Copyright JASSS

Douglas R. White (1999)

Controlled Simulation of Marriage Systems A synoptic outline

Journal of Artificial Societies and Social Simulation vol. 2, no. 3, <http://www.soc.surrey.ac.uk/JASSS/2/3/5.html>

* One-line Abstract.

The article presents and illustrates a new methodology for testing hypotheses about the departure of marriage choices from baseline models of random mating in an actual kinship and marriage network of a human population.

* Introduction

1.1 how to identify rules, structures, and strategies that give non-random inflections to mating systems

* A Network Approach

2.1 evaluating marriage structures in relation to marriage rules

2.2 The major hypothesis of the simulation - Case studies drawn from Indonesia, Sri Lanka, and Austria

* The Model

3.1 uniform marriage structure and partially ordered uniform marriage structure defined

* An Idealized Example for "Simple" Systems

4.1 4.3 Illustration of a "Simple" Partially ordered uniform marriage structure

4.4 generalization of this approach to determining marriage structures inductively regardless of the type of structure

* Limitations of the Classical Approaches to Studying Marriage Systems

5.1 marriage patterns structuralist concerns

5.2 statistical evaluation of marriage structures

5.3 previous work Hammel, Reed effects of demographic constraints

5.4 what are marriage rules or strategies? hierarchically embedded

5.5 how can network structures of kinship and marriage structure can be statistically disaggregated into components, including those of social choice versus demographic factors?

5.6 goodness of fit in a partially ordered marriage-rules model, where preferences or avoidances are thus always relative to some prior marriage probability model.

5.7 How the categories of partners are specified in such models: attributionallly versus relationally

* Defining the General Phenomena: Difficulties of the Attribute Approach

6.1 Regression model versus statistical normalization method: controlling demographic or numeric imbalance between groups in computing patterns of endogamy.

6.2 statistical normalization method: controlling demographic or numeric imbalance between groups in computing patterns of exogamy.

6.3 a more general approach: statistical control by a simulated random baseline

6.4 two fundamental problems with the attribute approach: (1) problem of arbitrary decisions about how to aggregate people into groups on the basis of attributes; (2) strict conditions for unambiguous rules of aggregation apply only to highly structured systems of marriage rules.

6.5 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 non-exclusive categories, and lack of consideration of relational or network-based rules or strategies.

* Specifying the General Phenomena in Network Terms

7.1 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. Relinking defined (relevant citations) and illustrated. Most marriage rules and strategies are formulated in terms of marriage with those with whom a pre-existing link is recognized such as common ethnicity, class, kinship, community, neighborhood, school ties, etc. In the general case ethnicity, class, community and school ties (varying in accordance with these three sociological variables) may be constituted to a large extent by patterns of relinking. The bias of our society, however, has been not to see marriage relationally but in terms of different social categories within a universalistic, anti-particularistic, egalitarian perspective.

7.2 endogamous "demes," on a scale of 2-20,000 or more persons, characteristic of modern urban populations and rural villages of modern states. Richards statistical method for assessing the occurrence of relinking in a population. Brudner and White approach to identifying class formation in rural Austria with the marriage strategy of relinking.

7.3 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.

* Decomposing Endogamy into its Constituent Relational Structures

8.1 The concrete form of endogamy that is constituted through marital relinking is not categorical endogamy but structural endogamy: 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.

8.2 Structural endogamy is a concept that implies emergent social boundaries, unlike the attribute approach. We can adopt techniques of network analysis to identify the boundaries of structural endogamy.

8.3 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 parent-child 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 p-graph.

8.4 The p-graph 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 p-graph 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 p-graph 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 p-graph is disregarded. Although parent-child links are directed, it is the undirected skeleton of the parent-child network among marriages that is relevant to structural endogamy. A structurally endogamous block of the skeleton of a p-graph is formally equivalent to a bicomponent of the graph since in a bicomponent two or more independent paths connect every pair of nodes.

8.5 Defining and illustrating the concepts of the components, bicomponents and tricomponents of graphs.

8.6 Using the illustration to conceptualize the measure of structural endogamy.

8.7 Structural endogamy 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), of necessity, imply some form of relinking and hence subblocks of structural endogamy.

8.8 strategic or rule-governed subsets of marriages will have non-random characteristics in comparison to a uniform-probability model at that level. 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. Sometimes small-scale rules and strategies may be masked in larger phenomena that resemble a near-random relinking model of structural endogamy.

8.9 smaller-scale relinking patterns to be found within structurally endogamous blocks: classification of relinkings by the type of prior personal connections. Prior personal connections may involve blood (parent/child or sibling/sibling connections) or marriage (husband/wife connection). A blood marriage (1-family) relinking is one where there exist prior personal connections between spouses involving only consanguinity (hence the spouses are blood related). An affinal (2-family) relinking is one where there exist prior personal connections involving only one prior marriage link. Relinkings of 1-, 2-, or k- families 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 rule-governed relinking, the density limits (high and low) of cluster of cycles, or global properties such as dual or segmentary dual organization. Global structures need to be evaluated by the characteristics of the circuits within them. In dual organization, for example circuits can be drawn as a bipartite graph where all the connections are between two supersets of nodes.

8.10 partially ordered character of marriage structures and appropriate methodologies for hierarchical decomposition

8.11 How to define the general phenomena of marriage rules and strategies empirically to make it possible to avoid mis-specification?

* Canonical Representation of Kinship and Marriage Network Data

9.1 Circuits of matrimonial relinking as canonical objects of study for marriage rules and strategies.

9.2 graph-theoretic algorithms identify structurally endogamous bicomponents of nodes involved in matrimonial relinking as a solution to the problem of precise decomposition of the aggregate social units potentially involved in marriage rules and strategies.

9.3 statistical tests and references; software programs used: (1) for computing frequencies of local structures such as blood marriages (2) for comparing actual structural endogamy versus simulated results (3) for computing within-block global structures such as dual organization (4) for analysis of 2-family relinking (5) and for estimating relatedness and inbreeding coefficients for individuals in a population. 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

10.1 Simulations that require a series of demographic parameters, to be matched to an empirical population for verisimilitude, are unnecessarily complex for the problems posed here of marriage structure under demographic constraints.

10.2

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 structurally controlled simulation, or the use of permutation tests. Say one has an observed network of parental links ordered by generation in a p-graph 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 structurally controlled 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[4].

10.3 an illustrative p-graph of three generations and 18 marriages and a permutation of the male links between adjacent generations

10.4 illustration of how the permutation might differ if both male and female links are permuted.

10.5 "Generation" is an empirical attribute of the p-graph marriage structure of a population because, unlike individuals (who have multiple marriages), each marriage is uniquely located in a partially-ordered generational structure defined by parent-child links. Because the p-graph is a partial order, it has a minimal number of generations. Use of the PGRAPH and Pajek programs to draw genealogical networks in p-graph format.

10.6 Controlled simulation is done within the PGRAPH kinship analysis program (and a forthcoming stand-alone Par-Sim program) that provides a graph of a kinship and marriage network of a given population.

* The Case Studies

11.1 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 has been characterized as having "loose structure" and no particular marriage rules or strategies other than nuclear family incest avoidance and status endogamy. The Sri Lankan village of Pul Eliya was originally characterized by Leach as having various types of low frequency blood marriages but was later discovered to have a bipartite marriage structure or cognatic dual organization.The Austrian village, with a proscription by the Catholic Church against blood marriages up to third cousins, was discovered by Brudner and White to have a high degree of matrimonial relinking related to a particular social class. Table 5 shows characteristics of the case studies

.


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

Hindu-Dravidian

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 with

Heir/nonheir social class

Marriage Structure

Status endogamy

Status endogamy

Cognatic dual (sided) organization

Class position defined by matrimonial relinking

Known Incest Prohibitions

Brother-sister

Brother-sister

Brother-sister

Brother-sister and 1st, 2nd cousins


* Decomposing Relinkings in the Case Studies in Structurally Endogamous Blocks,
with Significance Tests of Departure from Randomness

12.1

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, brother-sister marriages are disallowed. For the last case, 1st and 2nd cousin marriages are disallowed.

12.2 Comparisons of the simulated and actual data for the first two test cases - both from Dukuh village in Indonesia. The first case (A), a hamlet of Dukuh village, 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 within two generations. These are elite couples, with status endogamy, 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) that the commoners and elites share a lack of specific preferences for marriages with blood relatives, although their raw frequencies of behavior are vastly different due to demographic constraints.

 

12.3 In the third test case based on Edmund Leach's (1961) Pul Eliya data and restudied by Houseman and White (1998), we have a highly positive result regarding global "relinking" marriage rules.

12.4 In five of the six generations studied in this case, there are circuits of relinking among 1st and 2nd cousins or even closer relatives. The surfeit of actual over simulated relinkings is much higher for relatives linked within the second degree, with a ratio of 71:41, than for those linked within the third degree, where the ratio is close to a random regime. Pul Eliyans have a two to one (83:41) non-random surfeit of "close" relinkings, i.e., within one or two degrees. 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 (1998).

* Local Marriage Structures: Consanguineal Marriages

13.1 structural endogamy is examined at the level of local marriage structure to examine Dukuh hamlet and the Muslim elite that crosscuts the various hamlets of the village using the Par-Calc program 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 kin-type, 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 availability for marriage of a given type of relative (i.e., what is controlled is how many relatives there are of each type, which can otherwise drastically and spuriously affect the raw frequencies of different consanguineous marriages). Demographic biases on frequency of different types of cousins, for example, are fully controlled because the permutations within generations can only select, for example, those cross-cousins who are actually married in one's generation when generating the simulated marriage rates. Hence, structurally controlled simulation provides a direct solution to the problems that have plagued statistical inferences about marriage rules and strategies.

13.2 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 and Bartlett's 2x2x2 test, a generalization Fisher's exact significance test, to compare the 3-way differences between two 2x2 tables. The simulated data from these tests are from a single simulation; hence there are no fractional numbers in the S or TS columns. The methodology of a single run is methodologically conservative. When 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.

13.3 Examining 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. In Dukuh hamlet (which contains only a segment of the total elite network for the village), while there were no FBD marriages only one was found in the simulated data, with a probability, given the simulated data, of p = 0.59, that fails to reject the null hypothesis. Further, using 3-way interaction tests, 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. We 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.

13.4 The blood-marriage analysis for Pul Eliya shows that matrilateral cross-cousin 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 (1998), 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.

13.5 While the results of this section are based on analysis of consanguineal marriages, the analysis of local structures in marriage networks can also be extended to patterns of marital relinking with affinal relatives. Since there are many more combinatorial types of affinal relatives than consanguineal relatives, this involves greater statistical complexity, but appropriate models are under development (see Appendix).

* Complex Marriage Systems with "Sidedness" Rules

14.1 For Pul Eliya, there is a highly significant feature of the blood marriages versus the simulated marriages at the global level: All of the 18 actual non-MBD marriages have an even number of female links whereas only half of the simulated non-MBD 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. This indicates that the non-MBD marriages, taken as an ensemble, are not "random" and correspond to a marriage rule, namely, consistency with Dravidian viri-sidedness 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.

14.2 Can we test whether the marriage rule here is viri-sided (an even number of female links) as opposed to uxori-sided (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). The data show that generationally "skewed" marriages which are uxori-sided, such as FZ, MMBDDD, MMZSDD, and FMMFZSSD, never occur in actuality, while unsided marriages in the uxori-sided sense (such as MFMBDD, MMZSSD, MMZDDD, MFMBDDDD, MFMFZSSD, MFMFZDDD, FFMZDSSD, MFFZDSSD, MFMBDSSD) do occur, but each is sided in the viri-sided 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 viri-sidedness but 100% incompatible with uxori-sidedness where the two rules differ. Hence we can say that viri-sidedness is prescribed in blood marriages, not just preferred.

14.3 [not yet synopsized from here]

Can viri- and uxori-sidedness 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 an 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 viri-sidedness. Note, however, that there are a number of errors to viri-sidedness 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 viri-sidedness 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 11, we can accept the null hypothesis (p > .30) that Pul Eliyans disregard viri-sidedness completely when it comes to affinal relinking between 2 or more families, and choose spouses randomly in this respect.

14.4

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 11), which has an infinitesimally small probability by chance under the binomial hypothesis (p = 0.00000000003).

14.5

These results for Pul Eliya are especially interesting in that viri-sidedness 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 post-hoc 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 uxori-local 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 viri-sidedness were followed rigorously, turns out to have a property of "semi-complexity." 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 viri-sidedness in affinal relinking, there are adjustments of the viri-sided Dravidian kin terms (which apply the even-number of female links equally to affinal kin), where affinal relatives who are classified as siblings have their kin-terms readjusted to fit changing patterns of sidedness emergent in the marriage network through actual marriages that deviate from the viri-sided rule of affinal links. Leach (1961) describes numerous adjustments of kin terms to discrepancies that result when someone marries an affinal, classificatory sibling.

14.6

Sidedness in marriage systems is analyzed for Amazonian societies by Houseman and White (1998a ), and for indigenous Australian societies by Houseman (1997). Houseman and White (1996) develop other concepts for emergent properties of marriage networks based on dual organization or bipartite marriage structures, such as a "dividedness" concept which applies to certain Polynesian societies in which the parents of every husband and wife can be said to come from one of two opposite affinal "divides," but there is no tendency for these emergent affinal groupings to follow principles of unilineal descent.

14.7

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 viri-sidedness (p = 0.94) or uxori-sidedness (p = 0.31).

* Fully Complex Marriage Systems

15.1

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, " (6.1) and went on to argue for a better specification of the entire problem of marriage rules and strategies in "Network Terms," (7.1) 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.

15.2

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 argue that structural endogamy tends to define the boundaries of an Austrian rural class system that differentiates between principal heirs inheriting farmsteads and non-heir siblings who typically take up other occupations (workers, craftsmen, white collar) or emigrate from the village.

15.3

Table 13 examines Brudner and White's analysis of the evidence for non-random 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 non-random and certainly "strategic," but not prescribed. Hence there is marriage structure within this community despite of a near-absence 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.

15.4

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 (p=0.08) are on 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 (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 14, showing that higher-significance actual blood marriages have a greater tendency to occur on the father's side while higher-significance simulated blood marriages, with fathers distributed more randomly, tend to occur on both the father's and the mother's side (p=0.04).

The more non-random Actual Blood Marriages Favor the Fa's side

The more non-random avoidance of Actual Blood Marriages is more on the Mo's side

The more non-random Actual Blood Marriages favor the Fa's side while the more non-random simulated marriages favor the Mo's side

15.5

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 cross-tabulated predictors, one for farmer-farmer attribute endogamy (White et al. 1983), and the other for the correlation between farm heirs and structural endogamy. Although the heir/non-block 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 = 0.74, compared to correlations of 0.47 and 0.64 for attribute endogamy from language use and occupation, with p=.000006 and p=.000007 by Bartletts exact test for significance of difference in correlations).

15.6

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

16.1

By identifying social units such as structurally endogamous blocks, matrimonial sides, or emergent groups in which certain patterns of marriage occur, the simulation methodology allows a better identification of the links among kinship, economics and politics on the one hand, and among positions in the kinship and marriage network, language categories, and verbal norms, on the other. Each of the studies examined here 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 "two-sided" kinship terminologies that set up a contrast between non-marriageable and marriageable kin as if there is in place a matrimonial system of dual exchange.

16.2

These "two-sided" verbal formulae of Dravidian kinship terminologies are thought by most South Asianists to be merely ego-centered perspectives that have little or nothing to do with a moiety social structure at a group level. The comparison of actual marriage patterns with the simulated case where marriages are uniform random within generations of the Pul Eliyan local subcaste, combined with the network analysis of Houseman and White (1998a), does show two features that exemplify a moiety-like structure at the group level. At the social structural level, viri-sided 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 viri-sided. Here a second principle is asserted at the level of emergent social organization, where pragmatic social decisions are taken. Pul Eliyans "adjust" the sidedness of the non-blood kin "discrepant" marriages to bring them into alignment with a form of sidedness that is not associated with a strict rule of descent, but instead is associated with a variable rule of inheritance in which a 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, namely 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 among positions in the 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.

16.3

The Dukuh hamlet and elites and the Feistritz village are cases where marked social stratification is in play. This allows for much simpler statements of fit among 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 among kinship, economics and politics operate with 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.

16.4

Here, in the Austrian village of Feistritz, the structurally endogamous unit of the village defines a social boundary containing those marrying within, yet 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 siblings 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

17.1

Looking at marriage structure models from the long view, Lvi-Strauss is credited with assimilating both rules and strategies to a Von Neuman-Morgenstern 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; semi-complex 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 semi-prescriptive (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 community-level demand with institutional sanctions that causes heirs to farmsteads to marry into a self-defining, and thus a structurally emergent endogamous group -- a strategy that assures farmstead continuities in knowledge and property rights for both heir and spouse.

17.2

Santa Fe Institute perspectives on complexity, while not the subject of discussion here, are more compatible with the findings of the present study that social processes generate emergent structural forms and that some of the stability of living and social systems derives from processes and forms - to use Stuart Kaufmann';s analogy - hovering, ever changing, between deterministic order and aleatory chaos.

* Conclusions

18.1

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 structurally controlled simulation techniques (using permutation methods) for generating random baseline comparison models for individual cases, the analysis of marriage rules and strategies becomes analogous to log-linear or multiple regression analysis with interaction terms: a statistically decomposable problem.

Acknowledgements

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 co-discussant, particularly on the Austrian case study materials. Research preparatory to this article was funded by a matching National Science Foundation Award (SBR-9310033 "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, Franoise Heritier, 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 Grard 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 compilation of a new Pgraph program to carry out multiple simulations on the same dataset were done under National Science Foundation Award BCS-9978282 ("Longitudinal Network Studies and Predictive Cohesion Theory").

Notes

1 It is the phrasing of these prohibitions in terms of lineages that is seen as the link to Crow-Omaha "lineage-skewed" kinship terminologies.

2Paths 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.

3There 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 personal-computer version of the software and perhaps also through lack of motivation given Hammel's (1976b) finding.

4This approach is being pursued through a working group at the Santa Fe Institute.

5The 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. While studies such as that of White et al (1999) used this approach, it is computationally cumbersome and, as 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.

6If 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, structurally controlled simulation using permutation tests provides a means of refuting Hammel's argument concerning particular cases of MBD marriage.

Appendix: A List of Relevant Computer Programs

A web-site at http://eclectic.ss.uci.edu/~drwhite/pgraph.html provides guidance about program availability and documentation:

         PGRAPH - graphs and simulations, sidedness, segmentary dual organization, etc.

         Par-Sim - generates simulated data for p-graphs (available in 2000)

         Par-Calc - frequencies for marriages of blood relatives

         Par-Link - frequencies for two-family relinkings

         Par-Comp (incorporates Fisher tests) - for outcomes based on types (blood or relinking)

         Par-Bloc - analysis of structurally endogamous blocks of marital relinkings

         Par-Side (binomial test) - computes likelihood of sidedness

         Par-Coef - computes inbreeding and relatedness coefficients for individuals in a population

* 
References

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Copyright Journal of Artificial Societies and Social Simulation, 1999