- Email: [email protected]
Network modularity promotes cooperation
Journal of Theoretical Biology 324 (2013) 103–108
Contents lists available at SciVerse ScienceDirect
Journal of Theoretical Biology journal homepage: www.elsevier.com/locate/yjtbi
Letter to Editor
Network modularity promotes cooperation
a r t i c l e i n f o Keywords: Evolution Game theory Social network
abstract Cooperation in animals and humans is widely observed even if evolutionary biology theories predict the evolution of selfish individuals. Previous game theory models have shown that cooperation can evolve when the game takes place in a structured population such as a social network because it limits interactions between individuals. Modularity, the natural division of a network into groups, is a key characteristic of all social networks but the influence of this crucial social feature on the evolution of cooperation has never been investigated. Here, we provide novel pieces of evidence that network modularity promotes the evolution of cooperation in 2-person prisoner’s dilemma games. By simulating games on social networks of different structures, we show that modularity shapes interactions between individuals favouring the evolution of cooperation. Modularity provides a simple mechanism for the evolution of cooperation without having to invoke complicated mechanisms such as reputation or punishment, or requiring genetic similarity among individuals. Thus, cooperation can evolve over wider social contexts than previously reported. & 2012 Elsevier Ltd. All rights reserved.
1. Introduction Cooperation in animals and human is widely observed (Hill, 2002; Clutton-Brock, 2009; Dufour et al., 2009; Awata et al., 2010) even if evolutionary biology theories predict the evolution of selfish individuals (Darwin, 1859). The prisoner’s dilemma game has been repeatedly used to model cooperation in populations of selfish individuals (Rapoport and Chammah, 1965; Trivers, 1971; Axelrod and Hamilton, 1981). In the prisoner’s dilemma game, players have a choice between cooperation and defection. Each player receives a pay-off depending on his choice and the choice of the other player in the game. The highest pay-off for a player is achieved by defecting regardless of the decision of the other player (Rapoport and Chammah, 1965; Trivers, 1971). However, the total pay-off for the two players is the highest when they both cooperate (Rapoport and Chammah, 1965; Trivers, 1971). As the cost of the cooperative action increases in relation to its benefit, the percentage of cooperators in a population decreases (Hauert and Doebeli, 2004). Cooperation can evolve in a game context when players do not interact randomly, i.e. when their population is structured (Nowak and May, 1992; Ohtsuki et al., 2006). While in the traditional game players are equally likely to meet with other players (Axelrod and Hamilton, 1981), animals and humans live in structured populations in which their associations are not random (Underwood, 1981; Pepper et al., 1999; Newman, 2001; Lusseau, 2003; Croft et al., 2004). Computer simulations of cooperation games on lattices or networks have demonstrated that cooperation can evolve in structured populations when cooperators interact more frequently with each other than with defectors, and share the benefits of mutual cooperation (Rapoport and Chammah, 1965; Trivers, 1971). Some games were simulated on networks in which individuals could interact at different rates with others by controlling the frequency distribution of the 0022-5193/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jtbi.2012.12.012
number of partners individuals had (Wu et al., 2010; Cao et al., 2011). Modularity is a simple feature of all biological networks that influences heterogeneity in contacts between nodes in a network (Ravasz et al., 2002; Kashtan and Alon, 2005; Whitehead and Lusseau, 2012). Modularity describes the separation of networks into clusters and the degree with which those clusters interact, and can be estimated using a modularity coefficient (Q) ranging from 0 to 1 (Appendix A). A Q close to 1 indicates a network with a strong clustered structure in which interactions of individuals belonging to different clusters do not occur (Newman, 2006). Modularity can emerge without complicated rules of interactions (e.g. network motif, Milo et al., 2002; hierarchical organisation, Baraba´si et al., 2003) but simply from network nodes (individuals in our case) living in varying environments. For example, food availability or predator presence might affect individual interactions (Stanford, 1995; Heithaus and Dill, 2002). In addition, individual’s characteristics such as sex, age, and conditions can shape the way individuals interact with each other (Berman, 1982; McPherson et al., 2001; Ruckstuhl, 2007; Marcoux et al., 2010). Modularity is a key characteristic of social network topology that is not influenced by previously simulated network features (Reichardt and Bornholdt, 2007; Cao et al., 2011) but plays a key role in enabling the social behaviours of individuals (Sueur et al., 2011; Whitehead and Lusseau, 2012). This measure captures a key feature of social networks composed of different social units or communities (Girvan and Newman, 2002; Palla et al., 2005; Lusseau et al., 2006). Networks with high modularity should favour the evolution of cooperation (Voelkl and Kasper, 2009) because cooperation in animals tends to occur among individuals of the same social unit (Clutton-Brock et al., 2001; Awata et al., 2010). Even though network modularity can provide a measure of population’s structure, the two concepts are different. Measures of
104
Letter to Editor / Journal of Theoretical Biology 324 (2013) 103–108
population structure can be spatial, genetic or social (e.g. Hinde, 1976; Weir and Cockerham, 1984; Slatkin, 1987; Bohonak, 1999). In spatial and genetic structure, interactions between individuals are often modelled homogeneously, i.e. individuals interact with all individuals within their group and do not interact with individuals outside their group, or if they do, the rate of interaction is assumed to be equal among all groups (Killingback et al., 2006; Vainstein et al., 2007; Smaldino and Schank, 2012). While modularity might also arise from spatial or genetic structure, it allows us to model population structure more realistically. Modularity is a continuous measure; interactions between individuals vary continuously; individuals do not exclusively interact with individuals within their group and do not have to interact with all individuals in their group. In addition, the rate of interaction between groups is not the same for all groups (Newman and Girvan, 2004; Newman, 2006). Variation in the rates of association between members of a population, measured by association indices, also create structure in a population (Hinde, 1976; Whitehead, 1995; Bejder et al., 1998). Association indices can be binary or continuous, and are used to build sociograms and social networks (Lusseau, 2003; Whitehead, 1999; Croft et al., 2011). Different metrics are computed from
the association indices to describe the social structure of a population (e.g. Whitehead and Dufault, 1999) and modularity is one of them.
2. Material and methods Given the importance of modularity in social networks, we explored how it influences the evolution of cooperation in the 2-person prisoner’s dilemma game. We simulated games on weighted networks with varying modularity (Whitehead and Lusseau, 2012) in which vertices represented players and edges represented the associations between them (Fig. 1). For most analyses, network size was set to 100 players but the sensitivity of outcomes to network sizes was also evaluated by simulating games on network of size ranging from 20 to 500 (for details, see Appendix A). In addition, the effects of cluster size (from 2 to 33) and cost-to-benefit ratios (from 0.1 to 0.9) were examined (Appendix A). Games started with an equal number of cooperators and defectors, randomly located in the network. A player entered a game with all its neighbours that had an association higher than a threshold value set at the beginning of each
Fig. 1. Examples of weighted networks of 50 players (cooperators are represented by blue circles and defectors by red squares) with (a) and (c) low modularity of 0.12; and (b) and (d) high modularity of 0.62. Networks (a) and (b) show the initial distribution of cooperators and defectors in the networks; (c) and (d) show the distribution of cooperators and defectors at the end of 6000 rounds of 2-person prisoner’s dilemma games played with a threshold of 0.4 and a cost-to-benefit ratio of 0.2. The high modularity conditions of (b) and (d) limited interactions between players, which allowed for the evolution clusters of cooperators. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Letter to Editor / Journal of Theoretical Biology 324 (2013) 103–108
simulation. We used threshold values ranging from 0.1 to 0.9 (Appendix A) to make our simulations comparable with previously published studies in which the association between individuals was binary (not weighted: Ohtsuki et al., 2006; Nowak et al., 2010; Allen et al., 2012). The evolutionary dynamics followed the ’’death–birth’’ update rule (Ohtsuki et al., 2006). We ran each simulation for 6000 rounds because this number of rounds was sufficient to reach equilibrium (Appendix B). The percentage of cooperators was averaged for the last 1000 simulations. Results shown represent the average of 100 repetitions for all different parameter combinations (for more details about the methods, see Appendix A).
3. Results Cooperation evolved in networks with high modularity because clusters of cooperators could emerge in such networks. The percentage of cooperators becoming fixed in the population after the iterative games increased with modularity (Fig. 2). Cooperation could not evolve for modularity values close to zero except for high threshold values (Fig. 2). Increasing the threshold value above which two players enter in a game increased the frequency of cooperators (Fig. 2). Modularity did not affect the percentage of cooperators for modularity values above 0.4 (Fig. 2). The cost-to-benefit ratio (c/b) of the pay-off associated with a game is also crucial for the evolution of cooperation since a high cost of cooperation relative to benefit inhibits cooperation (Hauert and Doebeli, 2004). In our simulation, cooperation could evolve in games with high cost-to-benefit ratio by increasing the social network modularity (Fig. 3). For an equal cluster size, an increase in modularity, especially from 0 to 0.2, allowed for the evolution of cooperation with higher cost relative to benefit. Thus, modularity relaxed the condition under which cooperation could evolve. The size of clusters (communities) within social networks also influenced the evolution of cooperation. Cooperation was less likely to be fixed at the end of simulations when networks were composed of larger clusters (Fig. 4). This result was not affected by network size (Fig. 5a and b). In fact, the evolution of cooperation for networks of varying modularity and network size showed similar patterns when the average cluster size was kept constant (Fig. 5a). However, when we allowed the average cluster size
Fig. 2. Effect of network modularity on the evolution of cooperation at the end of prisoner’s dilemma games with varying threshold values and a cluster size of 10. High threshold values coupled with high modularity values favoured cooperation.
105
within the network to increase with the size of the network, cooperation was less likely to evolve in larger than in smaller networks (Fig. 5b).
4. Discussion Network modularity encouraged the evolution of cooperation in the prisoner’s dilemma game by limiting interactions between players. In general, increasing modularity limited the number of neighbours a player had, leading to higher chance for clusters of cooperators to develop. Thus, modularity favours cooperation by limiting the interactions of individuals to members of their community. This finding is in accordance with Ohtsuki et al. (2006) results that cooperation can evolve when the cost-to-
Fig. 3. Maximum cost-to-benefit ratio for cooperation to exist at the end of 2-person prisoner’s dilemma games with networks of varying modularity and cluster sizes. For any particular cluster size, increased modularity allowed cooperation to evolve at higher cost-to-benefit ratios. Note that it was not possible to create networks of high modularity for large cluster sizes.
Fig. 4. Effect of network modularity on the evolution of cooperation at the end of prisoner’s dilemma games with varying cluster sizes and a threshold value of 0.4. Cooperation was more likely to evolve in small clusters and/or with high modularity. Note that it was not possible to create networks of high modularity with large cluster sizes.
106
Letter to Editor / Journal of Theoretical Biology 324 (2013) 103–108
Fig. 5. Effect of network modularity on the evolution of cooperation at the end of prisoner’s dilemma games with varying network sizes. (a) When the average cluster size was 10 for all network sizes, the frequency of cooperators for different modularities is similar for networks of different sizes. (b) The number of clusters per network was kept constant such that the size of the cluster in each network increased with increasing network size. Cooperation was more likely to be fixed in small networks with small cluster sizes.
benefit ratio is smaller than the average number of neighbours of a player (c/bok, where k is the average degree or the average number of neighbours of a player). Increased threshold values also favoured cooperation by limiting the interactions between players. Consequently, increased threshold values limited the average number of neighbours of a player (k), providing an explanation for these previous findings (Ohtsuki et al., 2006). Lastly, increased modularity allowed the evolution of cooperation in games with high cost-to-benefit ratios. Hence, modularity widens the range of social systems in which cooperation can evolve. Cooperation can evolve without involving group selection or inclusive fitness. In our simulations, the fitness of individuals is based only on the payoff associated with the games played by that player and the fitness of other individuals in the network does not influence the fitness of one player. In addition, selection acted on the fitness of players only at an individual level. Thus, our results are different from those of previously published work on group and kin selection (Hamilton, 1964; Price, 1970, 1972; Wilson, 1975). While modularity might arise from spatial or genetic structure, other mechanisms such as preferential association of individuals can also generate modularity (Girvan and Newman, 2002; Newman, 2012). Thus, cooperation can evolve without spatial or genetic structure in populations. The size of the clusters within a network had a negative effect on the evolution of cooperation regardless of the size of the network. Therefore, the net benefit received from cooperation depends on cluster size in modular social networks, regardless of network size. Accordingly, individuals need to constrain their group size in order to maintain the benefits of cooperation. Thus, it appears that the evolution of cooperation can play a role in constraining the evolution of group size in social animals. This supports hypotheses developed in cognitive ecology that behavioural constraints on individuals, as opposed to cognitive constraints, play a key role in group fission (Barrett et al., 2007). Our
findings are also in accordance with theoretical and empirical work on optimal group size in cooperative species (Koenig, 1981; Creel and Creel, 1995; Clutton-Brock et al., 1999). Thus, network modularity, a prevalent feature of social networks, can explain the evolution of cooperation in social animals. We found that levels of modularity required to encourage cooperation are common in real-world social networks. Cooperation could evolve in networks with modularity coefficients larger than 0.4 regardless of the threshold for engaging in a game (and for some Q values lower than 0.4 depending on the threshold value and the size of the cluster) which is consistent with values of real-world social networks such as in several primate and marine mammal species (Kasper and Voelkl, 2009; Parra et al., 2011). Indeed, all species that are classically known for their cooperation live in modular social networks, even though the mechanisms behind this modularity differ (Connor and Whitehead, 2005; Wittemyer et al., 2005; Smith et al., 2010). Thus, network modularity provides a solution to the evolution of cooperation in social animals and humans.
Acknowledgements We thank four anonymous reviewers for constructive comments on a previous version of this manuscript. Funding for MM was provided by the Fonds de Recherche Nature et Technologies du Que´bec. This work also received funding from the MASTS pooling initiative (The Marine Alliance for Science and Technology for Scotland) and their support is gratefully acknowledged. MASTS is funded by the Scottish Funding Council (Grant Reference HR09011) and contributing institutions. Thanks to the University of Aberdeen for the usage of the RINH/BioSS Beowulf cluster.
Letter to Editor / Journal of Theoretical Biology 324 (2013) 103–108
Appendix A We used networks to represent populations of individuals involved in 2-person prisoner’s dilemma games. Vertices of the network represented players and edges represented the associations or ties between individuals. We constructed weighted networks to represent the associations among individuals because they are a more appropriate representation of social interactions than binary (edge present or absent) networks (Lusseau et al., 2008). The weight of the edges between individuals represented the proportion of time individuals spent interacting and ranged from 0 to 1. Networks with different modularities were constructed according to Whitehead and Lusseau (2012) by varying a parameter, m, the maximum association value for individuls among different clusters. First, individuals were randomly assigned to a cluster. The association among individuals within clusters was randomly chosen from a uniform distribution on the interval [0,1] and the association between individuals among different clusters was randomly chosen from a uniform distribution on the interval [0, m]. For m ¼1, the network was random because the association of individuals within clusters was similar to the association of individuals among clusters. A value of m ¼0 resulted in a network with completely disconnected clusters. We explored values for m ranging from 0.001 to 0.8. The modularity was calculated from Newman (2004) and we used the eigenvector method for the division of the network into communities (Newman, 2006). We explored different levels of modularity and their effect on the evolution of cooperation. The modularity Q was calculated using the following equation (Newman, 2004): Q¼
ki kj 1 X Aij d ci ,cj 2m ij 2m
where m is the number of ties in the network, Aij is the weighted ties between player i and j, ki is the sum of all the weighted ties of player i (degree), and the function d(ci , cj) is 1 if i and j are in the same cluster and 0 otherwise. Simulations investigating the effect of cluster size (Fig. 4) and of cost-to-benefit ratio (Fig. 3) performed on networks of 100 individuals were divided into 3, 5, 10, 20, 30, and 50 clusters of average size 33, 20, 10, 5, 3, and 2, respectively. For the simulations examining the effect of threshold values on cooperation, the cluster size was kept at 10 (Fig. 2). We also tested the effect of the size of networks by changing their sizes to 20, 50, 100, 200, and 500 in two different sets of simulations (Fig. 5). In the first set of simulations, the cluster size was kept at 10 for all the network sizes (Fig. 5a). In the second set of simulations, the average cluster size increased with the network size (clusters of 2, 5, 10, 20, 50; Fig. 5b). The games started with an equal number of cooperators and defectors, randomly located in the network. To enter a game, the association between two players had to be higher than a threshold (h) set constant for all the rounds of a game. Thus, a player could enter several games during each round. To explore how the value of the threshold influenced the evolution of cooperation, we ran simulations with h values ranging from 0.1 to 0.9 (Fig. 2a). For all the simulations apart from the simulation to investigate costto-benefit ratios, the pay-off matrix was set to have a cost-tobenefit ratio (c/b) of 0.2 (Table A1). We also ran simulations with varying pay-off matrices with cost-to-benefit ratios ranging from 0.1 to 0.9 (Fig. 3). The fitness of players was the cumulative pay-off from all the games played. After each round, one player was selected randomly to ‘die’ and its neighbours competed to replace him proportionally to their fitness, similar to the ‘death–birth’ update
107
Table A1 Payoff matrix for the prisoner’s dilemma game where the benefit (b) is greater than the cost (c).
Payoff to cooperator Payoff to defector
Cooperator
Defector
bc b
c 0
rule (Ohtsuki et al., 2006). Each simulation ran for 6000 rounds and the percentage of cooperators was averaged for the last 1000 simulations, after reaching an equilibrium. In all cases the proportion of cooperators in the population stabilised within the 6000 rounds. Results presented in the manuscript represent the average of 100 repetitions for all different parameter combinations. All simulations were performed using the statistical software R (R Development Core Team, 2010).
Appendix B. Supplementary Information Supplementary data associated with this article can be found in the online version at http://dx.doi.org/10.1016/j.jtbi.2012.12. 012.
References Allen, B., Traulsen, A., Tarnita, C.E., Nowak, M.A., 2012. How mutation affects evolutionary games on graphs. J. Theorl. Biol. 299, 97–105. Awata, S., Kohda, M., Shibata, J.-Y., Hori, M., Heg, D., 2010. Group structure, nest size and reproductive success in the cooperatively breeding cichlid Julidochromis ornatus: a correlation study. Ethology 116, 316–328. Axelrod, R., Hamilton, W.D., 1981. The evolution of cooperation. Science 211, 1390–1396. Baraba´si, A.L., Dezso, Z., Ravasz, E., Yook, S.H., Oltvai, Z., 2003. Scale-free and hierarchical structures in complex networks. AIP Conf. Proc. 661, 1–16. Barrett, L., Henzi, P., Rendall, D., 2007. Social brains, simple minds: does social complexity really require cognitive complexity? Philos. Trans. R. Soc. London B: Biol. Sci. 362, 561–575. ¨ Bejder, L., Fletcher, D., Brager, S., 1998. A method for testing association patterns of social animals. Anim. Behav. 56, 719–725. Berman, C., 1982. The ontogeny of social relationships with group companions among free-ranging infant rhesus monkeys I. Social networks and differentiation. Anim. Behav. 30, 149–162. Bohonak, A.J., 1999. Dispersal, gene flow, and population structure. Q. Rev. Biol. 74, 21–45. Cao, L., Ohtsuki, H., Wang, B., Aihara, K., 2011. Evolution of cooperation on adaptively weighted networks. J. Theorl. Biol. 272, 8–15. Clutton-Brock, T., 2009. Cooperation between non-kin in animal societies. Nature 462, 51–57. Clutton-Brock, T.H., Brotherton, P.N.M., O’riain, M.J., Griffin, A.S., Gaynor, D., Kansky, R., Sharpe, L., McIlrath, G.M., 2001. Contributions to cooperative rearing in meerkats. Anim. Behav. 61, 705–710. Clutton-Brock, T.H., Gaynor, D., McIlrath, G.M., Maccoll, A.D.C., Kansky, R., Chadwick, P., Manser, M., Skinner, J.D., Brotherton, P.N.M., 1999. Predation, group size and mortality in a cooperative mongoose, Suricata suricatta. J. Anim. Ecol. 68, 672–683. Connor, R., Whitehead, H., 2005. Alliances II. Rates of encounter during resource utilization: a general model of intrasexual alliance formation in fission–fusion societies. Anim. Behav 69, 127–132. Creel, S., Creel, N.M., 1995. Communal hunting and pack size in African wild dogs, Lycaon pictus. Anim. Behav. 50, 1325–1339. Croft, D.P., Krause, J., James, R., 2004. Social networks in the guppy (Poecilia reticulata). Proc. R. Soc. London B: Biol. Sci. 271, S516–S519. Croft, D.P., Madden, J.R., Franks, D.W., James, R., 2011. Hypothesis testing in animal social networks. Trends Ecol. Evol. 26, 502–507. Darwin, C., 1859. On the Origin of Species by Means of Natural Selection. John Murray, London, UK. Dufour, V., Pele´, M., Neumann, M., Thierry, B., Call, J., 2009. Calculated reciprocity after all: computation behind token transfers in orang-utans. Biol. Lett. 5, 172–175. Girvan, M., Newman, M.E.J., 2002. Community structure in social and biological networks. Proc. Natl. Acad. Sci. 99, 7821–7826. Hamilton, W.D., 1964. The genetical evolution of social behaviour I. J. Theorl. Biol. 7, 1–16. Hauert, C., Doebeli, M., 2004. Spatial structure often inhibits the evolution of cooperation in the snowdrift game. Nature 428, 643–646.
108
Letter to Editor / Journal of Theoretical Biology 324 (2013) 103–108
Heithaus, M.R., Dill, L.M., 2002. Food availability and tiger shark predation risk influence bottlenose dolphin habitat use. Ecology 83, 480–491. Hill, K., 2002. Altruistic cooperation during foraging by the Ache, and the evolved human predisposition to cooperate. Hum. Nat. 13, 105–128. Hinde, R.A., 1976. Interactions, relationships and social structure. Man 11, 1–17. Kashtan, N., Alon, U., 2005. Spontaneous evolution of modularity and network motifs. Proc. Natl. Acad. Sci. 102, 13773–13778. Kasper, C., Voelkl, B., 2009. A social network analysis of primate groups. Primates 50, 343–356. Killingback, T., Bieri, J., Flatt, T., 2006. Evolution in group-structured populations can resolve the tragedy of the commons. Proc. R. Soc. B 273, 1477–1481. Koenig, W., 1981. Reproductive success, group size, and the evolution of cooperative breeding in the acorn woodpecker. Am. Nat. 117, 421–443. Lusseau, D., 2003. The emergent properties of a dolphin social network. Proc. R. Soc. London B 270, S186–S188. Lusseau, D., Wilson, B., Hammond, P.S., Grellier, K., Durban, J.W., Parsons, K.M., Barton, T.R., Thompson, P.M., 2006. Quantifying the influence of sociality on population structure in bottlenose dolphins. J. Anim. Ecol. 75, 14–24. Lusseau, D., Whitehead, H., Gero, S., 2008. Incorporating uncertainty into the study of animal social networks. Anim. Behav. 75, 1809–1815. Marcoux, M., Larocque, G., Auger-Methe, M., Dutilleul, P., Humphries, M.M., 2010. Statistical analysis of animal observations and associated marks distributed in time using Ripley’s functions. Anim. Behav. 80, 329–337. McPherson, M., Smith-Lovin, L., Cook, J.M., 2001. Birds of a feather: homophily in social networks. Annu. Rev. Sociol. 27, 415–444. Milo, R., Shen-Orr, S., Itzkovitz, S., Kashtan, N., Chklovskii, D., Alon, U., 2002. Network motifs: simple building blocks of complex networks. Science 298, 824–827. Newman, M., Girvan, M., 2004. Finding and evaluating community structure in networks. Phys. Rev. E, 69. Newman, M.E.J., 2004. Analysis of weighted networks. Phys. Rev. E 70, 056131. Newman, M.E.J., 2001. The structure of scientific collaboration networks. Proc. Natl. Acad. Sci. 98, 404–409. Newman, M.E.J., 2006. Modularity and community structure in networks. Proc. Natl. Acad. Sci. 103, 8577–8582. Newman, M.E.J., 2012. Communities, modules and large-scale structure in networks. Nat. Phys. 8, 25–31. Nowak, M.A., May, R.M., 1992. Evolutionary games and spatial chaos. Nature 359, 826–829. Nowak, M.A., Tarnita, C.E., Antal, T., 2010. Evolutionary dynamics in structured populations. Philos. Trans. R. Soc. London B: Biol. Sci. 365, 19–30. Ohtsuki, H., Hauert, C., Lieberman, E., Nowak, M.A., 2006. A simple rule for the evolution of cooperation on graphs and social networks. Nature 441, 502–505. Palla, G., Derenyi, I., Farkas, I., Vicsek, T., 2005. Uncovering the overlapping community structure of complex networks in nature and society. Nature 435, 814–818. Parra, G.J., Corkeron, P.J., Arnold, P., 2011. Grouping and fission–fusion dynamics in Australian snubfin and Indo-Pacific humpback dolphins. Anim. Behav. 82, 1423–1433. Pepper, J.W., Mitani, J.C., Watts, D.P., 1999. General gregariousness and specific social preferences among wild chimpanzees. Int. J. Primatol. 20, 613–632. Price, G.R., 1970. Selection and covariance. Nature 227, 520–521. Price, G.R., 1972. Extension of covariance selection mathematics. Ann. Hum. Genet. 35, 485–490. R. Development Core Team, 2010. R: A Language and Environment for Statistical Computing. Rapoport, A., Chammah, A.M., 1965. Prisoner’s Dilemma. The University of Michigan Press, Ann Arbor.
Ravasz, E., Somera, A.L., Mongru, D.A., Oltvai, Z.N., Baraba´si, A.-L., 2002. Hierarchical organisation of modularity in metabolic networks. Science 297, 1551–1555. Reichardt, J., Bornholdt, S., 2007. Partitioning and modularity of graphs with arbitrary degree distribution. Phys. Rev. E 76, 015102. Ruckstuhl, K.E., 2007. Sexual segregation in vertebrates: proximate and ultimate causes. Integr. Comp. Biol. 47, 245–257. Slatkin, M., 1987. Gene flow and the geographic structure of natural populations. Science 236, 787–792. Smaldino, P.E., Schank, J.C., 2012. Movement patterns, social dynamics, and the evolution of cooperation. Theor. Popul. Biol. 82, 48–58. Smith, J.E., Van Horn, R.C., Powning, K.S., Cole, A.R., Graham, K.E., Memenis, S.K., Holekamp, K.E., 2010. Evolutionary forces favoring intragroup coalitions among spotted hyenas and other animals. Behav. Ecol. 21, 284–303. Stanford, C.B., 1995. The influence of chimpanzee predation on group size and anti-predator behaviour in red colobus monkeys. Anim. Behav. 49, 577–587. Sueur, C., Jacobs, A., Amblard, F., Petit, O., King, A.J., 2011. How can social network analysis improve the study of primate behavior? Am. J. Primatol. 71, 1–17. Trivers, R., 1971. The evolution of reciprocal altruism. Q. Rev. Biol. 46, 35–57. Underwood, R., 1981. Companion preference in an eland herd. Afr. J. Ecol. 19, 341–354. Vainstein, M.H., Silva, A.T., Arenzon, J.J., 2007. Does mobility decrease cooperation? J. Theorl. Biol. 244, 722–728. Voelkl, B., Kasper, C., 2009. Social structure of primate interaction networks facilitates the emergence of cooperation. Biol. Lett. 5, 462–464. Weir, B.S., Cockerham, C.C., 1984. Estimating F-statistics for the analysis of population structure. Evolution 38, 1358–1370. Whitehead, H., Lusseau, D., 2012. Animal social networks as substrate for cultural behavioural diversity. J. Theor. Biol. 294, 19–28. Whitehead, H., 1995. Investigating structure and temporal scale in social organizations using identified individuals. Behav. Ecol. 6, 199–208. Whitehead, H., 1999. SOCPROG programs: analysing animal social structures. Behav. Ecol. Sociobiol. 63, 765–778. Whitehead, H., Dufault, S., 1999. Techniques for analyzing vertebrate social structure using identified individuals: review and recommendations. Adv. Stud. Behav. 28, 33–74. Wilson, D.S., 1975. A theory of group selection. Proc. Natl. Acad. Sci. 72, 143–146. Wittemyer, G., Douglas-Hamilton, I., Getz, W.M., 2005. The socioecology of elephants: analysis of the processes creating multitiered social structures. Anim. Behav. 69, 1357–1371. Wu, B., Zhou, D., Fu, F., Luo, Q., Wang, L., Traulsen, A., 2010. Evolution of cooperation on stochastic dynamical networks. PLoS ONE 5, e11187.
Marianne Marcoux n, David Lusseau Institute of Biological and Environmental Sciences, Zoology Building, Tillydrone Avenue, University of Aberdeen, Aberdeen AB24 2TZ, Scotland, United Kingdom E-mail address: [email protected] (M. Marcoux) Received 9 July 2012 Available online 19 December 2012
n
Corresponding author. Tel.: þ44 1224 272789; fax: þ44 1224 272396.
Contents lists available at SciVerse ScienceDirect
Journal of Theoretical Biology journal homepage: www.elsevier.com/locate/yjtbi
Letter to Editor
Network modularity promotes cooperation
a r t i c l e i n f o Keywords: Evolution Game theory Social network
abstract Cooperation in animals and humans is widely observed even if evolutionary biology theories predict the evolution of selfish individuals. Previous game theory models have shown that cooperation can evolve when the game takes place in a structured population such as a social network because it limits interactions between individuals. Modularity, the natural division of a network into groups, is a key characteristic of all social networks but the influence of this crucial social feature on the evolution of cooperation has never been investigated. Here, we provide novel pieces of evidence that network modularity promotes the evolution of cooperation in 2-person prisoner’s dilemma games. By simulating games on social networks of different structures, we show that modularity shapes interactions between individuals favouring the evolution of cooperation. Modularity provides a simple mechanism for the evolution of cooperation without having to invoke complicated mechanisms such as reputation or punishment, or requiring genetic similarity among individuals. Thus, cooperation can evolve over wider social contexts than previously reported. & 2012 Elsevier Ltd. All rights reserved.
1. Introduction Cooperation in animals and human is widely observed (Hill, 2002; Clutton-Brock, 2009; Dufour et al., 2009; Awata et al., 2010) even if evolutionary biology theories predict the evolution of selfish individuals (Darwin, 1859). The prisoner’s dilemma game has been repeatedly used to model cooperation in populations of selfish individuals (Rapoport and Chammah, 1965; Trivers, 1971; Axelrod and Hamilton, 1981). In the prisoner’s dilemma game, players have a choice between cooperation and defection. Each player receives a pay-off depending on his choice and the choice of the other player in the game. The highest pay-off for a player is achieved by defecting regardless of the decision of the other player (Rapoport and Chammah, 1965; Trivers, 1971). However, the total pay-off for the two players is the highest when they both cooperate (Rapoport and Chammah, 1965; Trivers, 1971). As the cost of the cooperative action increases in relation to its benefit, the percentage of cooperators in a population decreases (Hauert and Doebeli, 2004). Cooperation can evolve in a game context when players do not interact randomly, i.e. when their population is structured (Nowak and May, 1992; Ohtsuki et al., 2006). While in the traditional game players are equally likely to meet with other players (Axelrod and Hamilton, 1981), animals and humans live in structured populations in which their associations are not random (Underwood, 1981; Pepper et al., 1999; Newman, 2001; Lusseau, 2003; Croft et al., 2004). Computer simulations of cooperation games on lattices or networks have demonstrated that cooperation can evolve in structured populations when cooperators interact more frequently with each other than with defectors, and share the benefits of mutual cooperation (Rapoport and Chammah, 1965; Trivers, 1971). Some games were simulated on networks in which individuals could interact at different rates with others by controlling the frequency distribution of the 0022-5193/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jtbi.2012.12.012
number of partners individuals had (Wu et al., 2010; Cao et al., 2011). Modularity is a simple feature of all biological networks that influences heterogeneity in contacts between nodes in a network (Ravasz et al., 2002; Kashtan and Alon, 2005; Whitehead and Lusseau, 2012). Modularity describes the separation of networks into clusters and the degree with which those clusters interact, and can be estimated using a modularity coefficient (Q) ranging from 0 to 1 (Appendix A). A Q close to 1 indicates a network with a strong clustered structure in which interactions of individuals belonging to different clusters do not occur (Newman, 2006). Modularity can emerge without complicated rules of interactions (e.g. network motif, Milo et al., 2002; hierarchical organisation, Baraba´si et al., 2003) but simply from network nodes (individuals in our case) living in varying environments. For example, food availability or predator presence might affect individual interactions (Stanford, 1995; Heithaus and Dill, 2002). In addition, individual’s characteristics such as sex, age, and conditions can shape the way individuals interact with each other (Berman, 1982; McPherson et al., 2001; Ruckstuhl, 2007; Marcoux et al., 2010). Modularity is a key characteristic of social network topology that is not influenced by previously simulated network features (Reichardt and Bornholdt, 2007; Cao et al., 2011) but plays a key role in enabling the social behaviours of individuals (Sueur et al., 2011; Whitehead and Lusseau, 2012). This measure captures a key feature of social networks composed of different social units or communities (Girvan and Newman, 2002; Palla et al., 2005; Lusseau et al., 2006). Networks with high modularity should favour the evolution of cooperation (Voelkl and Kasper, 2009) because cooperation in animals tends to occur among individuals of the same social unit (Clutton-Brock et al., 2001; Awata et al., 2010). Even though network modularity can provide a measure of population’s structure, the two concepts are different. Measures of
104
Letter to Editor / Journal of Theoretical Biology 324 (2013) 103–108
population structure can be spatial, genetic or social (e.g. Hinde, 1976; Weir and Cockerham, 1984; Slatkin, 1987; Bohonak, 1999). In spatial and genetic structure, interactions between individuals are often modelled homogeneously, i.e. individuals interact with all individuals within their group and do not interact with individuals outside their group, or if they do, the rate of interaction is assumed to be equal among all groups (Killingback et al., 2006; Vainstein et al., 2007; Smaldino and Schank, 2012). While modularity might also arise from spatial or genetic structure, it allows us to model population structure more realistically. Modularity is a continuous measure; interactions between individuals vary continuously; individuals do not exclusively interact with individuals within their group and do not have to interact with all individuals in their group. In addition, the rate of interaction between groups is not the same for all groups (Newman and Girvan, 2004; Newman, 2006). Variation in the rates of association between members of a population, measured by association indices, also create structure in a population (Hinde, 1976; Whitehead, 1995; Bejder et al., 1998). Association indices can be binary or continuous, and are used to build sociograms and social networks (Lusseau, 2003; Whitehead, 1999; Croft et al., 2011). Different metrics are computed from
the association indices to describe the social structure of a population (e.g. Whitehead and Dufault, 1999) and modularity is one of them.
2. Material and methods Given the importance of modularity in social networks, we explored how it influences the evolution of cooperation in the 2-person prisoner’s dilemma game. We simulated games on weighted networks with varying modularity (Whitehead and Lusseau, 2012) in which vertices represented players and edges represented the associations between them (Fig. 1). For most analyses, network size was set to 100 players but the sensitivity of outcomes to network sizes was also evaluated by simulating games on network of size ranging from 20 to 500 (for details, see Appendix A). In addition, the effects of cluster size (from 2 to 33) and cost-to-benefit ratios (from 0.1 to 0.9) were examined (Appendix A). Games started with an equal number of cooperators and defectors, randomly located in the network. A player entered a game with all its neighbours that had an association higher than a threshold value set at the beginning of each
Fig. 1. Examples of weighted networks of 50 players (cooperators are represented by blue circles and defectors by red squares) with (a) and (c) low modularity of 0.12; and (b) and (d) high modularity of 0.62. Networks (a) and (b) show the initial distribution of cooperators and defectors in the networks; (c) and (d) show the distribution of cooperators and defectors at the end of 6000 rounds of 2-person prisoner’s dilemma games played with a threshold of 0.4 and a cost-to-benefit ratio of 0.2. The high modularity conditions of (b) and (d) limited interactions between players, which allowed for the evolution clusters of cooperators. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Letter to Editor / Journal of Theoretical Biology 324 (2013) 103–108
simulation. We used threshold values ranging from 0.1 to 0.9 (Appendix A) to make our simulations comparable with previously published studies in which the association between individuals was binary (not weighted: Ohtsuki et al., 2006; Nowak et al., 2010; Allen et al., 2012). The evolutionary dynamics followed the ’’death–birth’’ update rule (Ohtsuki et al., 2006). We ran each simulation for 6000 rounds because this number of rounds was sufficient to reach equilibrium (Appendix B). The percentage of cooperators was averaged for the last 1000 simulations. Results shown represent the average of 100 repetitions for all different parameter combinations (for more details about the methods, see Appendix A).
3. Results Cooperation evolved in networks with high modularity because clusters of cooperators could emerge in such networks. The percentage of cooperators becoming fixed in the population after the iterative games increased with modularity (Fig. 2). Cooperation could not evolve for modularity values close to zero except for high threshold values (Fig. 2). Increasing the threshold value above which two players enter in a game increased the frequency of cooperators (Fig. 2). Modularity did not affect the percentage of cooperators for modularity values above 0.4 (Fig. 2). The cost-to-benefit ratio (c/b) of the pay-off associated with a game is also crucial for the evolution of cooperation since a high cost of cooperation relative to benefit inhibits cooperation (Hauert and Doebeli, 2004). In our simulation, cooperation could evolve in games with high cost-to-benefit ratio by increasing the social network modularity (Fig. 3). For an equal cluster size, an increase in modularity, especially from 0 to 0.2, allowed for the evolution of cooperation with higher cost relative to benefit. Thus, modularity relaxed the condition under which cooperation could evolve. The size of clusters (communities) within social networks also influenced the evolution of cooperation. Cooperation was less likely to be fixed at the end of simulations when networks were composed of larger clusters (Fig. 4). This result was not affected by network size (Fig. 5a and b). In fact, the evolution of cooperation for networks of varying modularity and network size showed similar patterns when the average cluster size was kept constant (Fig. 5a). However, when we allowed the average cluster size
Fig. 2. Effect of network modularity on the evolution of cooperation at the end of prisoner’s dilemma games with varying threshold values and a cluster size of 10. High threshold values coupled with high modularity values favoured cooperation.
105
within the network to increase with the size of the network, cooperation was less likely to evolve in larger than in smaller networks (Fig. 5b).
4. Discussion Network modularity encouraged the evolution of cooperation in the prisoner’s dilemma game by limiting interactions between players. In general, increasing modularity limited the number of neighbours a player had, leading to higher chance for clusters of cooperators to develop. Thus, modularity favours cooperation by limiting the interactions of individuals to members of their community. This finding is in accordance with Ohtsuki et al. (2006) results that cooperation can evolve when the cost-to-
Fig. 3. Maximum cost-to-benefit ratio for cooperation to exist at the end of 2-person prisoner’s dilemma games with networks of varying modularity and cluster sizes. For any particular cluster size, increased modularity allowed cooperation to evolve at higher cost-to-benefit ratios. Note that it was not possible to create networks of high modularity for large cluster sizes.
Fig. 4. Effect of network modularity on the evolution of cooperation at the end of prisoner’s dilemma games with varying cluster sizes and a threshold value of 0.4. Cooperation was more likely to evolve in small clusters and/or with high modularity. Note that it was not possible to create networks of high modularity with large cluster sizes.
106
Letter to Editor / Journal of Theoretical Biology 324 (2013) 103–108
Fig. 5. Effect of network modularity on the evolution of cooperation at the end of prisoner’s dilemma games with varying network sizes. (a) When the average cluster size was 10 for all network sizes, the frequency of cooperators for different modularities is similar for networks of different sizes. (b) The number of clusters per network was kept constant such that the size of the cluster in each network increased with increasing network size. Cooperation was more likely to be fixed in small networks with small cluster sizes.
benefit ratio is smaller than the average number of neighbours of a player (c/bok, where k is the average degree or the average number of neighbours of a player). Increased threshold values also favoured cooperation by limiting the interactions between players. Consequently, increased threshold values limited the average number of neighbours of a player (k), providing an explanation for these previous findings (Ohtsuki et al., 2006). Lastly, increased modularity allowed the evolution of cooperation in games with high cost-to-benefit ratios. Hence, modularity widens the range of social systems in which cooperation can evolve. Cooperation can evolve without involving group selection or inclusive fitness. In our simulations, the fitness of individuals is based only on the payoff associated with the games played by that player and the fitness of other individuals in the network does not influence the fitness of one player. In addition, selection acted on the fitness of players only at an individual level. Thus, our results are different from those of previously published work on group and kin selection (Hamilton, 1964; Price, 1970, 1972; Wilson, 1975). While modularity might arise from spatial or genetic structure, other mechanisms such as preferential association of individuals can also generate modularity (Girvan and Newman, 2002; Newman, 2012). Thus, cooperation can evolve without spatial or genetic structure in populations. The size of the clusters within a network had a negative effect on the evolution of cooperation regardless of the size of the network. Therefore, the net benefit received from cooperation depends on cluster size in modular social networks, regardless of network size. Accordingly, individuals need to constrain their group size in order to maintain the benefits of cooperation. Thus, it appears that the evolution of cooperation can play a role in constraining the evolution of group size in social animals. This supports hypotheses developed in cognitive ecology that behavioural constraints on individuals, as opposed to cognitive constraints, play a key role in group fission (Barrett et al., 2007). Our
findings are also in accordance with theoretical and empirical work on optimal group size in cooperative species (Koenig, 1981; Creel and Creel, 1995; Clutton-Brock et al., 1999). Thus, network modularity, a prevalent feature of social networks, can explain the evolution of cooperation in social animals. We found that levels of modularity required to encourage cooperation are common in real-world social networks. Cooperation could evolve in networks with modularity coefficients larger than 0.4 regardless of the threshold for engaging in a game (and for some Q values lower than 0.4 depending on the threshold value and the size of the cluster) which is consistent with values of real-world social networks such as in several primate and marine mammal species (Kasper and Voelkl, 2009; Parra et al., 2011). Indeed, all species that are classically known for their cooperation live in modular social networks, even though the mechanisms behind this modularity differ (Connor and Whitehead, 2005; Wittemyer et al., 2005; Smith et al., 2010). Thus, network modularity provides a solution to the evolution of cooperation in social animals and humans.
Acknowledgements We thank four anonymous reviewers for constructive comments on a previous version of this manuscript. Funding for MM was provided by the Fonds de Recherche Nature et Technologies du Que´bec. This work also received funding from the MASTS pooling initiative (The Marine Alliance for Science and Technology for Scotland) and their support is gratefully acknowledged. MASTS is funded by the Scottish Funding Council (Grant Reference HR09011) and contributing institutions. Thanks to the University of Aberdeen for the usage of the RINH/BioSS Beowulf cluster.
Letter to Editor / Journal of Theoretical Biology 324 (2013) 103–108
Appendix A We used networks to represent populations of individuals involved in 2-person prisoner’s dilemma games. Vertices of the network represented players and edges represented the associations or ties between individuals. We constructed weighted networks to represent the associations among individuals because they are a more appropriate representation of social interactions than binary (edge present or absent) networks (Lusseau et al., 2008). The weight of the edges between individuals represented the proportion of time individuals spent interacting and ranged from 0 to 1. Networks with different modularities were constructed according to Whitehead and Lusseau (2012) by varying a parameter, m, the maximum association value for individuls among different clusters. First, individuals were randomly assigned to a cluster. The association among individuals within clusters was randomly chosen from a uniform distribution on the interval [0,1] and the association between individuals among different clusters was randomly chosen from a uniform distribution on the interval [0, m]. For m ¼1, the network was random because the association of individuals within clusters was similar to the association of individuals among clusters. A value of m ¼0 resulted in a network with completely disconnected clusters. We explored values for m ranging from 0.001 to 0.8. The modularity was calculated from Newman (2004) and we used the eigenvector method for the division of the network into communities (Newman, 2006). We explored different levels of modularity and their effect on the evolution of cooperation. The modularity Q was calculated using the following equation (Newman, 2004): Q¼
ki kj 1 X Aij d ci ,cj 2m ij 2m
where m is the number of ties in the network, Aij is the weighted ties between player i and j, ki is the sum of all the weighted ties of player i (degree), and the function d(ci , cj) is 1 if i and j are in the same cluster and 0 otherwise. Simulations investigating the effect of cluster size (Fig. 4) and of cost-to-benefit ratio (Fig. 3) performed on networks of 100 individuals were divided into 3, 5, 10, 20, 30, and 50 clusters of average size 33, 20, 10, 5, 3, and 2, respectively. For the simulations examining the effect of threshold values on cooperation, the cluster size was kept at 10 (Fig. 2). We also tested the effect of the size of networks by changing their sizes to 20, 50, 100, 200, and 500 in two different sets of simulations (Fig. 5). In the first set of simulations, the cluster size was kept at 10 for all the network sizes (Fig. 5a). In the second set of simulations, the average cluster size increased with the network size (clusters of 2, 5, 10, 20, 50; Fig. 5b). The games started with an equal number of cooperators and defectors, randomly located in the network. To enter a game, the association between two players had to be higher than a threshold (h) set constant for all the rounds of a game. Thus, a player could enter several games during each round. To explore how the value of the threshold influenced the evolution of cooperation, we ran simulations with h values ranging from 0.1 to 0.9 (Fig. 2a). For all the simulations apart from the simulation to investigate costto-benefit ratios, the pay-off matrix was set to have a cost-tobenefit ratio (c/b) of 0.2 (Table A1). We also ran simulations with varying pay-off matrices with cost-to-benefit ratios ranging from 0.1 to 0.9 (Fig. 3). The fitness of players was the cumulative pay-off from all the games played. After each round, one player was selected randomly to ‘die’ and its neighbours competed to replace him proportionally to their fitness, similar to the ‘death–birth’ update
107
Table A1 Payoff matrix for the prisoner’s dilemma game where the benefit (b) is greater than the cost (c).
Payoff to cooperator Payoff to defector
Cooperator
Defector
bc b
c 0
rule (Ohtsuki et al., 2006). Each simulation ran for 6000 rounds and the percentage of cooperators was averaged for the last 1000 simulations, after reaching an equilibrium. In all cases the proportion of cooperators in the population stabilised within the 6000 rounds. Results presented in the manuscript represent the average of 100 repetitions for all different parameter combinations. All simulations were performed using the statistical software R (R Development Core Team, 2010).
Appendix B. Supplementary Information Supplementary data associated with this article can be found in the online version at http://dx.doi.org/10.1016/j.jtbi.2012.12. 012.
References Allen, B., Traulsen, A., Tarnita, C.E., Nowak, M.A., 2012. How mutation affects evolutionary games on graphs. J. Theorl. Biol. 299, 97–105. Awata, S., Kohda, M., Shibata, J.-Y., Hori, M., Heg, D., 2010. Group structure, nest size and reproductive success in the cooperatively breeding cichlid Julidochromis ornatus: a correlation study. Ethology 116, 316–328. Axelrod, R., Hamilton, W.D., 1981. The evolution of cooperation. Science 211, 1390–1396. Baraba´si, A.L., Dezso, Z., Ravasz, E., Yook, S.H., Oltvai, Z., 2003. Scale-free and hierarchical structures in complex networks. AIP Conf. Proc. 661, 1–16. Barrett, L., Henzi, P., Rendall, D., 2007. Social brains, simple minds: does social complexity really require cognitive complexity? Philos. Trans. R. Soc. London B: Biol. Sci. 362, 561–575. ¨ Bejder, L., Fletcher, D., Brager, S., 1998. A method for testing association patterns of social animals. Anim. Behav. 56, 719–725. Berman, C., 1982. The ontogeny of social relationships with group companions among free-ranging infant rhesus monkeys I. Social networks and differentiation. Anim. Behav. 30, 149–162. Bohonak, A.J., 1999. Dispersal, gene flow, and population structure. Q. Rev. Biol. 74, 21–45. Cao, L., Ohtsuki, H., Wang, B., Aihara, K., 2011. Evolution of cooperation on adaptively weighted networks. J. Theorl. Biol. 272, 8–15. Clutton-Brock, T., 2009. Cooperation between non-kin in animal societies. Nature 462, 51–57. Clutton-Brock, T.H., Brotherton, P.N.M., O’riain, M.J., Griffin, A.S., Gaynor, D., Kansky, R., Sharpe, L., McIlrath, G.M., 2001. Contributions to cooperative rearing in meerkats. Anim. Behav. 61, 705–710. Clutton-Brock, T.H., Gaynor, D., McIlrath, G.M., Maccoll, A.D.C., Kansky, R., Chadwick, P., Manser, M., Skinner, J.D., Brotherton, P.N.M., 1999. Predation, group size and mortality in a cooperative mongoose, Suricata suricatta. J. Anim. Ecol. 68, 672–683. Connor, R., Whitehead, H., 2005. Alliances II. Rates of encounter during resource utilization: a general model of intrasexual alliance formation in fission–fusion societies. Anim. Behav 69, 127–132. Creel, S., Creel, N.M., 1995. Communal hunting and pack size in African wild dogs, Lycaon pictus. Anim. Behav. 50, 1325–1339. Croft, D.P., Krause, J., James, R., 2004. Social networks in the guppy (Poecilia reticulata). Proc. R. Soc. London B: Biol. Sci. 271, S516–S519. Croft, D.P., Madden, J.R., Franks, D.W., James, R., 2011. Hypothesis testing in animal social networks. Trends Ecol. Evol. 26, 502–507. Darwin, C., 1859. On the Origin of Species by Means of Natural Selection. John Murray, London, UK. Dufour, V., Pele´, M., Neumann, M., Thierry, B., Call, J., 2009. Calculated reciprocity after all: computation behind token transfers in orang-utans. Biol. Lett. 5, 172–175. Girvan, M., Newman, M.E.J., 2002. Community structure in social and biological networks. Proc. Natl. Acad. Sci. 99, 7821–7826. Hamilton, W.D., 1964. The genetical evolution of social behaviour I. J. Theorl. Biol. 7, 1–16. Hauert, C., Doebeli, M., 2004. Spatial structure often inhibits the evolution of cooperation in the snowdrift game. Nature 428, 643–646.
108
Letter to Editor / Journal of Theoretical Biology 324 (2013) 103–108
Heithaus, M.R., Dill, L.M., 2002. Food availability and tiger shark predation risk influence bottlenose dolphin habitat use. Ecology 83, 480–491. Hill, K., 2002. Altruistic cooperation during foraging by the Ache, and the evolved human predisposition to cooperate. Hum. Nat. 13, 105–128. Hinde, R.A., 1976. Interactions, relationships and social structure. Man 11, 1–17. Kashtan, N., Alon, U., 2005. Spontaneous evolution of modularity and network motifs. Proc. Natl. Acad. Sci. 102, 13773–13778. Kasper, C., Voelkl, B., 2009. A social network analysis of primate groups. Primates 50, 343–356. Killingback, T., Bieri, J., Flatt, T., 2006. Evolution in group-structured populations can resolve the tragedy of the commons. Proc. R. Soc. B 273, 1477–1481. Koenig, W., 1981. Reproductive success, group size, and the evolution of cooperative breeding in the acorn woodpecker. Am. Nat. 117, 421–443. Lusseau, D., 2003. The emergent properties of a dolphin social network. Proc. R. Soc. London B 270, S186–S188. Lusseau, D., Wilson, B., Hammond, P.S., Grellier, K., Durban, J.W., Parsons, K.M., Barton, T.R., Thompson, P.M., 2006. Quantifying the influence of sociality on population structure in bottlenose dolphins. J. Anim. Ecol. 75, 14–24. Lusseau, D., Whitehead, H., Gero, S., 2008. Incorporating uncertainty into the study of animal social networks. Anim. Behav. 75, 1809–1815. Marcoux, M., Larocque, G., Auger-Methe, M., Dutilleul, P., Humphries, M.M., 2010. Statistical analysis of animal observations and associated marks distributed in time using Ripley’s functions. Anim. Behav. 80, 329–337. McPherson, M., Smith-Lovin, L., Cook, J.M., 2001. Birds of a feather: homophily in social networks. Annu. Rev. Sociol. 27, 415–444. Milo, R., Shen-Orr, S., Itzkovitz, S., Kashtan, N., Chklovskii, D., Alon, U., 2002. Network motifs: simple building blocks of complex networks. Science 298, 824–827. Newman, M., Girvan, M., 2004. Finding and evaluating community structure in networks. Phys. Rev. E, 69. Newman, M.E.J., 2004. Analysis of weighted networks. Phys. Rev. E 70, 056131. Newman, M.E.J., 2001. The structure of scientific collaboration networks. Proc. Natl. Acad. Sci. 98, 404–409. Newman, M.E.J., 2006. Modularity and community structure in networks. Proc. Natl. Acad. Sci. 103, 8577–8582. Newman, M.E.J., 2012. Communities, modules and large-scale structure in networks. Nat. Phys. 8, 25–31. Nowak, M.A., May, R.M., 1992. Evolutionary games and spatial chaos. Nature 359, 826–829. Nowak, M.A., Tarnita, C.E., Antal, T., 2010. Evolutionary dynamics in structured populations. Philos. Trans. R. Soc. London B: Biol. Sci. 365, 19–30. Ohtsuki, H., Hauert, C., Lieberman, E., Nowak, M.A., 2006. A simple rule for the evolution of cooperation on graphs and social networks. Nature 441, 502–505. Palla, G., Derenyi, I., Farkas, I., Vicsek, T., 2005. Uncovering the overlapping community structure of complex networks in nature and society. Nature 435, 814–818. Parra, G.J., Corkeron, P.J., Arnold, P., 2011. Grouping and fission–fusion dynamics in Australian snubfin and Indo-Pacific humpback dolphins. Anim. Behav. 82, 1423–1433. Pepper, J.W., Mitani, J.C., Watts, D.P., 1999. General gregariousness and specific social preferences among wild chimpanzees. Int. J. Primatol. 20, 613–632. Price, G.R., 1970. Selection and covariance. Nature 227, 520–521. Price, G.R., 1972. Extension of covariance selection mathematics. Ann. Hum. Genet. 35, 485–490. R. Development Core Team, 2010. R: A Language and Environment for Statistical Computing. Rapoport, A., Chammah, A.M., 1965. Prisoner’s Dilemma. The University of Michigan Press, Ann Arbor.
Ravasz, E., Somera, A.L., Mongru, D.A., Oltvai, Z.N., Baraba´si, A.-L., 2002. Hierarchical organisation of modularity in metabolic networks. Science 297, 1551–1555. Reichardt, J., Bornholdt, S., 2007. Partitioning and modularity of graphs with arbitrary degree distribution. Phys. Rev. E 76, 015102. Ruckstuhl, K.E., 2007. Sexual segregation in vertebrates: proximate and ultimate causes. Integr. Comp. Biol. 47, 245–257. Slatkin, M., 1987. Gene flow and the geographic structure of natural populations. Science 236, 787–792. Smaldino, P.E., Schank, J.C., 2012. Movement patterns, social dynamics, and the evolution of cooperation. Theor. Popul. Biol. 82, 48–58. Smith, J.E., Van Horn, R.C., Powning, K.S., Cole, A.R., Graham, K.E., Memenis, S.K., Holekamp, K.E., 2010. Evolutionary forces favoring intragroup coalitions among spotted hyenas and other animals. Behav. Ecol. 21, 284–303. Stanford, C.B., 1995. The influence of chimpanzee predation on group size and anti-predator behaviour in red colobus monkeys. Anim. Behav. 49, 577–587. Sueur, C., Jacobs, A., Amblard, F., Petit, O., King, A.J., 2011. How can social network analysis improve the study of primate behavior? Am. J. Primatol. 71, 1–17. Trivers, R., 1971. The evolution of reciprocal altruism. Q. Rev. Biol. 46, 35–57. Underwood, R., 1981. Companion preference in an eland herd. Afr. J. Ecol. 19, 341–354. Vainstein, M.H., Silva, A.T., Arenzon, J.J., 2007. Does mobility decrease cooperation? J. Theorl. Biol. 244, 722–728. Voelkl, B., Kasper, C., 2009. Social structure of primate interaction networks facilitates the emergence of cooperation. Biol. Lett. 5, 462–464. Weir, B.S., Cockerham, C.C., 1984. Estimating F-statistics for the analysis of population structure. Evolution 38, 1358–1370. Whitehead, H., Lusseau, D., 2012. Animal social networks as substrate for cultural behavioural diversity. J. Theor. Biol. 294, 19–28. Whitehead, H., 1995. Investigating structure and temporal scale in social organizations using identified individuals. Behav. Ecol. 6, 199–208. Whitehead, H., 1999. SOCPROG programs: analysing animal social structures. Behav. Ecol. Sociobiol. 63, 765–778. Whitehead, H., Dufault, S., 1999. Techniques for analyzing vertebrate social structure using identified individuals: review and recommendations. Adv. Stud. Behav. 28, 33–74. Wilson, D.S., 1975. A theory of group selection. Proc. Natl. Acad. Sci. 72, 143–146. Wittemyer, G., Douglas-Hamilton, I., Getz, W.M., 2005. The socioecology of elephants: analysis of the processes creating multitiered social structures. Anim. Behav. 69, 1357–1371. Wu, B., Zhou, D., Fu, F., Luo, Q., Wang, L., Traulsen, A., 2010. Evolution of cooperation on stochastic dynamical networks. PLoS ONE 5, e11187.
Marianne Marcoux n, David Lusseau Institute of Biological and Environmental Sciences, Zoology Building, Tillydrone Avenue, University of Aberdeen, Aberdeen AB24 2TZ, Scotland, United Kingdom E-mail address: [email protected] (M. Marcoux) Received 9 July 2012 Available online 19 December 2012
n
Corresponding author. Tel.: þ44 1224 272789; fax: þ44 1224 272396.