Abstract: network-inclusive Fitness
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 pairwisecohesion (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.
Network-inclusive Fitness :
Kinship networks from social and genetic perspectives
Testing 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
Hamilton’s rule: Genes for altruism should increase in
frequency when rB-C > 0 (where r < 1) and
B = 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 5875341 PMID 5875340.
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.
Network cohesion in kinship networks has the
potential to outperform biological kin selection.
Biological kinship (limited thresholds 0 ≤ r < 1)
Coefficients of relatedness rij Hamilton’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
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, bicomponents of
marriage ties linking descent lines, k-components
linking higher level lineages, all uniquely defined
mathematical groupings with distinct boundaries.
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. Kcomponents (k>2) are computed in R (igraph) and SAS: White and Moody 2003 Social
Cohesion and Embeddedness.� American Sociological Review 68:103-127.
kin-type-k = a pair of families with k node-disjoint connecting paths
a and b have k=four, not five node-independent connecting paths
c and d have k=two; the higher k, the more pairwise cohesion
DEFINITIONS
d
e
a
c
f
The bicomponent excludes node f.
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).
a-d-e-b disjoint from a-c-b
b
a-c-e-b not disjoint from a-c-b
because they share node c
CLARITY OF DEFINITIONS GIVES CLEAR BASES FOR PREDICTIONS
CONCERNING SELECTION OUTCOMES (offspring, resources)
Different but related measures of cohesion are appropriate at
different levels : “Interlineage cohesion uses k-components;
interfamily cohesion (thru marriage and descent) uses 2components, individuals use pairwise cohesion.” In each case we
are talking about behaviorally flexible subgroups bounded within
a network by inclusive levels of multiconnectivity, i.e., unique
maximal sets of individuals linked by multiple disjoint paths.
The flexibility and emergence of these sources of k-connectedcooperativity vastly exceed the slow-moving genetic processes.
In k-component groups inter-lineages redundancy of ties to other
groups is limited only by numbers of people in the groups. In 2component groups the direct links to the larger network through
(two) parents can grow in size indefinitely or shrink to provide
more restricted selective advantages.
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 marriag ties, far beyond
genetic relatedness, and k-component inter-lineage groups can
increase k and size indefinitely. For tests of theory, each of these
definitions is applied where appropriate.
Google: kinsources - 88 contributions of kinship network datasets and ethnographic
documentation to date @ http://kinsource.net/kinsrc/bin/view/KinSources
AMPLE DATA FOR THEORY TESTS ARE PROVIDED HERE
PREPARATION FOR THEORY TESTS BEGINS WITH CHOICE OF KINSHIP REPRESENTIONS
THAT ARE BEST FOR FACILITATING THE MEASUREMENT OF MULTICONNECTIVITY
Data and DAG (dir asym graph) Representation:
Multilevel Kinship Networks: Nodes are parents, edges are individuals
P-graphs link pairs of parents (biological or culturally defined) to their descendants
generations
Female
edge
Male edge,
ri,J=3/8
P-graphs are constructed from
standard genealogical data (GED)
files, using PAJEK or a number of
other programs. See:
http://eclectic.ss.uci.edu/~drwhite has
guides as to web-site availability
with documentation & sample
representations.
Male edge,
FaFa SiDaSon
ri,i =1
biological ri,J and social ki,J coeff of
relatedness are coded in edgeedge matrices (individuals as edges)
What MULTICONNECTIVITY BUYS YOU IN TERMS OF POTENTIAL
COOPERATION (for offspring, resources) IN KINSHIP NETWORKS
Most kinship networks have a giant bicomponent in which every
couple has multiple connections (with disjoint paths) to all others
in the bicomponent. Having no links to others inhibits cooperation.
Having only 1 path linking to this bicomponent in a p-graph (e.g., a
child to the parental couple, with single disjoint paths to all others)
is qualitatively different to having 2 disjoint paths to all others,
i.e., multiconnectivity.
This buys you information from multiple sources, introductions to
others through many channels, a multitude of independent paths
for indirect exchange, reinforcement for relations for others linked
through kinship ties, participation in a local culture where
knowledge diffuses easily, and there is more mediation of conflict.
Preview of examples for empirical tests: Do k-cohesion (networkinclusive fitness) and pairwise cohesion predict fitness in offspring
and resources for offspring of individuals for groups?
The examples illustrate differences in appropriate cohesion measures and
variation in R2 of different predictions.
Renaissance Florence: inter-lineage k-cohesion  lineage-pop growth R2~~.1? p=.01, .001
European farming village: marriage 2-cohesion  inheritors, others emigrate
R2=.54 p=.001 (an underestimation given partial sampling)
Turkish nomads: both inter-lineage k-cohesion and marriage 2-cohesion
R2=.95 p=.0001 marriage bicomponent  lineage population growth, stayers/leavers.
Australian Alyawarra marriage 2-cohesion spread over larger distances than at random,
systematically integrating larger inter-group populations for demic survival
R2=.81 p=.0001 (n339) for foragers other than (~4) Indigenous Australia (~6)
Pul Eliya Pairwise consanguineal cohesion  Old Fields inheritance (R2=.10, p= 0.02).
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:
•
Bicomponent: families – largest set of families (couples) 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.
Bicomponents calculated by Pajek.
Structural cohesion (k-components) are computed in R (igraph) and SAS.
Padgett Renaissance Florence regressions: family size 1282-1480
x
x
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.)
2nd example: Endogamous cohesion measures applied to
kinship networks: Brudner and White’s Austrian farmer
village: Bicohesive families retain farming property; other
siblings emigrate.
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.
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. Lilyan Brudner and Douglas White
Structural Endogamy social cohesion predictions
2nd example: Bicohesive core predicts Carinthian farm inheritance
R2=.29
Source: 1997 “Class, Property and Structural Endogamy: Visualizing Networked
Histories,” Theory and Society 25:161-208. Lilyan Brudner and Douglas White
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)
9
8
7
6 Generations 3
2
1
Data:
p-graph of the conicality of the nomad clan
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)
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):
Structural Endogamy Core of the nomad clan
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)
Does staying together as a clan depend on marital relinking?
Results: Testing the hypothesis for stayers versus leavers
R2=.90
Relinked
Non-Relinking
Marriages
Marriages
Totals
villagers who became clan members
2**
1**
clan Husband and Wife
148
0
“ Hu married to tribes with reciprocal exchange 12
14
“ Hu left for village life
13
23
“ Hu married to village wife (34) or husband (1) 11
24
“ Hu married to tribes w/out reciprocal exchange 2
12
“ members who left for another tribe
0
8
villagers not joined to clan
1
3**
* tribes
**non-clan by origin
Totals
189
85
Pearson’s coefficient R2=.90 without the middle cells
3
148
26
36
35
5
8
4
274
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. Family size for foragers (n=339)
averages ~4 elsewhere versus Indigenous
Australian size ~6 (R2=.81, p=.0001).u
5th Example: Pul Eliyan Sidedness (P-graph format)
(in-laws are cooperative, siblings competitive over inheritance
of land and irrigation resources through male or female,
creating gender ambiguity in sidedness)
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
R2=.10
involve claims on Old Fields inheritance (p= 0.02).
Summary: Pairwise and subgroup cohesion predictions
My motivation has been to test important propositions involving hypotheses
derived from PREDICTIVE COHESION THEORY that pairwise and subgroup
cohesion in kinship networks measuring different forms of potential
cooperativity among kin predict fitness variables for offspring and resources.
This theory competes very effectively with kin selection, which posits a
positive selection gradient for pairs of blood relatives if their inclusive fitness
r satisfies threshold rB - C > 0, r < 1 for 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. Thresholds for k typically run between 2 and 6.
Another concern is with an hypothesis that after calculating the pairwisecohesion (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.
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 whether biological 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.
The pairwise and social cohesion
approaches can also be used with ego
network survey data, as proposed by
White and Heady (2005).
end
“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
Which of these societies would you say has
the highest altruism?
• Alyawarra – no wars, extended cooperative
networks
• Nomads – occasional factionalism and excellent
dispute resolution
• Farmers – tight social controls, emigrants return
• Florence – rotating offices, high economic
cooperation within economic classes, but intense
occasional class warfare; competition between
lineages, some competition within lineages
• Sri Lankans (Pul Eliya) – intense conflicts over
land inheritance mediated by reciprocal exchange
Notes for comparison of the kin selection
and the pairwise cohesion models
Genotypes. Let Rij be 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*ij and 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
Kin-type-k pair. Let Kij be 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 pairwise cohesion, weighted by kin-type-k,
to produce inclusive reinforcement R*ij and 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.
Preliminaries:
1. Biological and social kinship
2. Kinship data and predictions
3. Representing kinship networks – White, Batagelj, Mrvar (Pajek 1999)\
4. DAGs for family networks
5. Measures of kinship cohesion (marriage and consanguinity; sexual selection?)
6. Empirical tests
7. Case studies for tests
Biological kinship (limited thresholds 0 ≤ r < 1)
Coefficients of relatedness rij Hamilton’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
Cohesion measures for endogamous kinship networks:
1.
Pairwise cohesion: individuals– how many node-independent
paths between each pair of nodes. (Not as effective as #2).
2.
Pairwise cohesion: families– how many node-independent paths
between each pair of nodes (couples or families)
3.
Bicomponent: families of couples – largest set of families 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.
(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.
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 models
A 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: marriage bicomponent 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
demic survival
e. Pul Eliya: Pairwise consanguineal cohesion  Old Fields inheritance and
irrigation rights.
Three multiconnectivity measures 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 multiconnectivity parallel genetic inclusive benefits.
Social benefits also apply to groups (structural cohesion -2 and 3)
Examples will show results of empirical predictions.
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.
What’s new: social kinship
The network network-inclusive fitness rule is based on coefficients of pairwise
cohesion relatedness kij for 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.
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.)