From a mathematical perspective many problems in anthropology concerned with the analysis of structures, patterns, and configurations are combinatorial in nature. There are three types of combinatorial problems:

1. The existence problem asks, “Is there a structure of a certain type?”

2. The counting problem asks, “How many such structures are there?”

3. The optimization problem asks, “Which is the best structure according to some criterion?” (Roberts 1984).

The minimum spanning tree problem (MSTP) is an optimization problem, well known in many fields; its history is detailed in Graham and Hell (1985). The problem has applications to the design of all kinds of networks, including communication, computer, transportation, and other flow networks. It also has applications to problems of network reliability and classification, among many others. Our purpose here is to describe some applications of the MSTP to anthropology – in particular to problems of size, clustering, and simulation in networks of various kinds. We proceed by presenting in a unified format the three standard MST algorithms of Kruskal (1956), Prim (1957), and Boruvka (1926a,b), describing the advantages and some of the applications of each one.

We first illustrate the MSTP intuitively, as follows. A large corporation with offices in many cities, v1, …, vn, wishes to determine the monthly telephone charge. All the distances d(vi, vj) are known and are distinct.

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.

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.

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. 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 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 (Krämer 1906; Krämer and Nevermann 1938; Erdland 1914; Mason 1947, 1954; Spoehr 1949a, b; Kiste 1974).

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.

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.

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);

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).

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:

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:

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 and Harary 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.

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, v1, …, vp. All the distances d(vi, vj) are known and are distinct. The corporation G does not wish to pay the phone company an amount proportional to the sum ∑d(vi, 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.