Abstract: network-inclusive Fitness

1. My motivation is to test an important proposition involving tests of the hypothesis that pairwise cohesion in kinship networks predicts many different forms of cooperativity among kin via network-inclusive fitness. This hypothesis competes with kin selection theory which posits a positive selection gradient for a pair of blood relative if their inclusive fitness r satisfies rB - C > 0, r < 1 the relatedness coefficient, B the benefit of the behavior and C the cost. Another concern is with an hypothesis that after calculating the pairwise-cohesion (node-independent paths in a kinship p-graph), taking the subset of upper u top generation t couples t < G, finding their respective k descendants in the G -t -1 lower generations, and storing the number of their common descendants in a matrix u x u. This hypothesis allows further tests of the prediction that the level k of pairwise-cohesion among pairs of ancestors in a DAG is correlated with higher numbers of shared descendants of the pairs. While software for pairwise cohesion tests near completion, empirical tests are given using existing cohesion measures for kinship subgroups. Abstract: network-inclusive Fitness

2. network-inclusive Fitness :Kinship networks from social and genetic perspectivesTesting theory that kinship cohesion theory competes with the ‘selfish gene’ and evolutionary games as explanations for altruism. Further investigation of genetic-social cohesion links needed. Douglas R. White, UC Irvine Family/Demographic Session on Biological and social aspects of Kinship. Session organizer: Patrick Heady, Saturday Dec 20, 2010

3. Hamilton’s rule: Genes for altruism should increase in frequency when rB-C > 0 (where r < 1) andB = the additional reproductive benefit gained by the recipient of the altruistic act, C = the reproductive cost to the individual of performing the act. r = the genetic relatedness of the recipient to the actor, the probability that a gene picked randomly from each at the same locus is identical by descent. Example of benefits: I help your son… at a cost…Hamilton, W.D. (1964). The Genetical Evolution of Social Behaviour, I, II. Journal of Theoretical Biology 7(1): 1-16, 17-52. PMID 5875341PMID 5875340.

4. Limits of biological explanations of cooperation • Heady (2007: 489) notes “the recognition by contemporary evolutionists that far more cooperation takes place than can currently be explained by arguments based on either tit-for-tat reciprocity or direct forms of kinship altruism (Henrich et al 2003)”. • Patrick Heady. 2007. Fertility as a process of social exchange. Demographic Research 17(16):465-496. • Henrich J et al. 2003 Group report: the cultural and genetic evolution of human cooperation. In P. Hammerstein (ed) Genetic and cultural evolution of cooperation, pp 445-468. Cambridge, Mass.: MIT Press.

5. Network cohesion in kinship networks has the potential to outperform biological kin selection. • Biological kinship (limited thresholds 0 ≤ r < 1) • Coefficients of relatedness rijHamilton’s kin selection: inclusive fitness for pairs of relatives with shared genotype: only if rB – C > 0 is the selection gradient positive for those individuals with the genotype pair. Predictions are tested and borne out in A. Hughes 1988, Evolution and Human Kinship. Social kinship (unlimited thresholds k > 1)Kinship cohesion as measured by pairwise connectivity, biconnectivity and multiconnectivity: predicting individual fitness and population growth or spread. It may also predict resources for support of offspring or of population growth or spread. And, with test data for predictions: We can compare results for biological and social kinship networks, i.e., relatedness coefficient networks versus cohesive coefficients for social networks

6. Social explanations give more possibilities of network cooperation, thresholds less restrictive • There are more forms of cooperation in species or societies with complex networks, and less limitation on how cooperation can be achieved through multiconnectivity. E.g., Among non-human primates, one gender is often the core spatial group, while the other gender joins another group for mating. • Marriage exponentially raises the numbers of relatives as interacting generations increase, for example; “fathers” ramify fractal branching. • There are many ways to increase multiconnectivity: pairwise between individuals, bicomponent through marriage ties linking descent lines, k-component through higher level lineages, all uniquely defined mathematical groupings with distinct boundaries.

7. kin-type-k = a pair of families with knode-disjoint connecting pathsa and b have k=four, not five node-independent connecting pathsc and d have k=two; the higher k, the more pairwise cohesion DEFINITIONS a-d-e-b disjoint from a-c-b a-c-e-b not disjoint from a-c-b because they share node c d e b a Bicomponent(or k-component) of a graph = a maximal subraph in which (1) every pair of nodes has 2+ (or k) node-disjoint paths, equivalent to (2) no separating set of nodes smaller than 2 (or k). c f The bicomponent excludes node f. Pairwise cohesion k is easy to compute for large networks with UCINET software, using White and Newman 2001 Fast Approximation Algorithms for Finding Node-Independent Paths in Networks. Santa Fe Institute Working Paper 01-07-035. MULTICONNECTED means connected through multiple disjoint paths Bicomponents (k=2) are easily computed in Pajek and other network software. K-components (k>2) arecomputed in R (igraph) and SAS: White and Moody 2003 Social Cohesion and Embeddedness.� American Sociological Review 68:103-127 .

8. CLARITY OF DEFINITIONS GIVES CLEAR BASES FOR PREDICTIONS • CONCERNING SELECTION OUTCOMES (offspring, resources) • Different but related measures of cohesion are appropriate at different levels : “Interlineagecohesion uses k-components; interfamily cohesion (thru marriage and descent) uses 2-components, individuals use pairwise cohesion.” In each case we are talking about bounded but behaviorally flexible subgroups in a network bounded by inclusive levels of multiconnectivity, i.e., maximal sets of individuals linked by multiple disjoint paths. • The flexibility and emergence of these sources of cohesion-cooperativity vastly exceed the slow-moving genetic processes. In k-component groups of inter-lineages ties the redundancy of ties to other groups is limited only by numbers of people in the groups. In 2-component groups the direct links to the larger network through (two)parents can grow in size indefinitely or shrink to provide more restricted selective advantages.

9. BUT PREDICTIVE COHESION THEORY FOR KINSHIP NEEDS TO BE TESTED IN EACH SPECIFIC CASE, involving an entire network. AS BETWEEN BIOLOGICAL SHARING OF GENES Vs. SOCIALLY COHESIVE SHARING THROUGH NETWORK TIES: While there are many genes and paired genotypes from the father and the mother, each set of genotypes for kin selection is limited by genetic relatedness rij< 1 for pairs of people with common ancestral genotypes. Whereas pairwise cohesion has no such limit:,2-component groups can grow indefinitely through married ties, far beyond genetic relatedness, and k-component inter-lineage groups can increase kand size indefinitely. For tests of theory, each of these definitions is applied where appropriate.

10. Google: kinsources - 88 contributions ofkinship network datasets and ethnographic documentation to date @ http://kinsource.net/kinsrc/bin/view/KinSources AMPLE DATA FOR THEORY TESTS ARE PROVIDED HERE I BEGIN PREPARATION FOR THEORY TESTS WITH CHOICE OF KINSHIP REPRESENTIONS THAT ARE BEST FOR FACILITATING THE MEASUREMENT OF MULTICONNECTIVITY

11. Data and DAG Representation: MultilevelKinship Networks: Nodes are parents, edges are individuals P-graphs link pairs of parents (biological or culturally defined) to their descendants P-graphs are constructed from standard genealogical data (GED) files, using PAJEK or a number of other programs. See: http://eclectic.ss.uci.edu/~drwhitehas guides as to web-site availability with documentation & sample representations. biological ri,Jand social ki,Jcoeff of relatedness are coded in edge-edge matrices (individuals as edges) Male edge,ri,J=3/8 Female edge Male edge, FaFaSiDaSon ri,i =1

12. Preview of examples for empirical tests: Does kin-type-k cohesion (network-inclusive fitness) predict fitness in offspring and resources for offspring of individuals for groups? • The examples illustrate differences in appropriate cohesion measures and variation in R2of different predictions. • p=.01, .001 Renaissance Florence: inter-lineage k-cohesion  lineage-pop growth R2=.?? • p=.001 European farming village: marriage 2-cohesion  inheritors, others emigrate • R2=.54 (an underestimation given partial sampling) • p=.0001 Turkish nomads: both inter-lineage k-cohesion and marriage 2-cohesion • R2=.95 marriage bicomponent (p=.0001)  lineage population growth, stayers/leavers. • Australian Alyawarra: marriage 2-cohesion spread over larger distances than at random, systematically integrating larger inter-group populations for demicsurvival • p=.0001 for Indigenous Australian family size (~6) versus foragers (n=339) elsewhere (~4) • PulEliya: Pairwise consanguineal cohesion  Old Fields inheritance (R2=.31, p= 0.02).

13. 1st example Lineage and marriage network fitness effects • Padgett, John. 2010. Open Elite? Social Mobility, Marriage, and Family in Florence, 1282-1494. Renaissance Quarterly 63: 357-411: predictors of population growth at the level of lineages defined by family name. Methods: • 3. Bicomponent: families – largest set of families (couples) with two or more node-independent paths between every pair. • 4. Structural (k-) cohesion: named family lines (e.g., patrilineages) – largest sets of lineages with k or more node-independent paths between every pair of lineages. Bicomponents calculated by Pajek. Structural cohesion (k-components) are computed in R (igraph) and SAS.

14. Padgett Renaissance Florence regressions: family size 1282-1480 Growth or decline in Florentine family size during the Renaissance, 1282-1317 (period Cohesive marriage cores predict lineage growth in family size in periods 3 (1352-79) through 7 (1458–80) (Padgett Table 5 OLS coef). Graphs at left show OLS predictive regression coefficients (N=1,697 distinct families), and, at right, the statistical significance of the OLS coefficients. Average household wealth predicts a small fraction of the variance in family growth or decline, compared to marriage-network structural cohesion between families. The negative effect of family size (larger families, less growth) has less effect than household wealth, except for period 4 (1379-1403). The measure used is structural cohesion  4 (White and Harary 2001). (The Black Guelf faction, after winning the struggle in 1301 (X) between Blacks and Whites, and the anti-Chiompi faction, after putting down the Chiompi revolt in 1378 (X) between Blacks and Whites, were also endogamous for a century.) x x

15. 2nd example: Endogamous cohesion measures applied to kinship networks: Brudner and White’s Austrian farmer village: Bicohesive families retain farming property; other siblings emigrate. • 2. Pairwise cohesion: interfamily node-disjoint paths between each pair of nodes (e.g., couples). These apply to marriage relinkings and to cycles created by common ancestry marriages. • Structural (k=2) endogamy bicomponent: largest set of families (couples) with two or more node-independent paths between every pair. Applies to edges in the bicomponent as well as the nodes. Bicomponents are calculated by Pajek/net/cohesion/bicomponents. Structural cohesion is computed in R (igraph) and SAS.

16. Structural Endogamy social cohesion predictions Social Class: Bicohesive core predicts Carinthian farm inheritance Source: 1997 “Class, Property and Structural Endogamy: Visualizing Networked Histories,” Theory and Society 25:161-208. LilyanBrudner and Douglas White

17. Structural Endogamy social cohesion predictions 2nd example: Social Class: CarinthianFarmers R2=.29 Source: 1997 “Class, Property and Structural Endogamy: Visualizing Networked Histories,” Theory and Society 25:161-208. LilyanBrudner and Douglas White

18. Structural Endogamy social cohesion predictions 3rd example: A Turkish Nomadic Clan as prototype of Middle Eastern segmented lineage systems: The Role of Marital Cohesion (interfamily bicomponents, 2-cohesive) p-graph of the conicality of the nomad clan Data: 9 8 7 6 Generations 3 2 1

19. Structural Endogamy social cohesion predictions 3rd example: A Turkish Nomadic Clan as prototype of Middle Eastern segmented lineage systems: The Role of Marital Cohesion (interfamily bicomponents, 2-cohesive) Structural Endogamy Core of the nomad clan Results: • The index of relinking of a kinship graph is a measure of the extent to which marriages take place among descendants of a limited set of ancestors. • For the nomad clan the index of relinking is 75%, which is extremely high by world standards. • This is a picture of the structurally endogamous or relinked marriages within the nomad clan (nearly 75% or all marriages are endogamous):

20. Structural Endogamy social cohesion predictions 3rd example: A Turkish Nomadic Clan as prototype of Middle Eastern segmented lineage systems: The Role of Marital Cohesion (interfamily bicomponents, 2-cohesive) R2=.90 Does staying together as a clan depend on marital relinking? Results: Testing the hypothesis for stayers versus leavers Relinked Non-Relinking Marriages Marriages Totals villagers who became clan members 2** 1** 3 clan Husband and Wife 148 0 148 “ Hu married to tribes with reciprocal exchange 12 14 26 “ Hu left for village life 13 23 36 “ Hu married to village wife (34) or husband (1) 11 24 35 “ Hu married to tribes w/out reciprocal exchange 2 12 5 “ members who left for another tribe 0 8 8 villagers not joined to clan 1 3** 4 * tribes **non-clan by origin Totals 189 85 274 Pearson’s coefficientr=.95 without middle cells

21. Woodrow W. Denham. Alyawarra, C. Australia. Only 23 married couples (16%) of 144 are blood related. Little emigration. More cohesive ties span distances than do controlled random marriages. They integrate three cross-cutting types of moieties as well as initiatory localities. 4th Example: Alyawarra girls marry at puberty (ca. 14) while boys remain in initiation groups until ~28. This accounts for the age skewing of 3 female generations per 2 male generations. Multiplied by 2 into 6x4=24 idealized kinship positions and 8 subsections.

22. 5thExample: Pul Eliyan Sidedness (P-graph format) (in-laws are cooperative, siblings competitive over inheritance of land and irrigation resources through male or female)

23. Pul Eliya – Leach’s Table 4 (1961:144-145) shows that all but 2 of the 13 consanguneal marriage links in the network (red lines below) that form “cohesive sociocentric sides” in the village involve claims on Old Fields inheritance (R= 0.31, p= 0.02). p<.0001). R2=.10

24. My motivation has been to test important propositions involving tests of the hypothesis derived from PREDICTIVE COHESION THEORY that pairwise and subgroup cohesion in kinship networks measuring different forms of cooperativity among kin predict fitness variables for offspring and resources. This theory competes very effectively with kin selection theory which posits a positive selection gradient for a pair of blood relative if their inclusive fitness r satisfies rB - C > 0, r < 1 the relatedness coefficient, B the benefit of the behavior and C the cost. Social cohesion theory has constraints on how connectivities of different types can be constructed but are in general more flexible. Another concern is with an hypothesis that after calculating the pairwise-cohesion (node-independent paths in a kinship p-graph), taking the subset of upper u top-generation-t couples t < G, finding their respective k descendants in the G -t -1 lower generations, and storing the number of their common descendants in a matrix u x u. This hypothesis allows further tests of the prediction that the level k of pairwise-cohesion among pairs of ancestors in a DAG is correlated with higher numbers of shared descendants of the pairs. Summary: Pairwise and subgroup cohesion predictions

25. Heady notes the recognition “by contemporary evolutionists that far more cooperation takes place than can currently be explained by” – the ‘selfish gene’ – i.e., “arguments based on either tit-for-tat reciprocity or direct forms of kinship altruism” (2007: 489). including Hamilton’s kin selection.Kinship cohesion competes at many levels with kin selection: it includes ties of marriage and broadens kinship network effects on inclusive fitness, and has much broader measurement parameters compared to the limited relatedness coefficients of the biological models. It might also be the case that genes influence kin selection in terms of kinship network cohesion. Mutation of mitochondrial genes exhibit rapid adaption to environmental conditions, for example: do they affect kinship connectivity? And sexual selection needs to be investigated as it affects inclusive cohesive kinship. It remains to investigate whetherbiological kin selection effects for inclusive fitness might favor and improve social cohesion cooperative effects, and the possibility that kinship cohesion is a part of sexual selection due to the fact that interfamily cohesion is a function of matrimonial endogamy, which goes beyond common ancestry and beyond investment in offspring.

26. The pairwise cohesion approach can also be used with ego network survey data, as proposed by White and Heady (2005). “Transforming Ethnographic Data and Analytical Problems into Network Data Suitable for Complementary Analysis and Theory” 2005. Douglas R. White and Patrick Heady Halle MPI for Social Anthropology http://eclectic.ss.uci.edu/~drwhite/pub/Halle-MPIb.pdf end

27. Method: Use Pajek to analyze a DAG (pgraph) to produce a matrix that has the number of common descendants of two ancestors. Then use UCInet to compute pairwise k-cohesion. Then correlate the entries in the two matrices. Does more pairwise cohesion predict more common descendants? Then regress matrix 1 on matrix 2 as vectors (this adds an intercept term) This approach can also be used with ego network survey data, as proposed by White and Heady (2005). “Transforming Ethnographic Data and Analytical Problems into Network Data Suitable for Complementary Analysis and Theory” 2005. Douglas R. White and Patrick Heady Halle MPI for Social Anthropology http://eclectic.ss.uci.edu/~drwhite/pub/Halle-MPIb.pdf Pairwise cohesion predictions

28. Notes for comparison of the kin selection and the pairwise cohesion models

29. Genotypes. Let Rijbe entries for persons i,j of a genetic relatedness matrix (rii=1), (rij<1). For a single locus and alleles G1,…,Gn, with joint frequencies p11,…,pnn. Behavior may affect the fitness a of the individual and of m-1 others. A genotype GxGy of person i will have a vector of effects on itself and on relatives with relatedness ri1,…,rim to the focal individual i. These effects are summed for genotypes over persons weighted by relatedness, to produce inclusive fitness R*ijand R* = I j pipjR*ij. Van Veelen (2007) proves that selection for relatives with a genotype GxGy where rB – C > 0 will not only maximize average fitness but fitness of those individuals. van Veelen, Matthijs. 2007. Hamilton's missing link. J Theor Biol. 246(3):551-4. PMID 17316699

30. Kin-type-k pair. Let Kijbe entries for persons i,j of pairwise cohesion rii=b, rij<b. Pairwise cohesion has G1,…,Gn joint frequencies p11,…,pnn. Behavior may affect reinforcement a of the individual and of m-1 others. A kin-type-k GxGy of person i comes with a vector of effects on itself and on relatives with relatedness ri1,…,rim to the focal individual i. These effects over a network are summed for pairwisecohesion, weighted by kin-type-k, to produce inclusive reinforcement R*ijand R* = I j pipjR*ij. Van Veelen’s (2007) proofs hold for evolutionary reinforcement for others with an empirically measured kin-type-k threshold that will not only maximize average reinforcement but reinforcement for those individuals.

31. Preliminaries: • Biological and social kinship • Kinship data and predictions • Representing kinship networks – White, Batagelj, Mrvar (Pajek 1999)\ • DAGs for family networks • Measures of kinship cohesion (marriage and consanguinity; sexual selection?) • Empirical tests • Case studies for tests Social kinship (unlimited thresholds k > 1)Kinship cohesion as measured by pairwise connectivity, biconnectivity and multiconnectivity: predicting individual fitness and population growth or spread. It may also predict resources for support of offspring or of population growth or spread. And, with test data for predictions: We can compare results for biological and social kinship networks, i.e., relatedness coefficient networks versus cohesive coefficients for social networks • Biological kinship (limited thresholds 0 ≤ r < 1) • Coefficients of relatedness rijHamilton’s kin selection: inclusive fitness for pairs of relatives with shared genotype: only if rB – C > 0 is the selection gradient positive for those individuals with the genotype pair. Predictions are tested and borne out in A. Hughes 1988, Evolution and Human Kinship.

32. Cohesion measures for endogamous kinship networks: • Pairwise cohesion: individuals– how many node-independent paths between each pair of nodes. (Not as effective as #2). • Pairwise cohesion: families– how many node-independent paths between each pair of nodes (couples or families) • Bicomponent: families of couples – largest set of families with two or more node-independent paths between every pair. • Structural (k-) cohesion: named family lines (e.g., patrilineages) – largest sets of lineages with k or more node-independent paths between every pair of lineages. • (3) Bicomponents are calculated by Pajek and Ucinet. (4) Structural cohesion (White and Harary 2001 The Cohesiveness of Blocks in Social Networks: Node Connectivity and Conditional Density. Sociological Meth.) is computed in R (igraph) and SAS.

33. Social cohesion rule: Coefficients of pairwise cohesion relatedness kij define network-inclusive fitness for pairs or subsets of relatives (ancestry or marriage) with a kin-type level k. Parallels between the two modelsA kin-type-k pair is a pair of families with k node-disjoint connecting paths.It has to be demonstrated empirically whether a kin-type-k measure predicts a positive selection gradient. • Whether kin-type-k cohesion predicts network-inclusive fitness, like Hamilton’s rule, requires empirical tests: • a. Padgett’s Renaissance Florence: inter-lineage k-cohesion lineage-pop growth • b. European farmers: marriagebicomponent 2-cohesion inherit versus emigrate • c. Turkish nomads: both k-cohesion(lineages) and 2-cohesion (families) • d. Australian Alyawarra: 2-cohesion (families)  spread over larger distances than random net; marriages integrate larger inter-group populations for demicsurvival • e. Pul Eliya: Pairwise consanguineal cohesion  Old Fields inheritance and irrigation rights.

34. Threemulticonnectivitymeasures for kinship networks are: 1.pairwise cohesion, 2. bicomponent groups, 3. group k-cohesion. • The first (1.) parallels kin selection coefficients. I.e., inclusive social benefits of multiconnectivityparallel genetic inclusive benefits. Social benefits also apply to groups (structural cohesion -2 and 3) • Examples will show results of empirical predictions. What’s new: social kinshipThe network network-inclusive fitness rule is based on coefficients of pairwise cohesion relatedness kijfor those with a kin-type-k, k an integer.The biological and kinship measurement frameworks are equivalent.Empirical evidence needed to identify where k for kin-type-k pairs or groups predict positive selection gradients. • What’s old: biological kinship Hamilton’s rule is based on Coefficients of relatedness rij Inclusive fitness applies to those who share genotype pairs Proof that: if rB – C > 0 for genotype pairs the selection gradient is positive.

35. Having looked at examples where different but related measures of cohesion at different levels are appropriate (pairwise cohesion between multiconnected individuals, 2-components limited by the fact that for a network of marriages, each marriage links directly to the larger network only through the two parents (then to other children, parents’ parents, parents’ siblings, etc.)