graph theory is closely tied to computer science in the design of programming algorithms. The accelerating interaction between these two fields has reached the point at which texts on graphs and combinatorics emphasize an algorithmic approach (Buckley and Harary 1990; Carre 1979; Roberts 1985), and there is now a specialty called "algorithmic graph theory" (Chartrand and Oellerman 1993; McHugh 1990).2 Algorithms are not only step-by-step procedures for arriving at solutions to problems; they are also aids to the imagination. In Chapter 3 we saw that MST algorithms provide greater insight into archaeological material than conventional seriation techniques. They also provide a 1 Needham (1975) has proposed the application of relation theory and combinatorics as solutions to the problem of polythetic categories, i.e., nontransitive analytical concepts, in anthropology. 2 In recognition of this trend, we now use the terms "node" and "edge" instead of "point" and "line."

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way to conceptualize evolutionary processes in social and linguistic networks. Graph theory, by virtue of its relation to many branches of mathematics, including several that we did not have occasion to utilize in this book, provides a rich, flexible, and unified model for the analysis of communication, kinship, and classification networks in anthropology and for the study of network structures in general. The essential importance of graph theory to research in the social, physical, and biological sciences, to engineering, and to the humanities, has been firmly established and can be vividly verified by a perusal of the research literature in these fields.

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Index

Admiralty Islands, 37-8, 121 age-area hypothesis, 143 Alexander, W. D., 96 Alkire, W. H., 33, 34, 38, 143, 145, 155, 157, 160, 161, 179, 205, 207, 248 Allen, J., 213 alpha index, 7, 49-50, 122 ambilateral lineage, 108-10 Ancestral Polynesian Society, 10, 106 arboresence, 43 arcs, 29, 219 symmetric pair of, 29 Arioi solution, 120 asymmetric connubium, 223 asymmetric marriage alliance, 91, 136, 145, 159, 161, 231; see also asymmetric connubium; fosterage; generalized exchange; matrilateral crosscousin marriage atoll social organization, 91, 105, 262 Audran, H., 60n5, 208 Austin, G., 233 Austronesian classification systems, 4, 35-43 expansion, 52 languages, 17-20,232 social organization, 15, 223 automorphism, 201-2

bilateral cross-cousin marriage, 21, 69, 135, 147, 149, 158; see also symmetric marriage alliance bilateral hypothesis, 15, 218, 222-31, 264 Birkhoff, G., 251 Bismarck Archipelago, 17, 87 Blust, R., 15, 16, 17, 19, 116,146, 218, 223-4, 225, 226, 231, 235, 241, 247, 249, 254, 260, 264 Boon,J. A., 40, 41,42-3 Borgatti, S. P., 12, 266 Boruvka, O., 7-9, 52, 55, 71-2, 78, 89, 263 Bott, E., 11, 69, 91, 95n4, 96n5, 107, 116nl2, 117, 119, 139nn8,9 Bourdieu, P., 37, 40, 41-2 Bowden, R., 161 Boyd, J. P., 2, 16, 232, 234, 238-41, 242, 248, 249, 253, 258, 266 Brainerd, G. W., 75 bridge, 25-6 Brookfield, H. C, 165 Bruner, J. S., 233 Buck, P. H. (Te Rangi Hiroa), 237 Buckley, F., 2, 12, 165, 169, 201-2, 266 Buganda solution, 120 Burrows, E. G., 157-8, 207

Barnes, J. A., 11, 107 Barraud, C, 42 Bartlett, H. K., 215 Bascom, W. R., 143, 147 Batak, 152 Beaglehole, E., 6 Beaglehole, P., 6 Beckett, J., 181, 182, 184n7 Bellwood, P., 18, 33, 34, 90, 106, 145, 223, 248 Bender, B. W., 16, 143-5, 146, 218, 247, 249 Biersack, A., 91, 95n4

Caillot, A. C. E., 209, 210 Capell, A., 69 Caroline Islands, 30-5, 139, 146, 157-8, 160-1, 205-7, 264 Carre, B., 116nl3,266 carrier, 219 Cartwright, D., 2 Caughey,J. L., 156, 161 Cayley, A., l l l n l l centrality, 12-13, 264 and accessibility, 194-5 betweenness, 12-13, 165,170-1,173, 174, 177-8, 185-8, 192-3 and cult centers, 198, 201

289

290

Index

centrality (continued)

degree, 12-13, 165, 168, 171, 173, 264 and eccentricity, 165, 178-9, 201-2 median (closeness), 12-13, 165, 169-70, 171, 173, 178-9, 192-3, 195-201, 264 and political power, 174-80, 264 and social stratification, 13, 193-4 in trade networks, 13, 168-78, 185-201,264 Chartrand, G., 266 chieftainship, 33-5, 37, 54, 68-74, 95-104, 106-7, 118-21, 126-9, 131-4, 136, 139-43,146-8, 150-2, 155-6, 157-60, 178-80, 193-4; see also conical clan; prestige good systems; social stratification chord, 48 Chorley, R. J., 50 Chowning, A., 248, 249 Clamagirand, B., 137 Clark, R., 16, 46, 218, 238, 240-1, 247, 249, 255, 259, 260, 264 close-proximity analysis, 52, 75-7, 81-3, 87, 88nl3,263 Codrington, R. H., 218, 232, 236 Cohen, R., 226 complement, 26 complete order, 114 complex marriage systems, 21, 61, 65-6, 151,256 conical clan, 10-11, 90-117, 118-20, 124, 125-34, 139-40, 145, 146-8, 149, 150-1, 153, 157, 159, 160-1, 162, 256, 265; see also chieftainship; prestige good systems; social stratification Cook,J., 117, 119 Corney, B. G., 207, 209 Crook, W. P., 240 Crow/Omaha systems, 21, 148, 154-5, 161-2,222,224,225 cutnode, 25 Cyclades, 81-7, 198-201 cycle rank, 6-7, 48-9 and exchange networks, 45-50, 263 cycles in a digraph, 219 in a graph, 25 independent, 6-7, 48-9 system of fundamental, 49 cyclomatic number, 49 Dahlberg, G., 62, 64, 66 Damas, D., 158 Damm, H., 206-7

D'Andrade, R. G., 260 Danielsson, B., 55, 58, 61, 63, 64, 65, 210 Davenport, W., 139 Davidson, J. M., 117 Davis, J. L., 13, 166, 194-5, 197-201 degree of a node, 24 Delos, 194, 197-201 Dempwolf, O., 19-20 Dening, G., 53, 209 Derrick, R. A., 68, 168, 173 descent ambilineal (ambilateral), 106, 108-9, 150, 223, 224, 236 bilineal, 116-17 cognatic (nonunilineal), 43-5, 53-4, 65, 121, 152, 223, 262 double, 105, 158, 224, 236 matrilineal, 91, 92, 110, 125, 126-9, 135, 140, 141, 143,145, 146-8, 149, 153-4, 156, 158, 222, 224, 236 patrilineal, 68, 69, 91, 92, 95-7, 99-101, 102-3, 106, 135n5, 136, 143, 150, 155-6, 158, 183, 222, 223, 224 unilineal (unilateral), 92-4, 107, 223 see also conical clan; ramage descent line system, 104 dialect groups, 9, 57-61 diameter of a graph, 178 digraphs, 14-16,29,219 connectedness category of, 15, 220-1, 230 converse of, 221, 226 covering, 252 as evolutionary models of kinship structures, 15-16, 226-31, 241-2, 246-7, 264 lattices, and semilattices, 15, 252-3 Dijkstra, E. W., 89 directional duality, 221, 253 disjunctive categories, 233, 234 distance cost, 196-7 in a digraph, 219-20 in a graph, 25 sum, 168-9 dominating sets, 13-14, 204, 205, 264 in alliance stuctures, 207-12, 264 independent, 205-7 minimal, 14, 205, 212 minimum, 14, 205, 217 «-step, 14,212,215,217 in political structures, 205-7, 264 in trade networks, 212-17, 264 Doran, E., 152-3 Douglas, B., 142 Dumont, L., 39, 42-3, 135n5

Index Duquenne, V., 266 Dyen, I., 145 Earle, T., 70n8, 139n8 early Austronesian society, 15, 116, 223-4, 231, 235 edges adjacent, 23 directed, 29, 219 independent set of, 71 Eggan, F., 260 Eilers, A., 124, 141 Ekholm,K., 116, 117 electrical networks, 6 elementary marriage systems, 21, 256 Ellis, W., 208 Elmendorf, W. W., 261 Emory, K. P., 53-5, 56, 60, 151nl3, 207, 208,209-10,211,212 endnode, 26 Epling, P. J., 2, 16, 232, 234, 238-41, 242, 249, 253, 258, 266 Erdland, A., 125, 126-7, 131, 132, 134, 135, 136, 139, 140-2, 257 Euler, L., 24, 90, 204 Evans-Pritchard, E. E., 4-5, 100, 107, 129-30 Everett, M. G., 266 Eyde, D. B., 4, 37-8, 40, 41-2 father-child metaphor, 31, 136, 147-8, 155-6 field of ethnological study, 145 Fiji, 120-1, 123-4, 165; see also Lau Islands Finney, B., 105 Firth, R., 90, 92, 94-5, 100, 102, 127, 130, 151nl3, 232, 236-8, 239, 240, 241, 249, 256, 260 Fischer, J. L., 148 Fison, L., 70 Five Queens Problem, 14 Foley,W.A., 17 Formosan languages, 17, 19 Fortes, M., 100, 129-30, 260 Forth, G., 42 Fortune, R. F., 215 fosterage, 151-2 founder-focused ideology, 106 founding marriages, 120, 137 Fox, J. J., 4, 38-9, 40, 41-2, 95n4 Fox, R., 151nl3, 162-4 Freeman, J. D., 65, 105 Freeman, L. C, 12, 168-71, 266 Freeman, O. W., 53, 63, 64 Fried, M. H., 91, 92n3, 107-8 Friederici, G., 56

291 Friedman, J., 1, 11, 31, 35, 92, 105, 107, 116-17, 121-2, 124, 160-1, 231 Frimagacci, D., 88 Fujimura, 38 Fustel de Coulanges, N. D., 92 Garanger, J., 121 Garrison, W. L., 7, 47-50 Geertz, C, 4 Gell, A., 120 genealogical method, 163, 260 generalized exchange, 21, 116 Sino-Tibetan axis of, 223 geodesic, 25 cost, 196 Gifford, E. W., 11, 91-2, 95-8, 100,102, 114-15, 117, 118, 119, 120,125, 127, 129-30, 132 Gilbert Islands, 43-4, 66, 118, 139, 146, 203, 256 northern, 149-52, 155, 160 southern, 152-3, 160 Gladwin, T., 139, 157, 206 Goldman, I., 91, 95n4 Goodenough, W. H., 3, 10, 43-4, 66, 90, 106, 126n2, 146, 151nl3, 152, 153-7, 162, 222-3, 259-60, 262 Goodnow, J. J., 233 Goody, J., 112-13, 120 Grace, G. W., 46, 143 Graham, L., 8-9, 51, 71nlO, 89 graphs, 3, 22-9 acyclic, 29 bipartite (bigraph), 26-7, 166, 202-3 coloring in, 266 complete, 25 component of, 25 connected, 25 domination number of, 206 embedded, 28 isomorphic, 26, 201 labeled, 24 vs networks, 7, 55 node symmetric, 202 oriented, 29, 252 planar, 28-9, 49-50 regular, 24 rooted, 27 self-centered, 201-3 self-median, 201-3 weighted, 7, 55 graph theory and abstract geometry, 265 algorithmic, 266 and combinatorics, 265-6 and computer science, 9-10, 263 and the logical theory of relations, 265

292

Index

graph theory (continued) and matrix algebra, 266 and network analysis, 2, 264-7 and recreational mathematics, 266 Graves, M. W., 1 Green, R. C, 1, 9, 10, 17, 20, 52, 57n3, 75, 87-9, 90 Greenberg, J. H., 234 Grimble, A. F., 66, 150, 151 Grimes, B. F., 17 Groves, M., 69 Guiart,J., 117 Gunson, N., 118nl4 Haddon, A. C., 13, 53, 123-4, 181, 182-3, 185, 186-91, 194, 207, 209 Haggett, P., 12, 49-50 Hakimi, S. L., 12 Hambruch, P., 141 Hammer, P. L., 212n2 Handy, E. S. C, 240 Hanlon, D., 68 harmonic vs disharmonic regimes, 135-6 Harris, D. R., 2, 13, 165, 180-1, 183-8 Hart, D., 165 Hasse, M., 196nlO Hasse diagram, 252 Hatanaka, S., 65 Hawaii, 104, 105-6, 108 Hawaiian systems, 15, 65, 218, 222-31, 264 Hayden, B., 120, 145-6nll Haynes, T. W., 14 Hedetniemi, S. T., 14, 204 Hell, P., 8-9, 51, 71nlO, 89 Henry, T., 61 Heritier, F., 21, 65, 154 hierarchical opposition, 42-3 high, low island relations, 30-5 Hjarn0, J., 122-3 Hocart, A. M., 4, 35-6, 37, 40, 41-3, 51, 66, 68, 69-71, 73-4, 110, 168n3, 175, 177 Hockett, C. F., 261 Hogbin, I., 90 Hoijer, H., 261 Hornell, J., 53, 123-4, 207, 209 Howell, S., 42 Hunt, T. L., 1, 45-6, 50, 165 hypergamy, 21, 98, 101 hypogamy, 21, 101 indegree, 219 India, 43 Indonesia, 38-40, 42-3, 95n4, 101, 137, 223, 263 institutions of mobility, 65

in-tree, 43, 263 Iroquois systems, 15, 218, 222, 224, 225, 226,229-31,264 Irwin, G. J., 1, 165, 192-3, 197, 213, 217 Island Oceania, 15 isolate, 219 isomorphism, 26 Jackson, F. H., 146 James, B., 88nl3 Japan, 125 Josselin de Jong, J. P. B. de, 145 Josselin de Jong, P. E. de, 135-6, 145 Kachin, 92, 99-101, 106,115-16, 129, 136,152 Kaeppler, A. L., 117, 120-1, 122, 123 Kenner, H., 90 kinship terms, 224-5, 260-1 Crow-Omaha, 148, 154, 161-2 Dravidian, 69 Generation-Hawaiian, 135, 222, 225 and marriage alliance, 135, 136-7, 148, 224, 260 Proto-Austronesian, 224 Proto-Malayo-Polynesian, 223-4, 264 Proto-Nuclear Micronesian, 258-9, 264 Proto-Oceanic, 16, 218, 232, 241, 247, 249-51,253-6,257-8,264 Proto-Polynesian, 107, 240-1 for siblings, 15-16, 107, 110, 126, 218, 223-4, 231-60, 261, 265, 266 Kirch, P. V., 1,10, 90, 92-4, 105, 106, 110, 117, 118, 120, 121-2, 132, 165, 248, 249, 255 Kirchhoff, G., 5-6, 45n7, 48 Kirchhoff, P., 11, 90, 91, 92-4, 101, 102, 107-8,130, 146, 151, 162, 218, 235 Kirk, J., 2, 16, 232, 234, 238-41, 242, 248, 249, 253, 258, 266 Kiste, R. C, 126 Knuth, D. E., 40 Konig, D., I l l Kosrae (Kusaie), 124, 146, 148-9, 155, 160-1, 259 Kotzebue, O. von, 139, 160, 179, 207 Kotzig, A., 89 Kramer, A., 105, 126, 131, 134, 159-60, 257 Krige, E. J., 101 Kroeber, A. L., 260 Kruskal, J. B., 52, 55-6, 78, 89, 263 Kuper, A., 129-30 Kuratowski, K., 29nl Laade, W., 185, 188, 189,191,194 Labby, D., 34

Index Lakemba, 66-75, 165, 166, 167, 171-2, 173, 174-8 Lambert, B., 150-2 Landtman, G., 185, 191 Lane, B., 148 Lane, R., 148 Lapita, 8, 87 design network, 87-9 exchange and voyaging network, 121-2, 255 society, 145, 146 Laskar, R., 204 lattices, 251-3 as models of kinship structures, 238, 255,258-60 Lau Islands, Fiji, 9, 13, 35-7, 43, 66-75, 122-4, 165, 166-8, 171-8 Lauer, P. K., 215 Lawrence, D., 181 Leach, E. R., 11, 92, 99-101, 107, 116nl2, 120, 130, 152 Lessa,W.A., 31-3, 34,207 Lester, R. H., 69, 122 Levison, M., 1 Levi-Strauss, C , 16, 21, 61, 62, 98, 125nl, 129, 135-6, 223, 226, 260 Lewis, D., 1, 87, 123-4 Lichtenberk, F., 142 Liep, J., 156 Lilley, I., 1, 192-3 Lingenfelter, S., 33-5 linguistic vs typological reconstructions, 16, 240-1, 247, 248, 259-60 Linnekin, J., 105—6 Lint, J. de, 226 Lounsbury, F. G., 141, 260 Lowie, R. H., 235, 260 Lucas, E., I l l Lucett, E., 53, 56, 207, 209, 210 Mabuchi, T., 101 McCabe, 45n7 McCarthy, F. D., 191 MacGilvray, J., 185 McHugh, J. A., 266 Mcintyre, M., 193, 215 McKinley, R., 161-2 McKinnon, S., 137 male vs female gifts, 39, 101 house areas, 37-8 Malayo-Polynesian languages, 17, 19 social organization, 15, 101, 218, 222-31, 264 Malinowski, B., 141, 213 Marble, D. F., 7, 47-50

293 Marck,J.C, 2, 66,210 Marcus, G. E., 162 Mariner, W., 96 marking rules, 233, 234, 241 marriage isolates, 53, 61-6 Marquesas Islands, 104 Marshall, M., 2, 15-16, 155nl6, 158, 218, 231-2, 234, 235, 238, 241-9, 253-5, 256-8, 264 Marshallese empires, 137-42 Marshall Islands, 11, 13, 91, 110, 124, 125-42, 146, 148, 150, 155, 158-60, 165,178-80,225,256,258 Mason, L., 11, 92, 126-30, 131-2, 133, 135, 137, 140, 141, 180, 257, 263 matrilateral cross-cousin marriage, 21, 69, 97-8, 99-101, 116,122, 135-7, 148, 151-2, 223-4 see also asymmetric marriage alliance matrix adjacency, 3-4, 195 cost, 196-7 distance, 195-6, 197, 230 reachability, 195-6, 230 similarity coefficient, 75-6, 83-4 universal, 221, 230 Maude, H. E., 152-3, 262 Mauss, M., 40 Mead, M., 105, 240 Meggitt, M. J., 189 Micro-Polynesia, 130, 156 Milicic, B., 43 Milke, W., 16, 19, 218, 223nl, 232, 238, 246, 247, 249-51, 253-6, 257, 260, 264 Minangkabau, 135-6 minimum spanning trees, 7-9, 51-2, 263, 265 algorithms, 55-6, 71-2, 78-9, 89, 263, 266-7 and archaeological classification, 77-8, 83-9, 263 clustering in, 58, 83-7, 88, 263 and dialect groups, 57-61, 66, 263 and marriage isolates, 61-6 and matrix methods, 78-80 as models of network evolution, 71-4, 89, 263 Moerenhout, J.-A., 208, 209, 211 Morgan, L. H., 260 Morgenstern, O., 204 mother's brother's widow marriage, 141 Munn, N. D., 215 Murdock, G. P., 11-12, 15, 17, 91, 151nl3, 153-4, 158, 162, 218, 22231, 232, 235-6, 241, 246, 256, 260-1, 263, 264

294

Index

myths of chiefly divine descent, 96, 99, 101, 103 of clan origins, 180 of the Stranger-King, 68-9 Nankanse systems, 15, 218, 224, 225, 231, 264 Nauru, 149 navigators, 160 Near Oceania, 17 Needham, R., 16n6, 38-9, 69, 266nl Nerlove, S. B., 58n4, 233-6, 240, 241, 242, 256, 261, 265 nested sets, 3, 4-5 network, 55 network-breaking model, 9, 52-3 Neumann, J. von, 204 Nevermann, H., 126, 131, 159-60, 257 Nietschmann, B., 184n7 Nietschmann, J., 184n7 Niven, I., 233 nodes adjacent, 23 central, 178 independent, 205 non-Austronesian languages, 17, 232 Norman, R. Z., 2 Nuclear Micronesian languages, 20, 143-5 Oberg, K., 92 Oceanic: hypothesis, 19-20 languages, 19-20 Oellerman, O., 266 Oliver, D. L., 134 100 mile overnight voyaging rule, 210 operations research, 12 Orans, 104n9 ordered pair, 14, 29, 251 ordinary node, 219 Ore, O., 204, 205, 212, 262 Ottino, P., 44, 53, 54, 61-2, 64-6, 207, 208,209,210,211 outdegree, 219 out-tree, 43 Pak, O.-K., 136n6 Panoff, M., 232 Papuan languages, 17 parent digraph, 43 Parker, P. L., 156 partially ordered set, 251 path in a digraph, 219 in a graph, 25

length of, 25 spanning, 25 patrilateral cross-cousin marriage, 98n6 Pawley, A., 2, 9, 11, 17, 19-20, 52, 57n3, 90,107,110,142-3,256 Penrose, R., 9n2 Peoples, J. G., 148 perpetual dichotomy, 4, 35-6, 42 Peterson, G., 143, 146, 147, 161 Pinker, S.,lln5 Pohnpei (Ponape) 124, 141, 146-8, 155, 158, 160-1, 256, 259 Pollock, N. J., 137, 139, 179-80 Powell, H. A., 141, 156 prestige good systems, 30-1, 35, 92, 116-17, 120-4, 159 Prim, R. C, 52, 55, 78-80, 89, 263 product cartesian, 28, 133 elementwise, 83 Proto-Austronesian language, 17 society, 223-4; see also early Austronesian society Proto-Malayo-Polynesian society, 15, 43, 106,218,222-31,262 Proto-Melanesian, 19-20 Proto-Nuclear Micronesian: language, 20, 126n2, 143, 145 society, 10, 15, 91, 101, 145-6, 149, 159, 160, 222, 231, 256, 263, 264 Proto-Oceanic language, 19-20 society, 10-11, 91, 106, 125, 142-3, 256, 263 Proto-Polynesian language, 20 society, 10, 263; see also Ancestral Polynesian Society Pukapuka, 5-6, 105 pyramidal descent groups, 110 Quackenbush, E. M., 157 Quiggin, A. H., 183n6 Radcliffe-Brown, A. R., 98n7, 129-30, 260 radiation model, 52 radius of a graph, 178 Ralik-Ratak, 165, 178-80; see also Marshall Islands ramage, 90, 94-5, 102-6, 150-1; see also conical clan Rappaport, R. A., 262 Rathje, W. F., 200 receiver, 219, 228 recursive complementarity, 4, 38-9, 42

Index recursive dualism, 4, 37-8, 42, 265 Reid, A. C, 1, 165, 174, 176-7 relational vs substantive contrast, 40 relations asymmetric. 251 irreflexive, 251 transitive, 251 Remote Oceania, 17 Renfrew, C. 9, 52, 75-8, 81-7 restricted exchange, 21 Richardson, M., 204, 215 Riesenberg, S. H., 143, 146-8 Rivers, W. H. R., 143, 232, 236, 260 Roberts, F. S., 10, 41, 265, 266 Robineau, C, 56 Robinson, W. S., 75 Romney, A. K., 58n4, 233-6, 240, 241, 242,256,260,261,265 Rosenstiehl, P., 89 Ross, I. C, 215 Ross,M., 17, 19n7 Routledge, D., 69 Rowlands, M. J., 116 Rudeanu, S., 212 Rynkiewich, M. A., 129, 130, 132, 137, 140 Sabatier, E., 151 Sabidussi, 202-3 Sabloff, J. A., 200 Sackett, J. R., 76 Sahlins, M. D., 10, 13, 36-7, 38, 40, 41-2, 68, 90, 91, 101-7, 108, 110, 115, 120nl5, 125, 129, 130, 138, 146, 151nl3, 160, 165, 166n2, 168, 175, 240, 262 Sarfert, E., 148, 206, 207 Samoa, 104, 118, 120-1, 122-4, 152 Sayes, S. A., 68 Schultz, E., 105 Schwartz, T. H., 30 search trees, 9-12, 262 breadth-first, 10, 11, 112-13, 163-4, 226, 228-30, 263 depth-first, 10, 11, 91, 110-16, 127, 163-4, 263 as evolutionary models of social structure, 228-30, 263 and genealogical knowledge, 11, 162-4 and maze solutions, 91, 111, 113, 266 as models of the conical clan, 10, 11, 91,110-16,127,263 segmentary lineage, 4-5, 10, 105, 107 Seligmann, C. G., 213-14, 215 semicomplex marriage systems, 21, 66, 154-5, 161-2, 256 semicycle, 220

295 semilattices, 16, 251-3 as models of kinship structures, 16, 238-9, 253-5, 264 semipath, 220 semiwalk, 220 Service, E. R., 91, 92, 101, 114, 231, 236n6 Sharp, A., 175 Shepard, R. N.58n4 sibling set marriages, 155 Simon, H., A. 1 sister exchange, 66, 149, 151, 154-5, 183 Slater, P., J. 12, 14 Smith, J., J. 248 social and spatial symbolism, 36-7, 130-1, 139, 147 social stratification and central location, 13, 160, 193-4 and control of inter-island networks, 159-60, 178-80 and descent, 222 and economic productivity, 104-5, 152-3, 160 form vs degree of, 104n9 on high vs low islands, 91, 105 on isolated islands, 158-61 and long-distance voyaging, 145-6 network determinants of, 159-61 see also chieftainship; conical clan; prestige good systems; tribute systems Society Islands, 58, 104 Southwold, M., 120 Speiser, F., 124 Spier, L., 235 Spiro, M. E., 157-8, 207 Spoehr, A., 126, 130, 134, 135, 136-7, 257 Spriggs, M., 121 square of a graph, 215 star, 27 Sterud, G., 9, 52, 75-8, 81-4 Stillfried, B., 136n6, 145 Stimson, J. F., 55, 57-61 Strathern, A. J., 189 subgraph, 24 spanning, 24 Sutter, J., 62, 64, 66 symmetric marriage alliance, 146, 161 systems of areal integration, 30 Tabah, L., 62, 64, 66 Tahiti, 58, 60, 105, 120 Tarjan, R., 11 Tarry, M. G., I l l , 113 Terrell, J., 2 Tessier, R., 56n2, 62-3 Thomas, N.I05

296

Index

Thompson, L., 13, 67, 68-71, 122, 165, 166-7, 171-4, 175,202 Tikopia, 100 Tindale, N. B., 215 Tobin,J. A., 159 Tonga, 91-2, 95-8, 100-1, 103, 104, 106, 114-15, 117-24, 125, 127, 129-30, 132-3, 136, 137-9, 142, 143, 159, 162, 175, 256 Tongan Empire, 117-21, 139 Tongan—Kachin solution, 120 Torres Strait, 13, 165, 180-94 Tory Islanders, 162-4 trade-horticulture hypothesis, 181, 183-8 trade networks Admiralty Islands, 213 Archaic Aegean, 194-201 Greater Lauan, 13, 174-8, 264 kula ring, 192-3, 201, 213-15, 217 Mailu, 192-3, 213, 216-17 Marshall Islands, 137-9, 264 Southeast-Solomons-Vanuatu, New Caledonia, 45-6, 50 southern Lau, 13, 166-8, 171-4, 202, 264 Tonga-Fiji-Samoa, 122-4, 201 Torres Strait, 13, 165, 180-94, 264 Vitiaz Strait, 192-3, 213 Western Motu, 213 trail, 25 Tran-Ngoc-Toan, 62 transmitter, 219 Trautmann, T. R., 69, 120 trees, 3, 29-30, 46-7, 263 binary, 4, 40-3 and classification systems, 40-3 and communication structures, 32-4 degenerate, 234 labeled, 111-12 ra-ary, 41 as models of cognatic descent, 43-5 as models of kinship categories, 234-5, 237, 239, 242, 243, 244, 254, 257, 258, 259 plane, 40, 111-12 rooted, 3, 30, 40, 110-12, 251, 263 spanning, 5-7, 47, 263 ternary, 41 see also minimum spanning trees; search

tribute systems Lakemban, 70-5 Yapese, 30-5, 205, 207 see also prestige good systems Trobriand Islands, 121, 141, 156 Truk (Chuuk) 66, 149, 153-7, 158-9, 222, 225, 256 Tryon, D. T., 17 Tuamotu Islands, 9, 44-5, 52-66, 203, 207-12, 264 Turkish solution, 120 Ueki, T., 148 Vanuatu (New Hebrides) 45-6, 50, 88-9, 121, 124, 142, 143 Vayda, A. P., 262 voyaging networks Archaic Aegean, 194-5, 197-201 Caroline Islands, 206-7 Lapita, 87 Marshall Islands, 138-9 Tuamotu Islands, 210-11 see also trade networks Wahlund, S., 62 Walikar, H. B., 204 walk in a digraph, 219 in a graph, 24-5, 195 vs path, 25 spanning, 219 Walker, D., 180 Wang, J. W., 146 Ward, R. G., 1 Watkins, J. J., 78-81 WebbJ.W., 1 Wedgwood, C. H., 149 White, D. R., 12 White, L. A., 91, 108-10, 115, 130 Wilkes, C, 123-4 Williamson, R. W., 240 Wilson, R.J., 78-81 Wilson, W. S., 148 Winkler, Captain, 139 Wouden, F. A. E. van, 101, 223 Yap, 30-5, 158 Yapese Empire, 30-5, 121, 122, 124, 205, 207 Young, J., 175