Network modularity promotes cooperation
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Axelrod and Hamilton, 1981). In the prisplayers have a choice between cooperatioplayer receives a pay-off depending on hisof the other player in the game. The highesachieved by defecting regardless of theplayer (Rapoport and Chammah, 1965; Trthe total pay-off for the two players is the h
, 1965ases inpopula
in structured populations in which their associations are not
that cooperation can evolve in structured populations when
(Sueur et al., 2011; Whitehead and Lusseau, 2012). This measure
should favour the evolution of cooperation (Voelkl and Kasper,
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Journal of Theoretical Biology 324 (2013) 103108with others by controlling the frequency distribution of the populations structure, the two concepts are different. Measures ofcooperators interact more frequently with each other than withdefectors, and share the benets of mutual cooperation (Rapoportand Chammah, 1965; Trivers, 1971). Some games were simulatedon networks in which individuals could interact at different rates
2009) because cooperation in animals tends to occur amongindividuals of the same social unit (Clutton-Brock et al., 2001;Awata et al., 2010).
Even though network modularity can provide a measure of
0022-5193/$ - see front matter & 2012 Elsevier Ltd. All rights reserved.
http://dx.doi.org/10.1016/j.jtbi.2012.12.012random (Underwood, 1981; Pepper et al., 1999; Newman, 2001;Lusseau, 2003; Croft et al., 2004). Computer simulations ofcooperation games on lattices or networks have demonstrated
captures a key feature of social networks composed of differentsocial units or communities (Girvan and Newman, 2002; Pallaet al., 2005; Lusseau et al., 2006). Networks with high modularity(Nowak and May, 1992; Ohtsuki et al., 2006). While in thetraditional game players are equally likely to meet with otherplayers (Axelrod and Hamilton, 1981), animals and humans live
topology that is not inuenced by previously simulated networkfeatures (Reichardt and Bornholdt, 2007; Cao et al., 2011) butplays a key role in enabling the social behaviours of individualscooperate (Rapoport and Chammahcost of the cooperative action increthe percentage of cooperators in aand Doebeli, 2004).
Cooperation can evolve in a gameinteract randomly, i.e. when theh, 1965; Trivers, 1971;oners dilemma game,n and defection. Eachchoice and the choicet pay-off for a player isdecision of the otherivers, 1971). However,ighest when they both; Trivers, 1971). As therelation to its benet,tion decreases (Hauert
xt when players do notulation is structured
and can be estimated using a modularity coefcient (Q) rangingfrom 0 to 1 (Appendix A). A Q close to 1 indicates a network witha strong clustered structure in which interactions of individualsbelonging 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,Barabasi et al., 2003) but simply from network nodes (individualsin our case) living in varying environments. For example, foodavailability or predator presence might affect individual interac-tions (Stanford, 1995; Heithaus and Dill, 2002). In addition,individuals characteristics such as sex, age, and conditions canshape 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 networkhas been repeatedly used to model cooperation in populations ofselsh individuals (Rapoport and Chammainto clusters and the degree with which those clusters interact,Letter to Editor
Network modularity promotes cooperation
a r t i c l e i n f o
a b s t r a c t
Cooperation in animals an
the evolution of selsh in
evolve when the game tak
interactions between indi
characteristic of all social
cooperation has never be
modularity promotes the
ing games on social netw
between individuals favou
for the evolution of coope
or punishment, or requiri
wider social contexts than
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 ofselsh individuals (Darwin, 1859). The prisoners dilemma game
journal homepage: www
Journal of Theumans is widely observed even if evolutionary biology theories predict
iduals. Previous game theory models have shown that cooperation can
lace in a structured population such as a social network because it limits
als. Modularity, the natural division of a network into groups, is a key
works but the inuence of this crucial social feature on the evolution of
investigated. Here, we provide novel pieces of evidence that network
ution of cooperation in 2-person prisoners dilemma games. By simulat-
s of different structures, we show that modularity shapes interactions
the evolution of cooperation. Modularity provides a simple mechanism
n without having to invoke complicated mechanisms such as reputation
enetic similarity among individuals. Thus, cooperation can evolve over
& 2012 Elsevier Ltd. All rights reserved.
number of partners individuals had (Wu et al., 2010; Cao et al.,2011).
Modularity is a simple feature of all biological networks thatinuences heterogeneity in contacts between nodes in a network(Ravasz et al., 2002; Kashtan and Alon, 2005; Whitehead andLusseau, 2012). Modularity describes the separation of networks
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 individualsare often modelled homogeneously, i.e. individuals interact withall individuals within their group and do not interact withindividuals outside their group, or if they do, the rate of interac-tion is assumed to be equal among all groups (Killingback et al.,2006; Vainstein et al., 2007; Smaldino and Schank, 2012). Whilemodularity might also arise from spatial or genetic structure, itallows us to model population structure more realistically.Modularity is a continuous measure; interactions between indi-viduals vary continuously; individuals do not exclusively interactwith individuals within their group and do not have to interactwith all individuals in their group. In addition, the rate ofinteraction between groups is not the same for all groups(Newman and Girvan, 2004; Newman, 2006). Variation in therates of association between members of a population, measuredby association indices, also create structure in a population(Hinde, 1976; Whitehead, 1995; Bejder et al., 1998). Associationindices can be binary or continuous, and are used to buildsociograms 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 apopulation (e.g. Whitehead and Dufault, 1999) and modularityis one of them.
2. Material and methods
Given the importance of modularity in social networks, weexplored how it inuences the evolution of cooperation in the2-person prisoners dilemma game. We simulated games onweighted networks with varying modularity (Whitehead andLusseau, 2012) in which vertices represented players and edgesrepresented the associations between them (Fig. 1). For mostanalyses, network size was set to 100 players but the sensitivityof outcomes to network sizes was also evaluated by simulatinggames on network of size ranging from 20 to 500 (for details, seeAppendix A). In addition, the effects of cluster size (from 2 to 33)and cost-to-benet ratios (from 0.1 to 0.9) were examined(Appendix A). Games started with an equal number of coopera-tors and defectors, randomly located in the network. A playerentered a game with all its neighbours that had an associationhigher than a threshold value set at the beginning of each
Letter to Editor / Journal of Theoretical Biology 324 (2013) 103108104Fig. 1. Examples of weighted networks of 50 players (cooperators are represented b(b) and (d) high modularity of 0.62. Networks (a) and (b) show the initial distribut
cooperators and defectors at the end of 6000 rounds of 2-person prisoners dilem
modularity conditions of (b) and (d) limited interactions between players, which alcolor in this gure legend, the reader is referred to the web version of this article.)e circles and defectors by red squares) with (a) and (c) low modularity of 0.12; and
of cooperators and defectors in the networks; (c) and (d) show the distribution of
games played with a threshold of 0.4 and a cost-to-benet ratio of 0.2. The high
d for the evolution clusters of cooperators. (For interpretation of the references to
simulation. We used threshold values ranging from 0.1 to 0.9(Appendix A) to make our simulations comparable with pre-viously published studies in which the association betweenindividuals was binary (not weighted: Ohtsuki et al., 2006;Nowak et al., 2010; Allen et al., 2012). The evolutionary dynamicsfollowed the deathbirth update rule (Ohtsuki et al., 2006). Weran each simulation for 6000 rounds because this number ofrounds was sufcient to reach equilibrium (Appendix B). Thepercentage of cooperators was averaged for the last 1000 simula-tions. Results shown represent the average of 100 repetitions forall different parameter combinations (for more details about themethods, see Appendix A).
Cooperation evolved in networks with high modularitybecause clusters of cooperators could emerge in such networks.The percentage of cooperators becoming xed in the populationafter the iterative games increased with modularity (Fig. 2).Cooperation could not evolve for modularity values close to zero
within the network to increase with the size of the network,cooperation was less likely to evolve in larger than in smallernetworks (Fig. 5b).
Network modularity encouraged the evolution of cooperationin the prisoners dilemma game by limiting interactions betweenplayers. In general, increasing modularity limited the number ofneighbours a player had, leading to higher chance for clusters ofcooperators to develop. Thus, modularity favours cooperation bylimiting the interactions of individuals to members of theircommunity. This nding is in accordance with Ohtsuki et al.(2006) results that cooperation can evolve when the cost-to-
Fig. 3. Maximum cost-to-benet ratio for cooperation to exist at the end of2-person prisoners 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-benet ratios. Note that it was not possible
to create networks of high modularity for large cluster sizes.
Letter to Editor / Journal of Theoretical Biology 324 (2013) 103108 105Fig. 2. Effect of network modularity on the evolution of cooperation at the end ofprisoners dilemma games with varying threshold values and a cluster size of 10.except for high threshold values (Fig. 2). Increasing the thresholdvalue above which two players enter in a game increased thefrequency of cooperators (Fig. 2). Modularity did not affect thepercentage of cooperators for modularity values above 0.4 (Fig. 2).
The cost-to-benet ratio (c/b) of the pay-off associated with agame is also crucial for the evolution of cooperation since a highcost of cooperation relative to benet inhibits cooperation(Hauert and Doebeli, 2004). In our simulation, cooperation couldevolve in games with high cost-to-benet ratio by increasing thesocial network modularity (Fig. 3). For an equal cluster size, anincrease in modularity, especially from 0 to 0.2, allowed for theevolution of cooperation with higher cost relative to benet. Thus,modularity relaxed the condition under which cooperation couldevolve.
The size of clusters (communities) within social networks alsoinuenced the evolution of cooperation. Cooperation was lesslikely to be xed at the end of simulations when networks werecomposed of larger clusters (Fig. 4). This result was not affectedby network size (Fig. 5a and b). In fact, the evolution of coopera-tion for networks of varying modularity and network size showedsimilar patterns when the average cluster size was kept constant(Fig. 5a). However, when we allowed the average cluster sizeHigh threshold values coupled with high modularity values favoured cooperation.Fig. 4. Effect of network modularity on the evolution of cooperation at the end ofprisoners 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 modularitywith large cluster sizes.
Letter to Editor / Journal of Theoretical Biology 324 (2013) 103108106benet ratio is smaller than the average number of neighbours ofa player (c/bok, where k is the average degree or the averagenumber of neighbours of a player). Increased threshold values
Fig. 5. Effect of network modularity on the evolution of cooperation at the end of prwas 10 for all network sizes, the frequency of cooperators for different modularities
kept constant such that the size of the cluster in each network increased with incr
small cluster sizes.also favoured cooperation by limiting the interactions betweenplayers. Consequently, increased threshold values limited theaverage number of neighbours of a player (k), providing anexplanation for these previous ndings (Ohtsuki et al., 2006).Lastly, increased modularity allowed the evolution of cooperationin games with high cost-to-benet ratios. Hence, modularitywidens the range of social systems in which cooperation canevolve.
Cooperation can evolve without involving group selection orinclusive tness. In our simulations, the tness of individuals isbased only on the payoff associated with the games played bythat player and the tness of other individuals in the networkdoes not inuence the tness of one player. In addition, selectionacted on the tness of players only at an individual level. Thus,our results are different from those of previously published workon group and kin selection (Hamilton, 1964; Price, 1970, 1972;Wilson, 1975). While modularity might arise from spatial orgenetic structure, other mechanisms such as preferential associa-tion of individuals can also generate modularity (Girvan andNewman, 2002; Newman, 2012). Thus, cooperation can evolvewithout spatial or genetic structure in populations.
The size of the clusters within a network had a negative effecton the evolution of cooperation regardless of the size of thenetwork. Therefore, the net benet received from cooperationdepends on cluster size in modular social networks, regardless ofnetwork size. Accordingly, individuals need to constrain theirgroup s...