Controlled Simulation of Marriage Systems
draft for publication in the Journal of Artificial Societies and Social Simulation
Douglas R. White
University of California, Irvine
Introduction
One of the issues in the study of family systems is how to identify rules, structures, and strategies that give nonrandom inflections to mating systems. Probability models include (uniform) random partner selection, in which each person in a group has an equal opportunity of being selected, as distinguished from probabilistic partner selection, a process that takes place randomly with certain probabilities. One of the questions raised by anthropological studies is whether there are determined structures, related to marriage rules and/or strategies, within which random marriage choices occur, such as uniform random partner selection for any given pair of positions in the structure. This article proffers a quantitative strategy for examining such a question. To the extent that it is successful, it could help to explain certain of the ties between kinship, economics, and political structure in more empirical terms.
A Network Approach
The problem of evaluating marriage structures in relation to marriage rules is complex because we may need to specify positions within a network of kinship and marriage relations that exists prior to any given marriage. Adding to the complexity is the variety of mating and marriage behaviors, including sequential marriages or polygamy. This complexity is further augmented by the fact that language categories, verbal and normative statements and differential sanctions may exist for proper or improper marriage choices and these different aspects of marriage "rules" may not correspond. To deal with this complexity, an approach to random baseline models is proposed that establishes for each case study a partially ordered classification of potential marriage partners stemming from their position within a network of kinship and marriage relations, and specifying those classes within which uniform random partner selection occurs. This is done by a simulation of marriage choices that imposes constraints on marriagability by classificatory positional relationships, and then assigns spouses within marriagability classes by random permutations under uniform probability of assortment.
The major hypothesis of the simulation is that once normalization is made against a random baseline suitable for each individual case, the observed agreement between the positions in the existing kinship and marriage network, language categories, and verbal norms may be greater than previously thought to be the case. Some verbal formulae are models "of" behavior, while others are models "for" behavior and may include adjustments for departures from established norms. Internal cultural variability may also turn out to be a key factor for understanding marriage patterns. Case studies drawn from Indonesia, Sri Lanka, and Austria are analyzed to test these ideas about random baseline norms in the context of establishing evidence for different marriage rules and strategies.
The Model
The key innovation of this study is a method for identifying a structure of social groupings on which to map uniform partner selection probabilities between pairs of positions in the structure and, by extension, how to decompose the statistical variance in marriage structures. A uniform marriage structure is a partition P of the marriages in U, the observed population of marriages, into subgroups in which uniform probabilities of partner selection are assigned as a function of the subgroup memberships of potential pairs of partners. Some models will take the form of a partially ordered uniform marriage structure where for a series S of subsets of marriages in U, any pair of distinct subsets X and Y in S will either be mutually exclusive or the set of elements of one subset will be contained in those set of the other, e.g., X Ì Y. S will relate to P in that S is made up of sets that are either a subset in a partition P or the union of two or more subsets in the partition P.
An Idealized Example for "Simple" Systems
Figure 1 gives in diagrammatic form an example of a "simple" partially ordered marriage classification structure of a matrimonial moiety system. The oval in Figure 1 is a maximal set E of marriages that are structurally endogamous in that every marriage in E has parentchild links to at least two other marriages in E (to parents of the husband, parents of the wife, and/or to one or more children’s marriages – where at least two of these are also in E). Sets 1, 2, …, n (horizontal boxes in Figure 1) represent mutually exclusive genealogical generations within U and E, and the sets a, b (semiovals in Figure 1) represent a moiety division within E such that for any marriage in E, if the parents of the husband come from one moiety, the parents of the wife come from the other moiety, and vice versa (hence sets a and b are also mutually exclusive). The sets a_{1}, b_{1}, a2, b_{2}, _{ …},_{ }a_{n}, b_{n} are mutually exclusive subsets of E that represent the intersection of the moiety division and the generational division. More precisely, let the set of sets M ={ a_{1}, b_{1}, a2, b_{2}, _{ …},_{ }a_{n}, b_{n} } consist of the Cartesian products of the set of sets A = {a, b} and the set of sets G = {1, 2, …, n}; that is, M = A X G = {a, b} X {1, 2, …, n}. The structure satisfies one set of criteria for a partially ordered marriage structure if we take the partition P to consist of the sets in M, all of which are in E, and the residual set UE (and, if wanted, its generational subsets). All the sets of marriages in A, G and M are either subsets in partition P or the union of two or more subsets in P.
The crucial feature of the model for statistical testing purposes is a moiety rule that allow us to place observed marriages in the structure and to determine how well observed behavior fits the uniform probability model in comparison to any other models that posit a simpler structure. Let us assign the moiety rule that children are members of their father’s moiety and belong to the succeeding generation. By this criterion each person can be placed in a premarital category included in the set M ={ a_{1}, b_{1}, a2, b_{2}, _{ …},_{ }a_{n}, b_{n} }, which, in the case of males, is predicted by the moiety rule to be the same as their postmarital category. Females, though, will preserve generation but switch moieties at marriage. In other words, males preserve generation and moiety whereas females preserve generation but switch moiety relative to their fathers.
(Insert Figure 1 about here)
A strict moiety model, with no other criteria affecting marriages, sets a uniform probability of marriage between any opposite sex pair of individuals x, y such that x Î a_{g} and y Î b_{g} where g Î {1, 2, 3, 4, 5, 6}, and a_{g}, b_{g} Î M; and all other marriage probabilities are zero. Given data for a marriage network, P, generated by a strict moiety model, we could determine inductively the set E of endogamous marriages and the subset G of generations, and within E we could determine inductively the moiety rules for A = {a, b}. Note that for marriages in E with one or more known parents, the structure of subsets G and A X G is partially ordered. Holding E and G constant, we can test the hypothesis that P is not a moiety structure by the null model where for any opposite sex pair of individuals x, y such that x Î g Î G and y Î g Î G the marriage probability is uniform. This is most easily done using a permutation test where married individuals in generation g are decoupled, and their couplings randomly permuted. Since individuals are classified by the set of moieties A = {a, b}, the null model for "no moieties" can then be compared statistically with the observed data and an appropriate correlation ("strength of the moiety tendency"), as well as significant tests, can be made. Another null model could be constructed for comparisons with different kinds of generational marriage structures.
The remainder of this paper provides a generalization of this approach to determining marriage structures inductively regardless of the type of structure, with the moiety structure as a special case.
Limitations of the Classical Approaches to Studying Marriage Systems
Anthropology has long concerned itself with marriage patterns, especially where variable cultural rules and marriage strategies shape mating regimes. Some of humanity’s fundamental social concerns are with marriage rules and strategies – ranging from incest and exogamy on the one hand to social norms and strategic interest on the other. Alliance theory takes the study of these features as central to understanding social organization. Their discussion in ethnographic case studies is nearly mandatory, and structuralists like LéviStrauss define them as paradigmatic to cultural systems. LéviStrauss’s theory of elementary, complex, and semicomplex kinship systems, briefly put, is that elementary systems prescribe "positive rules" of marriagability in terms of classes of relatives (including sections, moieties, etc.). Complex systems are characterized by "negative rules" such as incest prohibitions. Intermediate between them, semicomplex systems are defined by such extensive proliferation of "negative" proscriptions that individuals in similar kinship lines must disperse their marriage with other lines in a series of residual categories. This residue of marriageable relatives takes on the flavor of a prescriptive "elementary" system, but disperses marriages.
In spite of the centrality of these concerns with rules for marriage and against incest, anthropology has no established methodology for evaluating marriage strategies against random baselines. LéviStrauss understood correctly that terms such as marriage "preference" or marriage "avoidance" were relative terms when speaking statistically. If evidence for marriage preferences is established by "occurrence with greater frequency than expected by chance under the null hypothesis" then this evidence is entirely dependent on what probabilistic model is used for the null hypothesis. Thus, LéviStrauss preferred to speak of ideal models and to describe the models either as "simple" or "elementary systems" if they contained only deterministic rules (probability 1 or 0, respectively, of an individual marrying within a category), or as "complex systems" if all of their rules were probabilistic. Prescriptive marriage rules that imply proscription or avoidance of alternative marriage possibilities are thus "elementary" systems as ideal models, but LéviStrauss considered it improper to speak of "elementary" systems as having only certain preferences towards various marriage rules. Proscriptive marriage rules ("avoidance" of marriage with certain types of relatives), however, may proliferate to such an extent that they almost imply the existence of unstated prescriptive categories. LéviStrauss defines his "semicomplex" systems thusly. LéviStrauss’s reluctance to enter into a statistical evaluation of marriage structures in terms of "preferences" is similar, if not identical, to the problem of which "random baseline" model should be used to evaluate departures from null hypotheses.
Hammel (1976b) challenged LéviStrauss somewhat naively, by raising the problem of interpreting marriage frequencies. What he actually challenges is the use of raw frequencies of matrilateral cross cousin marriage (often taken as indexical of asymmetric and generalized exchange in "elementary systems"), relative to frequencies of other marriage types as evidence of marriage preferences. Evidence for preference or strategic behavior in marriage choices, however, is complicated by demographic constraints on marriage choices. Hammel shows the defects of reaching conclusions based on comparing raw frequencies of observed behavior. He concluded from simulation studies that many of the observed raw frequencies of matrilateral cross cousin marriage, upon which monographs and comparative theories have been built, are well within the range expected from a random distribution of marriages within a population if certain demographic features are held constant. He showed that age, status, or other systematic differences between potential husbands and wives create a relative age demographic bias that skews the potential marriage pool under a random mating regime given these demographic constraints so as to raise the raw frequency of matrilateral above that of patrilateral cross cousin marriage. This result undermines many of the writings on alliance theory in the U.S. and Britain, where marriage structures are often interpreted in terms of frequencies of different kinds of marriage choices. Given this kind of interpretation, to the extent that certain kinds of marriage frequency outcomes are shown to be the result of demographic effects or biases, the edifice of an empiricist alliance theory would seem to fall along with the demise of cherished assumptions about marriage rules and strategies that derive from behaviorist interpretations of LéviStrauss’s theory of marriage structures.
What, then, is a marriage rule or strategy? As in the moiety example given above shows, the question of how frequencies of behavior can provide evidence of marriage rules or strategies can be approached hierarchically within the partial order model. At a first level, for example, we might have rules of impossibility – proscriptions against marriage with dead people, for example, or people who have no interaction with members of the population. A second level might specify rules that establish an effective breeding population. Ethnicity or social class, for example, might establish high probabilities of endogamous marriages compared to low probabilities across such social boundaries. Within an effective breeding population there might be further specification of marriage probabilities given the respective ages or age classes of the potential partners, and so on.
Hammel’s (1976b) critique of LéviStrauss appears to be against "social choice" in marriage in favor of the idea that demographic structure affects partner selection. I address here two problems. One is how to assess the evidence for "social choice" (preferred marriage rules or strategies) given demographic baseline models that skew the probabilities of selecting different types of partners. Since demographic constraints and "social choice" are not mutually exclusive of one another, this aspect of the present work incorporates Hammel’s idea and is a continuation and extension of it. The other problem is a subtler one. It is not simply that network structures affect partner selection, as one anonymous reviewer put it. Rather, it is how "social choices" (preferred marriage rules or strategies) are aggregated, under demographic constraints, into network structures, and how the network structures of kinship and marriage structure can be statistically disaggregated into demographic and social choice components.
At each successive level in a partially ordered marriage model, goodness of fit can be established between probabilistic partner selection given the relative social positions of the potential partners as defined at that level, and the observed marriage frequencies. For the fixed probability of selection given these positions, partner selection is assumed to be random. Significant departures – rejection of the null hypothesis for the model at a given level – from uniform probabilities (in which each pair of persons in the respective groups have an equal opportunity of being selected as partners) may represent evidence of additional, as yet unspecified marriage rules or strategies. Preferences or avoidances are thus always relative to some prior marriage probability model. Two possibilities present themselves: (1) observed data may have a higher or lower uniform probability than predicted by a given model, or, (2) a further preferential or avoidance rule not already specified may demonstrate nonuniform probabilities between two categories of potential partners. Thus, a ‘preferential’ strategy needs to be evaluated by rejection of the null hypothesis by positive deviation from a given random baseline; an ‘avoidance’ strategy by a similar but negative deviation (and, potentially, a ‘randomizing’ strategy needs to be evaluated by its fit to a random baseline).
How are the categories of partners specified in such models? One possible answer is attributional: people with bundle X of attributes tend to marry those with bundle Y of attributes with a certain probability. A second possible answer stems from the relations of the social network. People related to others by relationship X tend to marry one another with a certain probability. Difficulties with the former and advantages of the latter are discussed in turn.
Defining the General Phenomena: Difficulties of the Attribute Approach.
The attribute approach usually entails a typological classification of a population, and analysis of various tendencies towards endogamy, exogamy, isogamy, anisogamy (hypergamy, hypogamy), or "alliances" between particular groups, based on such categories. Many models of marriage rules are formulated at this level, especially where lineages provide the typological attributes that classify the population. Attributional models, if they are handled statistically, are usually analyzed by one of two means. The first is straightforward regression analysis that considers how much of the variance in whom one marries (the dependent variable) is due to a specified set of attributes and interactions (the independent variables). A second approach developed by Romney (1971) takes into account demographic differences in the sizes of the various groups involved. Romney considers the aggregate intermarriage matrix for a given categorical typology (a matrix of marriage frequencies, say, between lineages, with wifegivers as rows and wifetakers as columns), in which different row and column sums reflect the relative demographic strengths of the social units in terms of numbers of men and women. He observes that if a large group were to have a preference to marry into a small group, they could not do so fully because of their numerical imbalance (for an optimization model approach relating more directly to preference orderings, see White 1973: 402410). Therefore, Romney removes the effects of demographic imbalance by a doublenormalization of the matrix to produce uniform marginal totals with proportional rates in the cells relative to "demographic balance" assumptions. This is a form of simulation or random baseline comparison since under independence assumptions the expected value is uniform in every cell in a doublenormalized matrix. For any 2 by 2 diagonal subtable of this matrix, such as Table 1, the cross product ratio alpha, a = ad/bc, is a measure of the tendency towards endogamy (a >> 1) versus exogamy (a << 1), with a =1 as the random baseline.
(Insert table 1 about here)
Further, with an off diagonal subtable such as Table 2, alpha is a measure of tendency of the XA and YB pairs to ally by intermarriage (a >> 1) or to avoid intermarriage (a << 1). Romney’s method takes only one type of demographic control into account: numeric imbalance between groups.
(Insert table 2 about here)
There is a more general approach, however, that can take other controls into account by a simulated random baseline, as shown in Table 3, and computation of a baselineadjusted ratio alphastar, a * = (ad/ bc)/(eh/fg) = adfg / bceh.
(Insert table 3 about here)
These exercises raise two fundamental problems with the attribute approach. One is that all of these approaches involve aggregation of individual level data in which there are a multitude of ways to aggregate people into groups on the basis of attributes. Even in the case of kinship groups it is not always clear where the boundaries of the exogamous sublineages might be located within the larger clans or lineages (e.g., How many generations deep does the prohibition extend? How to break the sublineages when they are hierarchically nested with different ranges of proscriptions for different members?). The only possibility of clear specification is (1) if people in the society have very precise rules demarcating the boundaries of endogamy, exogamy and alliance groupings, (2) if people, moreover, follow these rules to a high degree, and (3) if the marriage prohibitions fall into mutually exclusive sociological groupings, which is not usually the case even in lineage societies! If each of these conditions were met, however, we are approaching something like the "elementary" pole of marriage systems.
In the general case, the attribute approach ought to be able to model the underlying assumption of a complex marriage system, namely, that people’s marriage choices vary probabilistically across different categories of people. Trying to specify "rules or strategies" with appropriate attribute categories for such systems is fraught with the difficulties of potential specification errors resulting from incorrect aggregation, nested hierarchies or nonexclusive categories, and lack of consideration of relational or networkbased rules or strategies.
Specifying the General Phenomena in Network Terms
The bias of our society – a complex marriage system, in LéviStrauss’s terms – and our liberal professions is to see marriage in terms of different social categories within a universalistic, antiparticularistic, egalitarian perspective. Yet most marriage rules and strategies are formulated in terms of marriage with those with whom a preexisting link is recognized such as common ethnicity, class, kinship, community, neighborhood, school ties, etc. Some of these links, like connections through prior kinship or affinity, are directly relational. Some, like neighborhood, are category proxies for spatial proximity, that are more clearly specified in terms of pairwise proximities (distances) that create probability distributions of interaction. Interaction is a clear prerequisite to acquaintance, courtship, marriage, having children, etc. As for many of the remaining links that appear to be categorical (ethnicity, class, community, etc.), it would be extremely interesting in the general case (including "complex" marriage systems) to know whether ethnicity, class, and community (hence school ties varying in accordance with these three sociological variables) are not themselves constituted to a large extent by intermarriage. We can generalize and universalize LéviStrauss’s insight about the "closed form" of elementary systems by saying that the canonical forms of marriage alliance, marriage rules, and marriage strategies are those individual marriages that relink families already linked. To be more precise, marital relinking exists when partners are already related prior to their marriage. The concept of marital relinking was invented by French ethnographers such as Jola, Verdier and Zonabend (1970), and further explored by Segalen (1985), Richard (1993) and others.
Indeed, LéviStrauss (1966) speculated on the possibility of endogamous "demes" on a scale, say, of 220,000 or more persons, as characteristic of modern urban populations and rural villages of modern states. His speculations motivated French ethnographers to investigate and identify matrimonial relinking as a marriage alliance strategy employed in the "complex marriage systems" of French farming villages. Richard (1993) developed a statistical method for assessing the occurrence of relinking in a population. Brudner and White (1997) identify class formation in rural Austria with the marriage strategy of relinking.
To articulate the concept of marital relinking to marriage rules, strategies or structure, it can be said that the minimum necessary form of marriage alliance, rule or strategy, at the broadest level, involves some kind of endogamy, and that endogamy is minimally constituted through relinking. Note here the similarity to LéviStrauss’s idea that a marriage is not an alliance, an elementary form, unless it is in recognition of some preexisting relationship between the families. An alliance is not instituted by a simple marriage contract between man and wife but constituted upon a preexisting set of social relations, expectations, obligations, and/or privileges, existing between two groups, the givers and takers of the bride and reciprocally of the groom.
Decomposing Endogamy into its Constituent Relational Structures
The concrete form of endogamy that is constituted through marital relinking is not categorical endogamy but structural endogamy (White 1997; Brudner and White 1997). To be more precise, structural endogamy consists of a bounded set of marriages within which each pair of marriages is connected by two or more independent paths of parent/child links, and such that the set is maximal, that is, there are no other marriages that could be included in the set and satisfy the definition.
Structural endogamy provides the starting point for a simpler solution to the problem of specifying marriage rules and strategies than the attributional approach because it is a concept that implies emergent social boundaries. To apply the concept of structural endogamy we must resist the temptation to immediately aggregate our data. By doing so, and by retaining a definition of endogamy in terms of links among individual marriages, we can adopt techniques of network analysis to identify the boundaries of structural endogamy.
Structural endogamy is a relational concept that yields a unique decomposition or classification of sets of endogamous marriages in a population graph in which marriages – more generally, sexual unions – are represented by the nodes of a graph, and persons are represented by lines connecting marriages via parentchild links. The existence of sexual unions in a population implies that each of the two spouses in a given marriage links the marriage to a sexual union of their respective parents. In sociology a family of orientation is the unique, originary parental node of an individual, while the family of procreation is one of the several possible nodes of ego’s activity as parent. Every married person links a family of orientation to one or more families of procreation. If we generalize this idea and add extra nodes for unmarried children who link their "personal node" to that of their parents as their node of origin, we have the graph theoretic construction known as a pgraph (White and Jorion 1992).
The pgraph is an empirical representation of the reproductive and marital structure of a population. Some parental information will always be missing or outside the population boundaries, hence the representation is finite. Empirically speaking, a pgraph has two kinds of nodes: personal nodes for single children and marital nodes for marital unions of two children (generally of opposite sex and of different parents), and two kinds of arcs (directed edges): male and female, directed from parental nodes to child nodes. Formal speaking, a pgraph is an asymmetric and acyclic digraph with two kinds of arcs, and maximal indegree of 2, assuming arcs orient from parent to child (if the reverse, outdegree 2 is implied). In defining the boundaries of structural endogamy, however, the direction of the arcs in the pgraph is disregarded. Although parentchild links are directed, it is the undirected skeleton of the parentchild network among marriages that is relevant to structural endogamy. A structurally endogamous block of the skeleton of a pgraph is formally equivalent to a bicomponent of the graph since in a bicomponent two or more independent paths connect every pair of nodes.
Figure 2 illustrates the concepts of the components, bicomponents and tricomponents of graphs. A component of a graph is a subgraph of connected nodes that is maximal (as large as possible). A bicomponent (tricomponent) of a graph is a subgraph in which every pair of nodes is connected by two (three) or more independent paths. The figure shows a single graph that has three (disconnected) components, three bicomponents (two connected and one disconnected), and one tricomponent. In a bicomponent, every pair of nodes is necessarily connected by a cycle. The skeleton of a finite pgraph cannot be composed of tricomponents since this would imply that every node has at least one child (to reach the required degree of 3), which generates an infinite digraph. Hence structurally endogamous blocks of the skeleton of a pgraph are formally equivalent to bicomponents, never tricomponents. Two or more bicomponents can have at most one node in common, hence the same is true for structurally endogamous blocks. Thus, only single nodes or trees of linking nodes can connect bicomponents.
(Insert Figure 2 about here)
The graph in Figure 2 illustrates the concept of structural endogamy if we regard the nodes as marriages linked by undirected parent/child links. In the leftmost component of Figure 2 there is no structural endogamy (no bicomponent), since the removal of any node (marriage) increases the disconnectivity of the graph. In the middle component there are two overlapping blocks of structural endogamy. They have one marriage in common whose removal increases the disconnectivity of the graph. The component to the right has a bicomponent, and within it, three or more independent paths connect every pair of nodes in a tricomponent. A finite tricomponent of a pgraph cannot possibly represent biological kinship, so this subgraph must necessarily have some fictive or sociological constructed element that is not based on biology. Note that in a graph of linked marriages, a cycle implies a marriage that relinks marriages already linked by some prior path of personal connections.
Structural endogamy, then, is uniquely well suited – both theoretically and empirically – as a starting point for a precise and unambiguous specification of the problem of identifying the largest, bounded empirical units of endogamy within a genealogical network. It yields an aggregate social unit at the most generic and extensive level when decomposing the partially ordered levels of marriage structure. It is within this largest structural unit that more specific rules or strategies of marital relinking operate, since, by definition, every relinking defines a cycle that is inside a bounded unit of structural endogamy. All specific relational marriage rules and strategies, such as consanguineal marriages, redoubling of alliances between lineages, dual organization, etc., of necessity, imply some form of relinking and hence subblocks of structural endogamy.
If some subset of marriages is strategic or rulegoverned in some particular way and at some "level" in a partially ordered uniform marriage structure, we would expect this subset to have nonrandom characteristics in comparison to a uniformprobability model at that level. Marriages of similarage spouses, for example, appear nonrandom against a uniformprobability model across all age groups. In a partially ordered uniform marriage structure model, each test of marriage structure at a given level needs to be made independently, but holding constant the marriage structures at a more basic level. If, for example, after identifying a specific agebias in marriages, a uniform probability model predicts with high probability a narrow sizerange for the largest structurally endogamous block, this does not imply the lack of other marriage rules or strategies that might be identified against this randommodel backdrop. Sometimes smallscale rules and strategies may be masked in larger phenomena that resemble a nearrandom relinking model of structural endogamy.
What are some of the smallerscale relinking patterns to be found within structurally endogamous blocks? One way to classify new marital relinkings between those already connected is by the type of prior personal connections. Personal connections may involve blood (parent/child or sibling/sibling connections) or marriage (husband/wife connection). A blood marriage (1family) relinking is one where there exist prior personal connections between spouses involving only blood (hence the spouses are blood related). An affinal (2family) relinking is one where there exist prior personal connections involving only one prior marriage link. Multiple (k) family relinkings involve prior personal connections with multiple (k) prior marriage links. 1, 2, or k family relinkings are local structures, in that they are defined by characteristics of a single, marital relinking cycle. One or multiple sets of cycles may be characterized by effective limits on the genealogical depth involved in strategic or rulegoverned relinking, the density limits (high and low) of cluster of cycles, or global properties such as dual (Houseman and White 1998b) or segmentary dual organization (White and Houseman ms.). Other than the nodes or trees that connect blocks, all global structures need to be evaluated by the characteristics of the circuits within them. In dual organization, for example (see White and Jorion 1992, 1996), circuits can be drawn as a bipartite graph where all the connections are between two supersets of nodes .
The partially ordered character of marriage structures, and appropriate methodologies such as hierarchical decomposition, derives from the network property that all marriage structures (blood marriage taken as 1family relinking, redoubling of alliances taken as 2family relinking, etc.) are contained within (1) the larger spheres of the structurally endogamous blocks of relinking, and (2) subblocks with higher densities or other identifiable structural properties; otherwise, there remains only (3) the tree structure of the paths of articulation among the structurally endogamous blocks.
The question posed here is: How to define the general phenomena of marriage rules and strategies empirically to make it possible to avoid misspecification? That is, how to statistically decompose each of the possible levels of the problem and each of the possible "main effects" and "interaction effects," as it were, using the language of analysis of variance?
Canonical Representation of Kinship and Marriage Network Data
Circuits of matrimonial relinking have rarely been taken as canonical objects of study for marriage rules and strategies. To do so, one needs a representation of kinship and marriage networks in which matrimonial relinkings will always form identifiable circuits, and identifiable circuits are always and only associated with matrimonial relinkings. When such is the case, as with the pgraph representation, then graph theoretic analysis will be truly of value to the study of partner selection, population and demographic structure, marriage structures, kinship, and network approaches to social organization. The pgraph (White and Jorion 1992, 1996) is the only known graphic solution that fits these requirements. By taking as its nodes couples and single individuals, and as its arcs the concrete persons who link parental nodes to their descendants, there is no redundancy in the graph other than that created empirically by relinking marriages. All such redundancy then becomes a measure of cohesion related to "social structure" when patterns are recurrent and to "social organization" where new social choices or demographic constraints lead to diverse and potentially changing social structural outcomes. The pgraph is thus conceived as a measuring system for structural endogamy and other aspects of marriage systems, in which "relinking" and its recombinatory possibilities are major elements of structure.
The application of graphtheoretic algorithms to identify structurally endogamous bicomponents of nodes involved in matrimonial relinking represents a straightforward solution to the problem of precise decomposition of the aggregate social units potentially involved in marriage rules and strategies. These algorithms are capable, through depthfirst search procedures, of identifying all bicomponents in a network in a computational time that is a linear function of the number of nodes in the network (Gibbons 1985). Hence, the bicomponentfinding methodology is applicable to networks of any size, and further structural analysis of marriage rules and strategies will be bicomponentspecific. Populations of extremely large size can easily be decomposed into bicomponents and marriage structure can be described within each such unit separately.
Bicomponentdecomposition thus solves the basic problem of contexts for globalstructure specification of models for analysis, assuming that we know how to identify structure within bicomponents, which will be developed here. In the analyses that follow, I use programs for computing frequencies of local structures such as blood marriages (ParCalc: described in White and Jorion 1992), for comparing actual structural endogamy versus simulated results (ParBloc: described in Brudner and White 1997), and for computing withinblock global structures such as dual organization (PGRAPH: described in Houseman and White 1998a, b; White and Houseman ms.; PGRAPH is the pgraph drawing program). For the statistical tests used, see White (1994) and White, Pesner and Reitz (1983). In addition, programs exist for analysis of 2family relinking (ParLink: described in White and Skyhorse 1996), and for estimating relatedness and inbreeding coefficients for individuals in a population (ParCoef: forthcoming). These algorithms address the question posed of how to define the general phenomena of marriage rules and strategies empirically so as to avoid misspecification of marriage models. The two problems remaining are: (1) How to establish meaningful simulations for comparison of these results against demographic and random baselines? and (2) How to evaluate statistically the computational findings about structural characteristics or frequencies of various types of marriages against demographic and random baselines?
Simulating Comparative Random Baselines
Hammel (1976a) and Lang (1995) developed population simulations that are sufficiently disaggregated so that we can evaluate outcomes at the individual level in terms of the frequencies of different types of marriage under a random mating regime by specifying a range of demographic parameters and constraints. Hammel (1976b), however, eventually used his simulations in an act of magical "handwaving" that dismissed alliance theory as a serious enterprise. Lang’s (1995) simulation software, available for PCs, produces as output a population of men and women with birth dates and sibling sets through matrilineal links, where fatherhood is as yet unassigned. Hence, one could simulate a population under demographic constraints, and then assign marriage and/or paternity under another set of constraints (e.g., relative age, sibling or cousin avoidance, etc.). This approach, which requires a whole series of demographic parameters that would need to be matched to an empirical population for verisimilitude, is unnecessarily complex for the problems of marriage structure posed here.
There is an easier solution to the problem of simulation and to the complex problem of what constraints to use to establish random baseline "verisimilitude" to the empirical population under study. This is the approach of structural or permutational simulation. Say one has an observed network of parental links ordered by generation in a pgraph format. "Generation" is an empirical attribute of the marriage structure, and how generation is determined algorithmically is discussed below. How can one create a "control" population that shares as much structure as possible with this network but with one crucial difference, namely a random mating regime? Briefly, the idea of structural (or permutational) simulation is the following: Hold constant the ancestral tree generated by parental links through one gender, and then, within each successive generation, randomly permute the marriages that generate parental links through the other gender. Let us say that female descent lines will be held constant in the random baselines generated for the case studies in this article. Then, in each generation, one detaches the sons from their marriages, creating a marriage pool of potential mates who marry within this precise set of women. Husbands are then randomly reallocated from within this pool – they are by definition "marriageable" for that generation – to regenerate a total kinship and marriage network for this population with everything held constant (including sibling sets) except for a random marriage regime within each generation. One can also randomly reallocate both the son and the daughter marriages, keeping the sibling sets and marriage pools the same, but randomizing the ancestral structure in successive generations. The only additional parameters that need to be specified are the extent of prohibitions (e.g., brother/sister, and various prohibitions on first and second cousin marriages, for example). Because the simulated network is randomly constructed in order of successive generations, it is possible to specify rules, as each new generation is constructed, such as avoidance of certain kinds of cousins or other relatives (the network of relatives emerges from the random allocations of ties at earlier generations). Or, while not pursued here, one could use probability distributions defined in terms of classes of relatives, kinship distances from ego, and so forth.
Figure 3 shows an illustrative pgraph of three generations and 18 marriages and a permutation of the male links between adjacent generations. Note that the sibling sets are unchanged after permutation. The upper left hand node, for example, has one female and two male children (descending lines) in both the original and the permuted graph. In the permuted graph, males have been randomly assigned new spouses in the appropriate generation, but the female links between nodes are unchanged, as are the female lineages (female lineages are deliberately simplified in this figure by having each marriage produce exactly one daughter). The permutation represents a random mating regime where the number of marriages in each generation, the female lineages, and the size and gender composition of sibling sets in each generation are held constant. Note that the kinship links among nodes have now been radically altered. The marriage in the lower left in the original graph, for example, was a relinking between a man and his FaSiHuSiDa. After permutation, this marriage is one between a man and his FaBrDa.
(Insert Figure 3 about here)
Figure 4 illustrates how the permutation might differ if we permuted both male and female links. Note that again, the number of male and female children for every node in the original and permuted graphs is the same, since the permutations affect only the marriages of each person, not their parental node. This permutation, then, represents a random mating regime whether the number of marriages in each generation, and the size and gender composition of sibling sets in each generation are held constant. The structure of female as well as male lineages has been permuted. It is preferable to do simulations permuting only the links of one gender, however, because one additional feature is controlled (lineage structure of the nonpermuted gender) by further mimicking of the original data.
(Insert Figure 4 about here)
"Generation" is an empirical attribute of the pgraph marriage structure of a population because, unlike individuals (who have multiple marriages), each marriage is uniquely located in a partiallyordered generational structure defined by parentchild links. Because the pgraph is a partial order, it has a minimal number of generations. PGRAPH and the Pajek program (Batagelj and Mrvar 1997), both of which draw genealogical networks in pgraph format, use an algorithm for computing generations that minimizes the total sum of generational differences between the generational levels assigned to parental marriages and their children's marriages in the pgraph. In some societies, the average difference in generational level can be reduced to its minimum value of 1, in which case all marriages are unambiguously within the same generation. More commonly, there are some generational asymmetries in certain marriages, such as uncle/niece or FaFaBrWiDa marriage. Such marriages can still be placed within the overall minimum of distinct generations used to represent the marriage network, but they introduce a limited number of ambiguities as to whether to place certain marriages higher or lower in the structure. There is no such ambiguity for an uncle/niece marriage, but for a FaFaBrWiDa marriage, there is some ambiguity as to where to place the FaFaBr’s and FaFaBrWiFa’s marriage relative to the others. The generational algorithms resolve this ambiguity by locating the more ambiguous of the children's marriages first and placing them as close as possible to unambiguous parents’ marriages; then any ambiguous parents’ marriages that remain are placed as close as possible to their children's marriages. All such placements are made without violating the rule that parental marriages must be above – precede – those of their children. In most populations the marriage network is sufficiently dense that there is little ambiguity, and the variance in "random" placement within the minimal partial order constraint is very low.
Permutational simulation is done within the PGRAPH kinship analysis program (Windows version, 1999). Data can be entered in various formats and transformed to the native PGRAPH format described by White and Jorion (1992) and illustrated in Figure 5 for one of our case studies. The format consists of a parameter card with the number N of marriages and the dataset title, followed by five vectors each containing N blank separated numbers. The first gives the couple numbers of the husband's parents, where marriages are implicitly numbered 1N (hence the number j in position i of the vector indicates that couple j are the husband's parents for couple i). The second gives the couple numbers of the wife's parents. The third and fourth give the husband's and wife's individual id numbers, respectively. The fifth gives alternate numeric labels to each couple (if the same as the implicit numbers, the numbers are 1N as in this example).
Insert Figure 5 about here
Given the input data, the PGRAPH program creates a graph of a kinship and marriage network of a population. Once an initial graph of a network is brought to the screen, the program will execute specified options. Options "G" for generations, followed by "P" for permutations, and further options for keeping male or female ancestries constant, allows for permuting parentage in the other gender within each generation, and saving the results to a file (Pxxx.ves). The save file options S)lide PaJ)ek write the simulated dataset to a *.net file which is native format for Pajek software (Batagelj and Mrvar 1997) for large network and genealogical analysis, described by White, Batagelj and Mrvar (1999). Pajek can be used to analyze the size of components and bicomponents (done in the present article with the ParBloc algorithm) in terms of numbers of nodes and arcs. If one has data in Pajek format the Net2Ved program. Pajek will also read standard *.ged (genealogical exchange data) files and transform them to *.net format for the Net2Ved program.
Once data are displayed in the PGRAPH program, a whole series of simulations can be done, saving each run to *.ved and *.net files. Pajek can read, analyze, rename the *.net files and save the cumulative simulations. The ParCalc program (see below) can analyze the *.ved files for consanguineal marriage frequencies.
The Case Studies
The case studies of "complex" systems selected to exemplify the statistical decomposition of marriage rules and strategies using random baselines are three farming villages located in Indonesia, Sri Lanka, and Austria. The Javanese village (Schweizer 1989, White and Schweizer 1998) has been characterized as having "loose structure" and no particular marriage rules or strategies other than nuclear family incest avoidance and status endogamy. Rates of marriage amongst blood kin are very high among the elites. The Sri Lankan village of Pul Eliya was originally characterized by Leach (1961) as having various types of low frequency blood marriages but was later discovered to have a bipartite marriage structure or cognatic dual organization (Houseman and White 1998a). The Austrian village, with a proscription by the Catholic Church against blood marriages up to third cousins, was discovered by Brudner and White (1997, using data from Brudner 1969) to have a high degree of matrimonial relinking. Table 5 shows some of the characteristics of the case studies. Statistical identification of marriage patterns for these cases has been problematic since analyses have not been done against a random baseline.
(Insert Table 5 about here)
Decomposing Relinkings in the Case Studies in Structurally Endogamous Blocks, with Significance Tests of Departure from Randomness
We begin at the most general level by comparing simulation results with actual observations of the size of structurally endogamous blocks for each test case. In the simulations for each case, the female descent lines and the generational levels of the actual data are held constant. Then, in each generation, the sons in the actual datasets are detached from their marriages, creating a marriage pool of potential mates who marry within the precise set of women whose husbands were detached. Husbands are then randomly reallocated (equiprobability sampling without replacement) from within this pool to regenerate a total kinship and marriage network for this population with everything else held constant, including sibling sets. A different parameter is set for each test case to prevent marriages from violating known incest prohibitions. For all four cases, brothersister marriages are disallowed. For the last case, 1^{st} and 2^{nd} cousin marriages are disallowed.
Comparisons of the simulated and actual data for the first two test cases – both from Dukuh village in Indonesia – are summarized in Table 5. The first case (A), a hamlet of Dukuh village (Schweizer 1989), is characterized as having a "loose social structure" with status endogamy but minimal, specific marriage rules and strategies beyond incest prohibitions. The simulation test shows close similarity between the relinkings in a random marriage regime and the actual marriage network, except for six marriages, that relink sibling groups: The numbers of marriages such as these (in column 1 of Table 5) are marked with asterisks to show that they occur more frequently in the actual than in the simulated and randomized marriage network. These are elite couples, with status endogamy (White and Schweizer 1998), who reside within the hamlet. For the second case (B) – the Muslim elites in the village containing the Dukuh hamlet – the simulated and actual results agree perfectly. The relinkings (through blood marriages) are random although the pool of potential mates is greatly restricted by status endogamy among the smaller group of wealthy families, thus forcing an essentially random distribution of marriages to include a much higher proportion of blood marriages than occurs with the Dukuh commoners, who have a larger marriage pool. This result confirms the argument made by White and Schweizer (1998) even though they did not have recourse to statistical tests for their hypotheses.
(Insert Table 5 about here)
In Table 6, the third test case based on Edmund Leach’s (1961) Pul Eliya data and restudied by Houseman and White (1998a), we have a highly positive result regarding global "relinking" marriage rules. If we start one or two generations back from the last generation in Leach’s genealogies, we have a surfeit of 12 marriages (8+4) that relink in the parental generation. All the relinked marriages such as these, in column 1 of Tables 5 and 6 (and Table 13) are the result of sibling sets whose intermarriages form circuits. (One generation back is a circuit composed of eight sibling sets and two generations back is another circuit of four sibling sets. Since the "present" or most recent generation has not completed its marriages, sibling marriage circles may possibly be formed here as well. The fact that sibling circuits do not occur three or four generations back is more likely due to missing data than to true absence of relinking.)
In five of the six generations studied in this case (all but the earliest generation), there are circuits of relinking among 1st and 2nd cousins (asterisked in columns 2 and 3) or even closer relatives (although not necessarily at the same generational rank, e.g., cousins "once or twice removed" generationally). The surfeit of actual over simulated relinkings, however, is much higher for relatives linked within the second degree (column 2 – 1st cousin or closer circuits), with a ratio of 71:41 (summing to Starting from "back four" generations), than for those linked within the third degree, where the ratio of 114:105 is close to a random regime (summing again to "back four"). Pul Eliyans have a two to one (83:41) nonrandom surfeit of "close" relinkings, i.e., within one or two degrees (summing columns 1 and 2 to "back four" generations). Since the relinkings occur through linking relatives who are only one or two generations back (hence likely to be alive or salient at the time of marriage), we can conclude that Pul Eliyans are likely to be fully aware of, and knowledgeable about, affinal relinkings among their various families and compound groups. This is precisely what is argued by Houseman and White (1998a).
(Insert Table 6 about here)
Local Marriage Structures: Consanguineal Marriages
Table 7 moves from the global level of structural endogamy to the level of local marriage structure to examine Dukuh hamlet and the Muslim elite that crosscuts the various hamlets of the village. Here, the ParCalc program (White and Jorion 1992) was used to calculate the frequencies of consanguineal marriages in the actual and simulated networks. What is especially important about these calculations is that for each kintype, such as FBD for example, the program also computes the number of relatives that exist of this type. Hence, biases are controlled if the actual data have a different rate of occurrence of a given type of relative available for marriage. In addition, Hammel’s (1976b) "matrilateral" and similar biases are fully controlled because the permutations within generations can only select, for example, those matrilateral crosscousins who are actually married in one’s generation when generating the simulated marriage rates. Hence, structural (permutational) simulation is an astoundingly simple solution to the problems that have plagued statistical inferences about marriage rules and strategies.
In Table 7 and the following tables, Fisher’s exact significance test for dichotomous (2x2) tables is used to compare actual and simulated frequencies for the presence/absence of different types of marriage (White 1994), except where the binomial test is appropriate (Tables 810). Table 7 also uses Bartlett’s 2x2x2 test, a generalization Fisher’s exact significance test, to compare the 3way differences between two 2x2 tables (White, Pesner and White 1983). The simulated data from these tests are from a single simulation; hence there are no fractional numbers in the S or TS columns. A single run comparison is not only appropriate but also methodologically preferable for the use of the Fisher and Bartlett statistics. Two or more runs would double or triple the sample size of the simulated distributions, and where there are systemic (even if small) differences between the simulated and actual data, the inflation of sample size invariably yields greater significance. Hence the methodology of a single run is methodologically conservative, which is what is wanted for significance tests. If multiple simulations are averaged there is greater precision in the permutation results, but tests show that this will rarely affect accuracy at the integer level of measuring the frequencies expected by permutation, and rounding to the nearest digit will barely affect the significance tests. Nonetheless, multiple simulation runs with analysis by Pajek, ParBloc, ParCalc or other programs are fairly easily done (but not as yet automated) with his package of programs, and averages and standard deviations can be computed for the random baseline models across any number of simulations.
(Insert Table 7 about here)
Examining Table 7 for Dukuh hamlet versus the Muslim elites, we see that FBD marriage occurs once among the elites in the 4 cases where a FBD is present (a 25% marriage rate!), but this does not differ significantly from chance (p = 0.625 by Fisher’s Exact) from the simulated data, where no FBD marriages occur with only 3 such marriages possible. A second simulation to test this result (not shown here) obtained exactly the same rate in the random data as in the actual data, although the specific marriages resulting from permutation were different. Comparably, in Dukuh hamlet at large, while there were no FBD marriages (Dukuh contains only a segment of the total elite network for the village), only one was found in the simulated data, with a probability, given the simulated data, of p = 0.591 that fails to reject the null hypothesis. Further, using 3way interaction tests (White 1994), there is no indication (p = 1.0) of a significant difference between the elite and hamlet frequencies. Similar conclusions hold for MBD and FZDD, that are the only other actual blood marriages, and for ZD, which occurs only in the simulated data. Results are also shown for kintypes that exist but do not occur as blood marriages either in the actual or the simulated data. Hence, we may conclude that (1) such blood marriages as exist are not strategic or preferred but are either random or only a function of the status endogamy in smaller sized group of elites (in spite of their 25% and 50% rates of marriage with FZD and MBD relatives when they are members of the elite network!) and (2) controlling for status endogamy does not lead to any significant difference in rates of marriage with blood kin between Dukuh hamlet and the elites.
Table 8 shows the bloodmarriage analysis for Pul Eliya. Matrilateral crosscousin marriage is the only type of marriage among consanguineal relatives whose frequency is sufficiently high relative to the simulation results to reject the null hypothesis (p < 0.05). This conclusion accords with Houseman and White (1998a), who consider MBD marriages as a conscious marriage strategy related to the consolidation of wealth among families who are politically influential in the village. Leach (1961), on the other hand, did not make much of MBD marriage as strategic alliances. Although MBD marriages are infrequent (where ego has a MBD, ego marries MBD 12.5% of the time, compared to 50% for the Javanese Muslim elites), this rate is sufficiently higher than expected in a simulated random marriage regime to qualify as strategic. FZD, which occurs with a 7.7% rate, does not differ significantly (p = 0.32) from the expected random rate. Many other blood marriages occur, but none may be regarded as differing from expected random rates when taken individually.
(Insert Table 8 about here)
Complex Marriage Systems with "Sidedness" Rules
There is, however, one highly significant feature of the blood marriages versus the simulated marriages at the global level. As shown in the last two columns of Table 8 under the heading "Sided?" for actual and simulated marriages only, all of the 18 actual nonMBD marriages have an even number of female links whereas only half of the simulated nonMBD marriages do so. The probability of this occurring under a random regime whose character is estimated from this sample size is p = 0.002, but a better estimate is p = 0.000004, using the binomial test for a 50:50 expected distribution drawing 18 identical samples. Note that the simulation is performing exactly as expected for a random marriage regime since the chances of getting an even as opposed to an odd number of female links in a 1family marriage relinking is exactly 50:50. But this also indicates that the nonMBD marriages, taken as an ensemble, are not "random" and correspond to a marriage rule, namely, consistency with Dravidian virisidedness defined as "marrying an affine" where blood relatives are converted to affines by the rule that an odd number of female links makes the relative an affine. The blood marriages, then, are strictly consistent with the Dravidian kinship terminology of Pul Eliya.
(Insert Table 9 about here)
Can we test whether the marriage rule here is virisided (an even number of female links) as opposed to uxorisided (an even number of male links)? We can do so by removing all those marriage with someone in one’s own generation, where the two definitions of sidedness are necessarily identical (same generation blood marriages have an even number of linking relatives, and subtracting an even number of male links from an even number of total links always yields an even number of female links). Table 10 shows that generationally "skewed" marriages which are uxorisided, such as FZ, MMBDDD, MMZSDD, and FMMFZSSD, never occur in actuality, while unsided marriages in the uxorisided sense (such as MFMBDD, MMZSSD, MMZDDD, MFMBDDDD, MFMFZSSD, MFMFZDDD, FFMZDSSD, MFFZDSSD, MFMBDSSD) do occur, but each is sided in the virisided sense (p = 0.02 and, using the binomial test of 50:50 expected, p = 0.002). Given the generational depth of up to four generations to the linking ancestors, we can say that Pul Eliyans are definitely aware of sidedness within their personal kindreds, and their marriages with blood kin are 100% compatible with virisidedness but 100% incompatible with uxorisidedness where the two rules differ. Hence we can say that virisidedness is prescribed in blood marriages, not just preferred.
(Insert Table 10 about here)
Can viri and uxorisidedness be tested independently of blood marriages? The idea here is to identify the number of elementary cycles in each graph theoretic block of the network (in the Pul Eliya case there is only one such block). Since n nodes require n  1 edges to be connected as a tree without cycles, each additional edge adds an extra elementary cycle, so that the formula for the number of elementary cycles is k  n + 1, where k is the number of edges in the block. Each elementary cycle has a 50: 50 chance of having a even or odd number of male links, and the same for the number of female links. Hence the likelihood of getting the observed number of sided versus unsided cycles can be computed from the binomial distribution (White and Jorion 1996) – the same formula used to test whether a given coin is fair with no preference for heads or tails. Table 11 shows the results of the binomial test for the Pul Eliyan network, and gives p = 0.008 for rejection of the null hypothesis of no virisidedness. Note, however, that there are a number of errors to virisidedness that (as we have seen) do not come from blood marriages but from affinal relinking. Thus, we can say that the Pul Eliyans are not strict about a rule of virisidedness when it comes to affinal relinking between 2, 3 or more families. In fact, if we remove the blood marriages from the count of balanced cycles in Table 11A, we can accept the null hypothesis (p > .30) that Pul Eliyans disregard virisidedness completely when it comes to nonbloodrelated affinal relinking between 2 or more families, and choose spouses randomly in this respect.
(Insert Table 11 about here)
Houseman and White (1998a) show that lacking brothers, Pul Eliyan female heirs to residential compounds and associated land and water rights take the place (and sidedness) of males in the marriage networks. The authors use this concept to identify what they call "ambilateral sidedness." Their criteria for ambilateral sidedness (see Houseman and White (1998a) for a definition and discussion), gives a perfect 35:0 hit rate in predicting the sidedness of marriages (Table 11B), which has an infinitesimally small probability by chance under the binomial hypothesis (p = 0.00000000003).
These results for Pul Eliya are especially interesting in that virisidedness is prescribed in blood marriages, but absent (and aleatory) in affinal relinking, yet there is an apparently determinate ambilateral pattern that follows inheritance rules linked to the affinal relinkings. The determinacy, however, is posthoc in that while female inheritance in an agnatic line lacking a male heir "converts" the daughter from the side opposite her father to the father’s side (where her brothers should be; a pattern associated with a special form of binna uxorilocal marriage), there are some marriages whose assignment is indeterminate a priori but nonetheless consistent in the emergent pattern of sidedness. Hence, what might appear to be an "elementary" marriage system if virisidedness were followed rigorously, turns out to have a property of "semicomplexity." Indeed, this is not a unilineal descent system, and the Pul Eliya lack hereditary matrimonial moieties. Ambilateral sidedness here follows principles of cognatic inheritance. Hence, we cannot say that there is an ambilateral sidedness marriage rule, but rather a strategic motivational schema that is oriented to an "emergent" sidedness that keeps principles and pragmatics of inheritance in line, but is also consistent with Dravidian kinship terminology. But since there are violations of virisidedness in affinal relinking, there are adjustments of the virisided Dravidian kin terms (which apply the evennumber of female links equally to affinal kin), where affinal relatives who are classified as siblings have their kinterms readjusted to fit changing patterns of sidedness emergent in the marriage network through actual marriages that deviate from the virisided rule of affinal links. Leach (1961) describes numerous adjustments of kin terms to discrepancies that result when someone marries an affinal, classificatory sibling.
What about sidedness for Dukuh hamlet and Javanese Muslim elites? Table 12 shows the sidedness test, independent of blood marriage, for the elites (the Dukuh hamlet results are similar, and are not shown). There is no evidence for the statistical significance either of virisidedness (p = 0.94) or uxorisidedness (p = 0.31).
(Insert Table 12 about here)
Fully Complex Marriage Systems
Finally, what of the application of this paradigm for analysis of marriage rules and strategies to fully complex marriage systems, such as in European societies? In the sections on "Defining the General Phenomena," this article began with "Difficulties of the Attribute Approach, " and went on to argue for a better specification of the entire problem of marriage rules and strategies in "Network Terms," namely through considering relinking as the "elementary" but universal form of endogamy, and structural endogamy through relinking as the universal form taken by marriage rules and strategies. What was proposed was basically a theory of kinship and marriage networks in which relinking lies at the root of much of what we call ethnicity, class, community, and other seemingly "categorical" variables that have traditionally – but ambiguously – been used to define some of the outer limits of endogamy.
As a theory – call it a "relinkage theory of social class and ethnicity "– this is a speculative idea because extensive network data (on networks of size 2,000 to 200,000, for example, as possible endogamous "demes" in urban societies) are neither easily available nor readily yield to analysis. What evidence do we have from European societies? The inspiration for relinkage theory comes from the findings of Brudner and White (1997) on an Austrian farming village where they show that structural endogamy tends to define the boundaries of an Austrian rural class system that differentiates between principal heirs inheriting farmsteads and nonheir siblings who typically take up other occupations (workers, craftsmen, white collar) or emigrate from the village.
Table 13 examines Brudner and White’s analysis of the evidence for nonrandom relinking within the Austrian village network of about 3,000 people. The construction of this table is identical with that of Tables 5 and 6 for Dukuh and Pul Eliya. What it shows is a surfeit of 42 marriages over the last three generations that relink within the depth of a single generation, whereas no relinkings occur in the simulated "random marriage" regime within such a short time span. In the last two generations there is a surfeit of 56 (total of 74) marriages over 18 expected. In the last generation there is a surfeit of 38 (total of 70) over 32 expected. Relinking with families where the links involve more than three generations depth, however, converge to randomness. From this it is apparent that shallow relinking (within 3 generations or less) is nonrandom and certainly "strategic," but not prescribed. Hence there is marriage structure within this community despite of a nearabsence of any kind of blood marriage (9 out of 2491 marriages), at least by links within people’s memory, which supply the major source of the data.
(Insert Table 13 about here)
Is there evidence of mild preference or avoidance, not only for proximal kin up to third cousins (proscribed by the Catholic Church), but for more distant kin? Table 14 analyzes the frequencies of actual blood marriages compared to the simulation model. The nine actual blood marriages, especially those that differ most from random expectations in terms of biases towards certain kinship types (Hammel’s "handwaving" problem) in the actual data, show a tendency (subTable 14a, p=0.08) for the father’s side (side here is used in an ordinary sense rather than that of Dravidian sidedness). Among these, one is with a first cousin, three with a second cousin once removed, and one with a third cousin. Similarly, the more significant avoidances, compared to expectations from the random model, are on the mother’s side (subTable 14b, p=0.06). This is in keeping with the common European idea that women often know more about kinship relations than men, and here it may be that women tend more to be the keepers of kinship prohibitions. There may, however, be a strategic interest in relinking on the father’s side in light of the heavier inheritances that typically pass through males. These results are summarized in Table 14c, showing that highersignificance actual blood marriages have a greater tendency to occur on the father’s side while highersignificance simulated blood marriages, with fathers distributed more randomly, tend to occur on both the father’s and the mother’s side (p=0.04).
(Insert Table 14 about here)
Finally, can we answer the question of whether attribute endogamy or structural endogamy is a better indicator of class formation? Table 15 attempts to do so by comparing the strength and the significance of the differences between two crosstabulated predictors, one for farmerfarmer attribute endogamy (White et al. 1983), and the other for the correlation between farm heirs and structural endogamy. Although the heir/nonblock cell in the latter table can only be estimated, the two correlations do not look all that distinct, although the network hypothesis may fare slightly better (the correlation between relinking and heirship, adjusted for missing data, is r=.74, compared to correlations of .47 and .64 for attribute endogamy from language use and occupation).
(Insert Table 15 about here)
There is support, then, for the idea that structural endogamy might provide a clue to marriage patterns, rules, and strategies in complex marriage systems. Richard’s (1993) findings on French villages support this view, and he again finds occupational correlates of relinking that are probably also concomitant to differential social class formation.
Links between Kinship, Economics and Politics
By identifying social units, such as structurally endogamous blocks, or matrimonial sides, or emergent groups in which certain patterns of marriage occur, the simulation methodology allows a better identification of the links between kinship, economics and politics on the one hand, and between positions in the existing kinship and marriage network, language categories, and verbal norms. Each of the present studies offers a case in point. Pul Eliya offers a dramatic example. Societies in the "Dravidian kinship" culture area of South Asia (numbering in the tens of millions of people) have "twosided" kinship terminologies that set up a contrast between nonmarriageable and marriageable kin as if there is in place a matrimonial system of dual exchange. These verbal formulae are thought by most South Asianists to be merely egocentered perspectives that have little or nothing to do with social structure at a group level, such as moieties. The present comparison of actual marriage patterns with what would occur if marriage where uniform random within generations of the Pul Eliyan local subcaste, combined with the network analysis of Houseman and White (1998a) shows two features that exemplify a moietylike structure at the group level. At the social structural level, virisided marriages among blood kin are strictly prescribed. This is not sufficient to produce moieties, however, since there are "discrepant" marriages outside the circle of blood kin that are not virisided. Here a second principle is asserted at the level of emergent social organizational, where pragmatic social decisions are taken: Pul Eliyans "adjust" the sidedness of the nonblood kin "discrepant" marriages to bring them into alignment with a form of sidedness that is not associated with a strict rule of descent, but is rather associated with a variable rule of inheritance in which household "successor" may be women if the current generation in the household lacks an appropriate male heir. Decisions about the variable "sidedness" of female heirs gives rise to an emergent dual organization at the group level that correlates perfectly with elements of Pul Eliyan ideology that are widespread in the Dravidian culture area: the value placed on exchange marriages as an expression of the political equality among different household and lineage groups. The substantiated preference for MBD marriage further cements the behavioral expressions of these economic and political values. Hence, the fit between positions in the existing kinship and marriage network, language categories, and verbal norms is far greater than previously thought to be the case. Certain aspects of the kinship system are strictly prescribed, however, such as appropriate marriages among blood kin, while others are indeterminate and emerge only through decisions and behaviors taken among variable alternatives. Some of the verbal formulae in play are not strictly models "of" behavior but models "for" behavior and may include adjustments for departures from norms that apply in one domain (blood kinship) but not another. Internal variability turns out to be a key factor for understanding the connection between marriage patterns, kinship ideology, economics and politics.
The Dukuh hamlet and elites and the Feistritz village cases, where marked social stratification is in play, allow much simpler statements of fit between kinship, economics and politics. In the Dukuh case, elites and commoners behave very differently but do not consider themselves to differ culturally and they operate under a more general norm of moral equality (any one family, for example, may have rich and poor relatives, with the wealthier helping and often taking as clients their poorer bretheren). A general norm of status endogamy is found to apply at each level of the status hierarchy, and the simulation results show that the apparent difference in marriage behavior – that the elites are more likely to marry close blood relatives – are due only to the smaller size of their social circles where there is less chance of not marrying a relative. In the Feistritz case, the connections between kinship, economics and politics operate on a very different hierarchical inflection of status and property: whereas the Dukuh have equal division of property among sons (and also among daughters, who receive half that of their male counterparts under Islamic norms), the Feistritz farmers pass their farmsteads to single heirs (paying quitclaim inheritances to other children). Here, the structurally endogamous unit of the village defines a social boundary containing those marrying within and multiply connected through kinship links to the entire Slovenian farming community of the Gailtal valley. The connection between marriage structure and economics is striking: those within the structurally endogamous core of the village – emergent from marriage choices – belong to a social class of propertied farmers, and those outside this core, even if they are the sibling of heirs, rarely belong to this social class. The farmers constitute an economic block, a political block, a social class (as is well documented in other studies), and a kinship unit constituted not by consanguinity but by an emergent pattern of structurally endogamous marriages.
Models: Systems, Rules, Strategies and Emergence
Looking at marriage structure models from the long view, LéviStrauss is credited with assimilating both rules and strategies to a Von NeumanMorgenstern game theoretic conception of society. "Structure" defines the rules and constraints of the games and strategies are taken accordingly. His classification of marriage system models as elementary (=generalized prescriptive rules, 0/1 probabilities constraining strategies) or complex (=limited proscriptive rules, variably probabilistic strategies; semicomplex where the proscriptive rules are maximally generalized to nearly prescribe a distributive structure) does not do justice to cases like the Pul Eliya (Houseman and White 1998a) which are semiprescriptive (for blood relatives) with dependence on strategy for the emergence of global structure out of networked interaction. Nor does it do justice to cases like Feistritz (Brudner and White 1997) where an apparent complexity of open ended marriage strategies (with proscriptions against blood marriages) masks a nearly prescriptive communitylevel demand with institutional sanctions that heirs to farmsteads marry into a selfdefining and thus emergent structurally endogamous group, a strategy that assures farmstead continuities in knowledge and property rights for both heir and spouse.
Santa Fe Institute perspectives on complexity, while not the subject of discussion here, are more compatible with the findings of the present study: social processes generate emergent structural forms, and some of the stability of living and social systems derives from processes and forms – to use Stuart Kaufmann’s analogy – that hover, ever changing, between deterministic order and aleatory chaos.
Conclusions
The approach to marriage rules and strategies presented here has applications to, and implications for, the study of social class, community organization, wealth consolidation, transmission of political office, elite structural endogamy, ethnic integration, and the testing of alliance theories and specific alliance models in different ethnographic cases. Four such cases were used to exemplify the approach. Characterized as a network problem, and using structural simulation (permutational simulation) techniques for generating random baseline comparison models for individual cases, the analysis of marriage rules and strategies becomes analogous to loglinear or multiple regression analysis with interaction terms: a statistically decomposable problem.
Acknowledgments
This paper, dedicated to Tom Schweizer, was written for and presented at the Institute of Anthropology, University of Cologne, Colloquium on Current Ethnological Research, July 1, 1996, at the invitation of Professor Thomas Schweizer, with Lilyan A. Brudner as contributor and codiscussant, particularly on the Austrian case study materials. Research preparatory to this article was funded by a matching National Science Foundation Award (SBR9310033 "Network Analysis of Kinship, Social Transmission and Exchange: Cooperative Research at UCI / UNI Cologne / Paris CNRS") and an Alexander von Humboldt (Transatlantic Cooperation) Award. A considerable debt of gratitude is owed to Thomas Schweizer, Ulla Johansen, Vincent Duquenne, Clemens Heller, Alain Degenne, Françoise Hèritiér, François Héran, Paul Jorion and Michael Houseman for opening the path through French anthropology and mathematical social science to the discrete structure analysis of marriage and kinship networks. Thanks to Michael Houseman and Thomas Schweizer for editorial suggestions, to Gérard Weisbuch for pointing out some problems of presentation, and to an anonymous JASSS journal reviewer for many excellent suggestions as to clarity of presentation. The final revisions of this paper, and the Windows version of PGRAPH for multiple simulation test output directly to Pajek *.net files, were done under National Science Foundation Award BCS9978282 ("Longitudinal Network Studies and Predictive Cohesion Theory").
Appendix: A List of Relevant Computer Programs
A website at http://eclectic.ss.uci.edu/~drwhite/pgraph.html provides guidance as to program availability and documentation:
PGRAPH  graphs and simulated data for pgraphs, sidedness, segmentary dual organization, etc.
Net2Ved  transforms Pajek *.net files to PGRAPH *.ved files.
ParCalc  frequencies for marriages of blood relatives
ParLink  frequencies for twofamily relinkings
ParComp (incorporates Fisher tests)  for outcomes based on types (blood or relinking)
ParBloc  analysis of structurally endogamous blocks of marital relinkings
ParSide (binomial test)  computes likelihood of sidedness
ParCoef  computes inbreeding and relatedness coefficients for individuals in a population
References:
Batagelj, Vladimir and Andrej Mrvar. 1997. Networks / Pajek: Program for Large Networks Analysis. University of Ljubljana, Slovenia. http://vlado.fmf.unilj.si/pub/networks/pajek/.
Brudner, Lilyan A. 1969. The Ethnic Component of Social Transactions. Ph.D. Dissertation: University of California, Berkeley.
Brudner, Lilyan A. and Douglas R. White. 1997. Class, Property and Structural Endogamy: Visualizing Networked Histories. Theory and Society 25:161208.
Gibbons, Alan. 1985. Algorithmic graph theory. Cambridge: University Press.
Hammel, Eugene.1976a. The SOCSIM demographicsociological microsimulation program: operating manual. Berkeley: Institute of International Studies, University of California.
1976b. The Matrilateral Implications of Structural CrossCousin Marriage. pp. 145168, in, Ezra B. W. Zubrow, editor, Demographic Anthropology: A Quantitative Approach. Albuquerque: School of American Research, University of New Mexico Press.
Houseman, Michael, and Douglas R. White. 1998a. Ambilateral Sidedness among the Sinhalese: Marriage Networks and Property Flows in Pul Eliya, pp. 5989 in Kinship, Networks and Exchange, edited by Thomas Schweizer and Douglas R. White. Cambridge University Press.
 1998b. "Taking Sides: Marriage Networks and Dravidian Kinship in Lowland South America," pp. 214243 in, Maurice Godelier and Thomas Trautmann, eds., Transformations of Kinship. Washington, D.C.: Smithsonian Institution Press.
Jola, Tina, Yvonne Verdier and Françoise Zonabend. 1970. Parler famille. L’homme 10/3:526.
Lang, Hartmut. 1995. Demographic MicroSimulation. Paper presented at the Nanterre conference of the Working Group on Kinship and Computing, September, 1995.
Leach, E. R. 1951. The Structural Implications of Matrilateral CrossCousin Marriage. Journal of the Royal Anthropological Institute 81:2355.
 1961 [1968]. Pul Eliya: A Village in Ceylon. Cambridge: University Press.
LéviStrauss, Claude. 1966. The Future of Kinship Studies. Proceedings of the Royal Anthropological Institute for 1965. pp. 1322.
Richard, Philippe. 1993. "Étude des renchaînements d’alliance," Mathématique, Informatique et Science humaines 123:535.
Romney, A. K. 1971. Measuring of Endogamy. pp. 191213, in, Explorations in Mathematical Anthropology, edited by Paul Kay. Cambridge: MIT Press.
Schweizer, Thomas. 1989. Reisanbau in einem javanischen Dorf: Eine Fallstudie zu Theorie und Methodik der Wirtschaftsethnologie. Cologne: Böhlau Verlag.
Segalen, Martine. 1985. Quinze générations des BasBretons. Parenté et société dans le pays bigouden sud. 17201980. Paris: Presses Universitaires Français.
White, Douglas R. 1973. Mathematical Anthropology, pp. 369446, in, John J. Honigmann, editor, Handbook of Social and Cultural Anthropology. Chicago: RandMcNally and Co.
 1994. FisherB: A Program for Exact Significance Tests for Two and ThreeWay Interaction Effects. World Cultures 8(2):4043.
 1997. Structural Endogamy and the graphe de parenté. Mathématique, Informatique et Sciences Humaines. 137:107125.
White, Douglas R., and Michael Houseman. book ms. Balance in Kinship Networks: Cognatic Dual Organization.
White, Douglas R., and Paul Jorion. 1992. Representing and Analyzing Kinship: A Network Approach. Current Anthropology 33:454462.
. 1996. Kinship Networks and Discrete Structural Analysis: Formal Concepts and Applications. Social Networks 18:267314.
White, Douglas R., Robert Pesner, and Karl Reitz. 1983. An Exact Test for 3Way Interactions. Behavior Science Research 17:103122.
White, Douglas R., and Thomas Schweizer. 1998. Kinship, Property Transmission, and Stratification in Rural Java, pp. 3658 in Kinship, Networks and Exchange, edited by Thomas Schweizer and Douglas R. White. Cambridge University Press. In Press.
White, Douglas R., and Patricia Skyhorse. 1996. PGRAPH User’s Manual. University of California Irvine.
Marry: 
women of A 
women of B 
Table 1: Normalized Endogamy (diagonal) 
Men of A 
a 
b 
a = ad/bc, endogamy ratio >> 1 
Men of B 
c 
d 
Marry: 
women of A 
women of B 
Table 2: Normalized Alliance (offdiagonal) 
men of X 
a 
b 
a = ad/bc, endogamy ratio << 1 
men of Y 
c 
d 
Actual data 
Simulated data 
Table 3: Simulated Normalization 

a 
b 
e 
f 
a * = adfg / bceh, endogamy ratio 

c 
d 
G 
h 
Endogamy >> 1, Exogamy << 1 
Table 4: Case Studies and their Characteristics 

Case Studies: 
1A 
1B 
2 
3 
Village 
Dukuh hamlet 
Dukuh Elites 
Pul Eliya 
Feistritz 
Country 
Indonesia 
Indonesia 
Sri Lanka 
Austria 
Religion 
Muslim/Hindu 
Muslim 
HinduDravidian 
Catholic 
Descent 
Bilateral 
Bilateral 
Bilateral 
Bilateral 
Residence 
Dispersed 
Clustered 
Diga (patrilocal) v. Binna (uxorilocal) 
Stem 
Inheritance 
Equal Division, 2:1 male/female 
Equal Division, 2:1 male/female 
Agnatic 
Impartible farmsteads 
Class/Caste 
Elites/commoners 
Elites 
Varna subcaste 
Heir/nonheir social class 
Marriage Structure 
Status endogamy 
Status endogamy 
Cognatic dual sided organization 
Class position by matrimonial relinking 
Known Incest Prohibitions 
Brothersister 
Brothersister 
Brothersister 
Brothersister and 1^{st}, 2^{nd} cousins 
No. of marriages in dataset 
N=110 
N=45 
N=105 
N=2580 
Table 5: Comparison of Relinking Frequencies for Actual and Simulated Data, for (A) Dukuh hamlet, Indonesia (Schweizer 1989), and (B) Muslim elite of the Village containing Dukuh Hamlet (White and Schweizer 1998) 

Magnitude of Structural Endogamy with ancestors back 1, 2, ..., g generations 

Back: 
1 
2 
3 
4 
5 
6 
(A) Dukuh 

Starting from: 

Present generation 

Actual 
0 
27 
43 
45 
45 

Simulated 
0 
32 
42 
42 
42 

Back one generation 

Actual 
6* 
6 
17 
17 

Simulated 
0 
14 
14 
14 

(B) Elites 

Starting from: 

Present generation 

Actual 
0 
4 
9 
10 
10 

Simulated 
0 
4 
9 
10 
10 

Back one generation 

Actual 
0 
7 
8 
8 

Simulated 
0 
7 
8 
8 

*=greater than chance 
Table 6: Comparison of Relinking Frequencies for Actual and Simulated Data for Pul Eliya (Houseman and White 1998) 

Magnitude of Structural Endogamy with ancestors back 1, 2, ..., g generations 

1 
2 
3 
4 
5 
6 
7 
8 

Pul Eliya 

Starting from: 

Present generation 

Actual 
0 
34* 
55* 
62 
73 
81 
94 
94 
Simulated 
0 
26 
48 
63 
71 
84 
93 
93 
Back one generation 

Actual 
8* 
30* 
33 
45 
54 
67 
67 

Simulated 
0 
15 
40 
46 
60 
69 
69 

Back two generations 

Actual 
4* 
7* 
7* 
38 
50 
50 


Simulated 
0 
0 
0 
37 
46 
46 

Back three generations 

Actual 
0 
0 
4* 
25 
25 



Simulated 
0 
0 
0 
29 
29 

Back four generations 

Actual 
0 
4* 
15 
15 

Simulated 
0 
0 
17 
17 

Back five generations 

Actual 
0 
5 
5 

Simulated 
0 
10 
10 

*=greater than chance 
Table 7: Test of Actual versus Simulated Marriage among Consanguineal Kin, Dukuh Hamlet and Village Elites (conclusion: no preferred marriages, only status endogamy) 

key: 
A = frequency of actual marriages with a given type of relative S = frequency of simulated random marriages with a given type of relative TA = total of actual relatives of this type TS = total of simulated relatives of this type p = probability (Fisher Exact) 


Javanese elites 
Dukuh Hamlet 
3way 


A 
S 
TA 
TS 
p= 
type 
A 
S 
TA 
TS 
p= 
type 
Test 

1: 
1 
0 
4 
3 
.625 
FBD 
0 
1 
9 
12 
.591 
FBD 
1.00 

2: 
1 
2 
2 
3 
.714 
MBD 
1 
0 
11 
16 
.429 
MBD 
1.00 

3: 
2 
1 
3 
2 
.714 
FZDD 
0 
0 
11 
0 
 
FZDD 
1.00 

4: 
0 
1 
6 
7 
.571 
ZD 
0 
0 
18 
24 
 
ZD 
1.00 

5 
0 
0 
11 
11 
 
Z 
0 
0 
36 
43 
 
Z 

6 
0 
0 
4 
4 
 
BD 
0 
0 
22 
27 
 
BD 

7 
0 
0 
2 
2 
 
ZSD 
 
 
 
 
 
 

8 
0 
0 
3 
3 
 
BDD 
0 
0 
8 
8 
 
BDD 

9 
0 
0 
3 
3 
 
ZDD 
 
 
 
 
 
 

10 
0 
0 
4 
4 
 
FZ 
0 
0 
21 
27 
 
FZ 

11 
0 
0 
1 
1 
 
FZSD 
 
 
 
 
 
 

12 
0 
0 
3 
3 
 
FZD 
0 
0 
13 
14 
 
FZD 

13 
0 
0 
3 
3 
 
FBDD 
0 
0 
3 
2 
 
FBDD 

14 
0 
0 
5 
4 
 
MZ 
0 
0 
18 
23 
 
MZ 

15 
0 
0 
2 
2 
 
MZSD 
 
 
 
 
 
 

16 
0 
0 
4 
4 
 
MZD 
0 
0 
13 
14 
 
MZD 

17 
0 
0 
1 
2 
 
MBDD 
0 
0 
6 
5 
 
MBDD 

18 
0 
0 
2 
3 
 
MZDD 
 
 
 
 
 
 

19 
 
 
 
 
 
 
1 
0 
1 
0 
 
FFBDD 

20 
 
 
 
 
 
 
0 
0 
6 
10 
 
BSD 

21 
 
 
 
 
 
 
0 
0 
5 
6 
 
FBSD 

22 
 
 
 
 
 
 
0 
0 
6 
7 
 
FFZ 

23 
 
 
 
 
 
 
0 
0 
4 
5 
 
FFBD 

24 
 
 
 
 
 
 
0 
0 
3 
2 
 
FFBSD 

25 
 
 
 
 
 
 
0 
0 
2 
3 
 
FFZD 

26 
 
 
 
 
 
 
0 
0 
6 
5 
 
MBSD 

27 
 
 
 
 
 
 
0 
0 
10 
11 
 
MFZ 

28 
 
 
 
 
 
 
0 
0 
7 
7 
 
MFBD 

29 
 
 
 
 
 
 
0 
0 
4 
3 
 
MFBSD 

30 
 
 
 
 
 
 
0 
0 
7 
9 
 
MFZD 

31 
 
 
 
 
 
 
0 
0 
4 
2 
 
MFBDD 

32 
 
 
 
 
 
 
0 
0 
10 
11 
 
MMZ 

33 
 
 
 
 
 
 
0 
0 
5 
4 
 
MMBD 

34 
 
 
 
 
 
 
0 
0 
3 
1 
 
MMBSD 

35 
 
 
 
 
 
 
0 
0 
6 
5 
 
MMZD 

36 
 
 
 
 
 
 
0 
0 
2 
1 
 
MMBDD 
Table 8: Test of Actual versus Simulated Marriage of Consanguineal Kin for Pul Eliya (conclusion: MBD is a preferred marriage) 

Type of 
Actual Freq. 
Simul Freq. 
Total Actual 
Total Simul 
Fisher Exact 
Blood Marriage 
ViriSided? 

Mar. 
p= 
Type 
Pgraph notation 
Actual 
Simul 

12: 
5 
0 
40 
38 
.042 
MBD 
GF=FG 
yes 

2: 
3 
1 
39 
40 
.317 
FZD 
GG=FF 
yes 

1: 
0 
1 
56 
57 
.508 
FZ 
GG=F 

no 
3: 
0 
1 
6 
6 
.538 
FFFZDSD 
GGGG=FGFF 

no 
4: 
1 
0 
3 
1 
.800 
FFMZDSSD 
GGGF=FGGFF 
yes 

5: 
0 
1 
5 
3 
.444 
FFMBDSDD 
GGGF=FFGFG 

no 
6: 
1 
0 
18 
15 
.558 
FMBSD 
GGF=FGG 
yes 

7: 
0 
1 
17 
12 
.433 
FMBDD 
GGF=FFG 

no 
8: 
2 
1 
18 
12 
.661 
FMZDD 
GGF=FFF 
yes 

9: 
0 
1 
9 
5 
.399 
FMMBSSD 
GGFF=FGGG 

no 
10: 
0 
1 
4 
5 
.600 
FMMFZSSD 
GGFFG=FGGF 

yes 
11: 
0 
1 
6 
3 
.400 
FMMFZDSD 
GGFFG=FGFF 

yes 
13: 
0 
1 
25 
27 
.528 
MBSD 
GF=FGG 

yes 
14: 
1 
0 
14 
10 
.600 
MFZDD 
GFG=FFF 
yes 

15: 
1 
0 
7 
3 
.727 
MFFZDSSD 
GFGG=FGGFF 
yes 

16: 
1 
0 
8 
4 
.692 
MFFZDSD 
GFGG=FGFF 
yes 

17: 
1 
0 
8 
2 
.818 
MFMBDSSD 
GFGF=FGGFG 
yes 

18: 
1 
0 
9 
3 
.769 
MFMBDD 
GFGF=FFG 
yes 

19: 
1 
0 
3 
0 
1.000 
MFMBDDDD 
GFGF=FFFFG 
yes 

20: 
1 
0 
8 
2 
.818 
MFMFZSSD 
GFGFG=FGGF 
yes 

21: 
1 
0 
3 
0 
1.000 
MFMFZDDD 
GFGFG=FFFF 
yes 

22: 
1 
0 
13 
8 
.636 
MMZSSD 
GFF=FGGF 
yes 

23: 
1 
0 
15 
13 
.551 
MMBDD 
GFF=FFG 
yes 

24: 
0 
1 
11 
5 
.352 
MMZSDD 
GFF=FFGF 

no 
25: 
0 
1 
11 
5 
.352 
MMBDDD 
GFF=FFFG 

no 
26: 
1 
0 
11 
4 
.749 
MMZDDD 
GFF=FFFF 
yes 

Table 9: nonMBD marriages, Correlating Actual versus Simulated with Dravidian Viri Sided/Unsided Marriage (p=.0004; p=.000004 using the binomial test of 50%:50% expected)
ViriSided 
Unsided 

Actual 
18 
0 
Simulated 
5 
7 
Table 10: nonMBD generationally "skewed" marriages, Correlating Actual versus Simulated with Dravidian UxoriSided/Unsided Marriage (p=.02; p=.002 using the binomial test of 50%:50% expected)
UxoriSided 
Unsided 

Actual 
0 
9 
Simulated 
4 
3 
Table 11: Test of Sidedness for Pul Eliya (Programs: PGRAPH and ParSide) 

Number of Elementary Cycles: 25 

A.Virisidedness 
Actual 
Expected 
Balanced Cycles (Even length) 
25 
17.5 
Unbalanced Cycles (Odd Length) 
10 
17.5 
p=.008 

B. Amblilateralsidedness (women adjusted by inheritance rules) 
Actual 
Expected 
Balanced Cycles (Even length) 
35 
17.5 
Unbalanced Cycles (Odd Length) 
0 
17.5 
p=.00000000003 
Table 12: Test of Sidedness for Javanese Muslim Elites (Programs: PGRAPH and ParSide) 

Number of Elementary Cycles: 4 


Actual 
Expected 
Balanced Cycles (Even length) 
1 
2 
Unbalanced Cycles (Odd Length) 
3 
2 
p=.94 


Actual 
Expected 
Balanced Cycles (Even length) 
3 
2 
Unbalanced Cycles (Odd Length) 
1 
2 
p=.31 
Table 13: Comparison of Relinking Frequencies for Actual and Simulated Data, from Brudner and White, 1997 ‘Visualizing Networked Histories,’ Theory and Society 

Magnitude of Structural Endogamy with ancestors back 1, 2, ..., g generations 

Back: 
1 
2 
3 
4 
5 
6 
7 
8 
9 
10 
11 
12 
Starting from: 

Present generation 

Actual 
8* 
16* 
70* 
179 
257 
318 
349 
363 
376 
390 
399 
405 
Simulated 
0 
0 
32 
183 
273 
335 
365 
382 
388 
397 
397 
403 
Back one 

Actual 
8* 
58* 
168 
246 
308 
339 
353 
366 
380 
389 
395 

Simulated 
0 
18 
168 
255 
320 
347 
366 
372 
381 
381 
387 

Back two 

Actual 
26* 
115* 
178 
243 
278 
292 
305 
319 
328 
334 


Simulated 
0 
98 
194 
262 
291 
310 
316 
325 
325 
331 


*=greater than chance 
Table 14: Test of Actual versus Simulated Marriage of Consanguineal Kin for Austrian Village (conclusion: maternalside blood marriages are avoided, paternal side not) 

key: 
A = frequency of actual marriages with a given type of relative S = frequency of simulated random marriages with a given type of relative TA = total of actual relatives of this type TS = total of simulated relatives of this type p = probability (Fisher Exact) 

A 
S 
TA 
TS 
P= 

Type of mar 
Actual Frequency 
Simulation 
Total Actual 
Total Simul 
Fisher Exact 
Blood Marriage 

riage 
Freq. 
type 
degree 

7: 
1 
0 
32 
34 
.492 
FFMBDD 
2nd + 
1: 
1 
0 
165 
144 
.535 
FBD 
1st 
14: 
1 
0 
34 
33 
.514 
FMMBSD 
2nd  
8: 
1 
0 
31 
28 
.533 
FFMZDD 
2nd + 
17: 
1 
0 
12 
11 
.541 
FMMMBSSD___ 
3rd + median 
13: 
1 
0 
6 
5 
.583 
FMFMZSDD 
3rd ++ 
16: 
1 
0 
4 
3 
.625 
FMMFZDDSD 
4th 
23: 
1 
0 
3 
2 
.666 
MFMMBSSDD 

24: 
1 
0 
54 
22 
.714 
MMBSD 

26: 
1 
0 
8 
3 
.750 
MMMBSDD 
The more nonrandom Actual Blood Marriages weakly favor the Fa's side
SubTable 14a. 
Mo's side 
Fa's side 

<.55 (median) 
0 
5 

>.55 
3 
2 
p=.08 

25: 
0 
1 
23 
5 
.206 
MMFBSSD 
3rd 

21: 
0 
1 
10 
3 
.285 
MFMFBSSD 
3rd + 

20: 
0 
1 
16 
6 
.304 
MFMBDSD 
3rd 

19: 
0 
1 
6 
2 
.333 
MFFFBDSD 
3rd + 

5: 
0 
1 
20 
13 
.411 
FFFZSDD 
3rd 

18: 
0 
1 
7 
4 
.416 
MFBDDDD 
2nd  

9: 
0 
1 
5 
3 
.444 
FFMFBDDSD 
4th 

15: 
0 
1 
21 
16 
.447 
FMMZSSD____ 
3rd median 

11: 
0 
1 
46 
37 
.452 
FMBSSD 

2: 
0 
1 
105 
91 
.467 
FBSD 

6: 
0 
1 
17 
16 
.499 
FFFZDDD 

10: 
0 
1 
4 
3 
.500 
FFMFBDSDD 

12: 
0 
1 
10 
10 
.523 
FMFFZSSD 

3: 
0 
1 
16 
21 
.578 
FBSDDD 

4: 
0 
1 
10 
13 
.583 
FFFBSSDD 

22: 
0 
1 
1 
1 
.666 
MFMFBDSDD 
The more nonrandom avoidance of Actual Blood Marriages is more on the Mo's side
SubTable 14b. 
Mo's side 
Fa's side 

<.42 
5 
3 

>.42 
1 
7 
p=.06 
The more nonrandom Actual Blood Marriages favor the Fa's side while the more nonrandom simulated marriages favor the Mo's side
SubTable 14c. 
Mo's side 
Fa's side 

Actual 
0 
5 

Simulated 
5 
3 
p=.04 
Table 15: Comparison of Magnitude of Attribute versus Structural Endogamy as Predictors
A. attribute endogamy: language
marriage 
Wi bilingual 
monolingual 
Hu bilingual 
96 
16 
monolingual 
7 
15 
r=.47
B. attribute endogamy: occupation
marriage 
Wi's fa: farmer 
nonfarmer 
Hu's fa: farmer 
80 
7 
nonfarmer 
15 
34 
r=.64
C. structural endogamy
Marriage 
block members 
nonblock 
Heirs 
173 
117 (adjusted est.40) 
Residents 
25 
281 
r=.55, adj. r=.74
3way tests of difference A&B p=.46
B&C p=.??, B&C adj., p=??
A&C p=.??, A&C adj., p=??
(note: I need to recompile the 3way program with larger dimension statements to complete these 3way tests)
Figure 1: Illustration of a "Simple" Partially ordered uniform marriage structure
Figure 2: Components, bicomponents and tricomponents of graphs, illustrating structural endogamy in graphs where marriages are the nodes and persons connecting marriages via parentchild links are the lines (direction of lines ignored)
Original Graph 
à à à 
Permuted Graph 
Figure 3: Illustration of permuting male links in a pgraph with 3generations (top row is the first generation of marriages, solid arrows represent sons descending from parental marriages to form a marriage with a wife, the latter shown as a broken arrow descending from her parents)
Original Graph 
à à à 
Permuted Graph 
Figure 4: Illustration of permuting male and female links in a pgraph with 3generations
105 Pul Eliya vectors by couple: HuPa, WiPa, HuId, WiId, Couple ID
2 105 1 5 45 7 8 10 11 0 0 0 0 15 0 15 18 22 23 21
0 0 0 12 26 0 16 0 25 14 33 33 35 33 0 35 42 40 0 41
0 44 0 0 0 0 1 19 50 0 50 49 40 38 56 0 56 14 61 14
32 30 64 0 36 67 36 66 39 69 66 34 74 63 63 75 57 75 72 69
80 83 0 0 0 85 14 43 92 92 93 94 0 0 97 94 0 0 97 97
98 97 98 51 0
0 0 0 3 0 4 9 11 10 0 0 10 1 13 0 17 19 0 0 19
0 105 0 20 16 9 28 9 27 31 0 37 10 39 0 23 35 39 41 43
0 0 48 0 44 1 0 0 0 0 0 40 49 7 7 8 14 52 60 0
63 61 0 0 0 0 0 69 0 66 72 7 72 57 0 58 58 25 29 0
6 30 0 0 0 87 0 0 88 88 90 0 92 0 96 0 0 94 104 101
0 103 0 0 0
1 146 2 4 60 6 7 9 11 14 13 15 0 18 20 19 22 25 26 24
27 0 28 30 31 0 35 0 37 39 41 41 42 41 48 42 51 46 0 54
48 56 0 58 0 58 0 62 63 122 64 65 67 69 71 0 71 74 77 74
80 79 82 83 85 86 85 87 89 90 93 95 97 99 99 101 104 105 108 110
112 114 116 116 118 119 121 123 127 127 129 130 0 133 135 132 134 0 137 137
142 140 141 143 147
0 0 0 3 0 5 8 10 12 0 0 12 16 17 0 21 23 0 0 23
0 145 0 29 32 33 34 36 38 40 0 44 43 45 49 47 52 45 53 55
50 0 61 59 57 0 59 0 0 0 0 66 68 70 70 73 72 75 76 0
81 78 0 84 84 0 0 88 0 91 92 94 96 98 100 102 103 106 107 109
111 113 115 117 117 120 0 0 124 125 126 0 128 0 131 0 0 136 144 138
0 139 0 0 0
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 48 49 50 51 52 53 54 55 56 57 58 59 60
61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80
81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100
101 102 103 104 105
Figure 5: PGRAPH format for Pul Eliya data
ENDNOTES
^{1 It is the phrasing of these prohibitions in terms of lineages that is seen as the link to CrowOmaha "lineageskewed" kinship terminologies. 2 Paths that are not contained in circuits are trees which can always – trivially – be mapped onto a bipartite graph by assigning a simple alternation of connected nodes to supersets. 3 There is a consequent dearth of current applications of his programs to the issues of marriage rules and strategies, partly caused by a failure to supply a personalcomputer version of the software and perhaps also through lack of motivation given Hammel’s (1976b) finding. 4 This approach is being pursued through a working group at the Santa Fe Institute. 5 Actually, there will often be multiple prior relations between spouses in such graphs. 6 While children are not detached from their parental node in the simulation, what the simulation permutes is who are to be the parents of each sibling group. 7 The more standard approach of Monte Carlo estimation is to run many simulations, build a statistical distribution empirically, and then locate where the observed sample is located in this distribution to give a probability estimate. This is computationally cumbersome and is in the present case unnecessary. The computer programs used here can of course be used to verify that the present approach converges with Monte Carlo results, but that will not be undertaken here. 8 If Leach was later stung by Hammel’s (1976b) critique, with its mocking title of the Leach 1951 arguments about MBD as a strategic alliance among the Kachin, the present permutational simulation tests provide a means of refuting Hammel’s argument concerning particular cases of MBD marriage. }