Physica A 389 (2010) 4708–4714

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Physica A

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Heterogeneity of allocation promotes cooperation in public goods gamesChuang Lei a,∗, Te Wu b, Jian-Yuan Jia a, Rui Cong a, Long Wang a,ba School of Mechano-electronic Engineering, Xidian University, Xi’an 710071, Chinab State Key Laboratory for Turbulence and Complex Systems, Center for Systems and Control, College of Engineering, Peking University, Beijing 100871, China

a r t i c l e i n f o

Article history:Received 12 May 2010Received in revised form 1 June 2010Available online 10 June 2010

Keywords:Evolution of cooperationPublic goods gameHeterogeneity

a b s t r a c t

We investigate the effects of heterogeneous investment and distribution on the evolutionof cooperation in the context of the public goods games. To do this, we develop a simplemodel in which each individual allocates differing funds to his direct neighbors basedupon their difference in connectivity, because of the heterogeneity of real social ties.This difference is characterized by the weight of the link between paired individuals,with an adjustable parameter precisely controlling the heterogeneous level of ties. Bynumerical simulations, it is found that allocating both too much and too little funds todiverse neighbors can remarkably improve the cooperation level. However, there existsa worst mode of funds allocation leading to the most unfavorable cooperation inducedby the moderate values of the parameter. In order to better reveal the potential causesbehind these nontrivial phenomena we probe the microscopic characteristics includingthe average payoff and the cooperator density for individuals of different degrees. Itdemonstrates rather different dynamical behaviors between the modes of these two typesof cooperation promoter. Besides, we also investigate the total link weights of individualsnumerically and theoretically for negative values of the parameter, and conclude thatthe payoff magnitude of middle-degree nodes plays a crucial role in determining thecooperators’ fate.

© 2010 Elsevier B.V. All rights reserved.

1. Introduction

Cooperation is widespread in natural and social systems [1,2]. Understanding the emergence and persistence of coopera-tive behavior among selfish individuals remains a fascinating challenge. Evolutionary game theory has provided a powerfultheoretical framework for addressing the intricate phenomenon of cooperation [3–5]. As the most suitable paradigms, theprisoner’s dilemma game (PDG) [6–12] and the snowdrift game (SG) [13–15] have been widely employed to interpret thecooperation conundrum through pairwise interactions. In particular, the public goods game (PGG) [16] usually conductedin economic experiments [17], well capturing the collective cooperation action in the case of group interactions of multipleagents, has also attracted much attention of scientists of different fields for studying the evolutionary dynamics [18–20].In a typical mode of the PGG played by N agents, all of them decide simultaneously whether or not to contribute to a com-mon pool. Closely similar to the PDG (or SG), two pure strategies are available. A cooperator (C) donates a cost c to the pool,while a defector (D) donates nothing. The total investment summed over all participants is multiplied by an enhancementfactor r , and then equally distributed among all these players irrespective of their initial contributions. As a consequence,each defector gets the same benefit of a cooperator at no cost. Under natural selection agents with more successful strat-egy have more capability of reproducing newly identical ones. Therefore, egoistic defectors would outperform cooperatorswhen r < N for any given well-mixed group [21–23]. Whereas this is at odds with the phenomenon of the flourishing co-operative behavior commonly observed in human society. To clearly reveal the potential conundrum in the context of the

∗ Corresponding author.E-mail address: chuang_lei@hotmail.com (C. Lei).

0378-4371/$ – see front matter© 2010 Elsevier B.V. All rights reserved.doi:10.1016/j.physa.2010.06.002

C. Lei et al. / Physica A 389 (2010) 4708–4714 4709

PGG, much efforts has been expended on studying the evolution of cooperation by virtue of varying mechanisms in morecomplex topologies.Hauert et al. have introduced the voluntary participation instead of the compulsorymode into the public goods game and

concluded that cooperation can be enhanced greatly [24]. In Ref. [25], the authors further introduced the autarkic ‘‘loner’’ asa third strategy besides cooperation and defection in a square lattice, which leads to a scenario of the cyclic dominance ofthe strategies. More recently, Wu et al. investigated the effect of the constant group size [26], in which interaction partnerswere picked up based upon their degrees and reputations respectively, on the evolution of cooperation. It was found thatlarge group can substantially improve the cooperation level in the most range of enhancement factor in both selectionregimes. Most notably, it has been well recognized that diversity and heterogeneity play a decisive role in the evolutionof cooperation. In Ref. [27], social diversity, associated with the number and the size of the PGG in which each individualparticipates, has been introduced and the results shown that cooperation is enhanced effectively. Subsequently, the authorsfurther verified that diversity can promote cooperation, while too much diversity would be desirable for cooperation in theevolutionary PGG [28]. Furthermore, heterogeneities of teaching activity of individuals [29,30], of payoff allocation [31], andof investment [32], have been developed as potential promoters of cooperation, with noticeable success.It is worth noting that, in most previous models of the PGG [24–28], both the investment and the distribution are almost

identical with respect to each individual. Due to the existence of social diversity [33], however, each individual evaluatesthe relationship between him and his neighbors being different. For example, in Ref. [34], the authors investigated theeffect of edge weight’s heterogeneity on the evolution of cooperation where the cooperators’ fate is nontrivially dependenton the value of weighting factor. Recently, ‘‘tag-based cooperation’’ [35–38] has been developed as an newly importantresearch line in evolutionary game theory. In these models, evolutionary dynamics of cooperation proceeds based onphenotypic tags associated with the similarity between individuals. Furthermore, in Refs. [39,40], the authors found thatmoderate tolerance can result in the optimal cooperation level, in which the focal individual either only interacts with hisneighbors whose reputations [26,41,42] are within his tolerance range, or severs unfavorable interactions and searches forthe favorable ones [40,41] based on his tolerance threshold. These conditional interactions to a degree can be seen as a kindof inhomogeneous phenotypic tags which facilitates the evolution of cooperation.In this paper we incorporate heterogeneity characterized by the phenotypic distance in the degrees of nodes into the

PGG, because of individuals treating their diverse neighbors differently with rational manner, i.e., the difference of verticesinduced by their degrees leads to their social ties being different. Generally speaking, individuals usually paymore attentionto interact with those having the similar degree rank, but on the other hand, paired individuals with great distance in thedegree may bring more benefits for each other. Specifically, in line with most already studied PGGs, we assume that it iscompulsory for individuals to attend all activities centered on their neighbors and themselves. The investment of a donorfor each PGG in which he participates, however, is inequal, which depends on the relationship between paired individualsassociated with their degrees, so does the distribution of the payoff. We define this relationship as the weight of the linkwhich is different from the weighting version in Ref. [34]. The evolution of cooperative behavior in the PGG is addressed ina quantitative manner, with a parameter accurately controlling the influence of weighting intensity. Interestingly, by usingMonte Carlo simulations, we find that cooperation can be enhanced greatly when individuals are preferential to allocate(including to invest and to distribute) more funds towards ones either with similar degrees or with widely different degreesamong their neighbors. Nevertheless, there are moderate parameters leading to a worst circumstance for cooperators tosurvive. In what follows, we will describe in detail this minimal model, and present the main findings as well as thecorresponding explanations.

2. The model

We consider the evolutionary PGG in well-known Barabási–Albert (BA) scale-free networks [43], with each node beingoccupied by one individual. To construct such a network, we start from m0 fully connected vertices. At each time step, anew vertex is added withm links being preferentially attached to different vertices in the existing network. We repeat thisprocess until there are N individuals in the population. Here, we set m = m0 = 2. Thus the average connectivity of thenetwork can be given as 〈k〉 ' 2m = 4, and the degree distribution is P(k) ∼ 2m2k−3. Subsequently, each link is assigned aweight wij, associated with the degrees (ki and kj) of paired individuals (i and j), without changing once a special networkis established.In this minimal public goods game, each individual i participates in ki + 1 PGGs that centered on i and his ki neighbors,

instead of selectively playing part of them. Each individual invests a quantity Q for each PGG in which he engages. Differentfrom previous studies [26,27], however, the focal individual i contributes the amount, Qi,j = wijsi, for the PGG centered onneighbor j. Where si is the strategy of individual i, namely, si = 1 if playing C and si = 0 if playing D. Correspondingly, thepayoff of individual i associated with the PGG centered on neighbor j is given by

pi,j = rwij∑lwlj

kj∑l=0

Ql,j − Qi,j, (1)

where r is the multiplication factor in the PGG and, l runs over all j’s links with l = 0 standing for j himself. In Eq. (1), thelink’s weight, wij, is set as wij = (ks/kl) β , where ks(kl) is the smaller (larger) one between ki and kj. In particular, wij = wji

4710 C. Lei et al. / Physica A 389 (2010) 4708–4714

Fig. 1. Cooperator density ρc as a function of the multiplication factor r for different values of β . Each data point results from an average of over 100different realizations.

and wii = 1. Although the diversity of links of individuals is constant due to the static population structure we adopt,depending on the value of β being tunable, the heterogeneity can be either intensified or weakened. For β > 0 individualstend to invest and distribute more funds to ones with similar degree ranks. The choice β = 0 corresponds to the traditionalversion of PGG, in which individuals treat each other without discrimination. While β < 0 implies that individuals preferto allocate more funds to ones with a large degree difference. Following common practices, individuals performing moresuccessfully are more capable of replacing the bad ones. After each game round, the strategies of all individuals are updatedsynchronously according to the Fermi function. In other words, individual i will adopt the strategy sj of neighbor j chosenrandomly with a probability that depends upon the payoff difference:

Wsi→sj =1

1+ exp[(Pi − Pj)/κ], (2)

where κ characterizes the selection intensity in the strategy adoption process. Following previous studies [44–46], we setκ = 0.1 for all simulations. Pi, Pj denote the total payoff of individual i and j, respectively, which is obtained by summingover all the PGGs which one participates in.

3. Simulations and discussions

We carried out all the simulations within a population of N = 3000 individuals and have confirmed that the resultsremain qualitatively unaffected for varying population size. Initially, the two strategies (i.e., C and D) are randomlydistributed among the playerswith equal probability 0.5. The key quantity for characterizing the cooperative behavior of thepopulation is the density of the cooperation ρc , which is defined as the frequency of cooperators in the dynamic equilibriumstate. ρc is obtained by averaging over the last 1000 generations of the entire 6000 ones. In our study, in order to ensurethe validity of the quantity of compulsory participation in allocating funds with respect to each PGG, the weighting factor βbeing a series of discrete values is moderately limited in the interval [−3, 5].Fig. 1 reports the cooperator density ρc as a function of the multiplication factor r for different values of β . One can see

that for an arbitrary value of r , different values of β can affect the final cooperators’ fate dramatically. As a whole, ρc almostalways increases monotonically as r increases, consistent with the previous result in the context of the PGG [22]. Moreover,a careful inspection of the results reveals that, for β ≤ 0, there is a quick increase in ρc once a suitable r is reached forcooperators to survive, and decreasing β can better promote cooperation. Contrary to the above situation, however, largerβ (> 0) can result in the best promotion of cooperation. And ρc increases mildly across the range of parameter r .To qualify the effects of β on the evolution of cooperation more precisely, we report ρc as a function of β for different

values of r in Fig. 2. Clearly, there exists a valley of cooperation induced by moderate values of β for all the studied r .Most interestingly, the larger the value of r , the narrower the valley of cooperation, i.e., the number of unfavorable β forcooperators to survive tapers off. Beyond the valley, the cooperation level can be enhanced greatly, and even the scenario ofall cooperation emerges. These phenomena indicate that both too large and too small values of β can favor the emergenceof cooperation. In other words, both of the two extreme allocations of funds, occurring mainly between paired individualswhose phenotypic ranks of degrees are either very similar or far different, can provide better environments for the evolutionand flourishing of cooperation. Different from other related studies in which too much diversity will be a handicap forcooperators [28,47], however, the reverse is true in our model of heterogeneity and altruistic cooperators would benefitfrom the very diverse allocations of funds in the context of the PGG.Now, let us first give some intuitive explanations for the phenomenon of enhanced cooperation for β > 0. Without loss

of generality, we assume that there are no paired individuals who have the same degree. In this case, consider the extreme

C. Lei et al. / Physica A 389 (2010) 4708–4714 4711

Fig. 2. Cooperator density ρc as a function of β for different values of r .

a b

Fig. 3. Average payoff (a) and cooperator density ρc (b) as a function of degree for different values of β (>0) in the steady state, all of which are assignedwith r = 2. In particular, in panel (b), the cooperation levels of the whole population are 0.84, 0.66 and 0.33 from the upper curve to the lower one,respectively. Each data point results from an average of over 100 different realizations.

situation of β →∞, each individual just invests funds to himself and does the samewhen distributing the payoff due to theweights of all the links being near zero, which can be viewed as the appearance of self-sufficiency. As a result, cooperatorshave the same payoffs being equal to r − 1 while defectors no cost no payoffs. According to the replicator dynamics,individuals would take the strategy of those ones who have better performance, i.e., defectors imitate the cooperativestrategy with a high probability, which quickly results in a steady state of the evolution where all the individuals holduniformly the C strategy. Similarly, back to the actual network topology, for a given value of β(� ∞), each individualalso allocates more funds to himself and the neighbors who have the close degree rank. This also leads to the free-riderdefectors receiving less help (payoffs) than the altruistic cooperators. The advantage of defectors exploiting the cooperatorsis annealed by increasing β . Therefore, for larger β (>0), cooperation can evolve and prevail finally while defectors evencan be completely wiped out from the population.To gain further insights into the effect of β on the cooperative behavior, the inspection of some microscopic phenomena

is indispensable for the above investigations. We scrutinize the average payoffs of individuals and the cooperator densityρc as a function of degree k, respectively, for different values of β (>0) in the steady state. The corresponding results arereported in Fig. 3.We can see that individuals of different degrees have roughly equal average payoffs in Fig. 3(a). The reasonis that individuals mainly acquire the payoffs from the PGG centered on themselves, which becomes more apparent forincreasing β . Defectors failing to parasitize on the altruistic cooperators, learn the strategy of best performing cooperators.This results in a positive feedback effect for cooperators on the evolution of cooperation with respect to increasing β . As aresult, the average payoffs of all the individuals are also pulled higher via the enhanced cooperation for fixed r . Interestingly,in Fig. 3(b), the curve of ρc shows a similar scenario compared to that of the average payoff. In fact, individuals being locatedon the scale-free networks can be viewed as some unsociable loners under such heterogeneous allocation of funds becausethey treat weakly their relationships with negligible weights. In this sense, individuals have no difference in degree rank. Asa result, the cooperation level is roughly equivalent for individuals of different degrees. Cooperators can resist the invasion of

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a b

Fig. 4. (a) Total link weights of individualsWi as a function of degree for different values of β , in which the solid curves are the theoretical results fromEq. (3), and the dotted curves are the numerical ones. (b) Change of each link weight with decreasing β , in a subgraph of population with three cooperativeindividuals whose degrees are 2, 10 and 100, respectively.

a b

Fig. 5. Average payoff (a) and cooperator density ρc (b) as a function of degree for different values of β (<0) in the steady state, all of which are assignedwith r = 2. In panel (b), the cooperation levels of the whole population are 0.89, 0.67 and 0.32 from the upper curve to the lower one, respectively.

defectors by way of self-protection. Therefore, for increasing β (>0), cooperation can be enhanced and the parasitic abilityof defectors gets weaker and weaker until vanishing.In what follows, we will focus on why decreasing β (<0) can also favor the emergence of cooperation. Now that the

heterogeneous allocation induced by the link weight plays a decisive role in the evolution of cooperation, we examine thetotal weights of all the links (Wi) of individuals as a function of degree for β < 0, as shown in Fig. 4(a). Meanwhile,Wi canalso be approximatively evaluated theoretically. Once a network is constructed,Wi of individual iwhose degree is ki can beexpressed as:

Wi ' kiE(wij)+ 1 = ki

∫ ki

kmin

(kjki

)βP(kj)dkj + ki

∫ kmax

ki

(kikj

)βP(kj)dkj + 1, (3)

where E(wij) is the expectation ofwij, and P(kj) is the degree distribution of nodes. It shows that the theoretical analysis canprecisely predict the tendency of the distribution of the total link weights of nodes, although there exists slight deviation forlowest-degree nodes. In fact, the total link weights are the amount of the accumulated investments for all the PGGs inwhichthe focal individual acting as a cooperator participates. On the other hand, in a sense, this distribution also manifests one’sability to obtain the payoffs. As shown in Fig. 4(b), consider a subgraph including three cooperators with different degrees.Due to the change of each link weight from panel (1) to panel (4), for decreasing β , it can be calculated that the payoffs ofthe node of middle degrees 10 relatively decrease compared to that of the other two ones.To validate the above point, we also numerically probe the microscopic behavior, i.e., the average payoff and the

cooperator density of different degrees, for β < 0. The corresponding results are reported in Fig. 5. Different from the resultsfor β > 0, both the average payoff and the cooperator density are nontrivially dependent on the social ties of individuals.For large β (=− 0.6), the payoff increases monotonously as the degree increases in Fig. 5(a). For small β (=− 1), however,

C. Lei et al. / Physica A 389 (2010) 4708–4714 4713

Fig. 6. Cooperator density ρc as a function of β for different values of r , in which the payoffs of individuals are normalized by their degrees. Each datapoint results from an average of over 100 different realizations.

individuals withmiddle degrees have the lowest payoffs. In fact, the low ebb formiddle-degree nodes getting payoffs wouldbecome more sunken for further decreasing β (e.g., the case in the panel (4) of Fig. 4(b)). In Fig. 5(b), it can be observedthat the minimum ρc appears on the middle-degree nodes, which agrees with other related studies in Refs. [28,48]. In otherwords, lowest-degree nodes are not the easiest ones to be invaded by defectors, and cooperative hubs are still strong enoughto dominate their neighbors being mostly lowest-degree nodes. Moreover, with increasing β , the nodes of more degreehierarchies have been invaded by defectors. This is because middle-degree nodes can still get higher payoffs matching withtheir degrees for larger β (= − 0.6), which leads to these nodes well parasitizing under the shroud of cooperators, thusacting as defectors with a high probability. While this parasitic advantage in acquiring the payoffs would be annealed bydecreasing β , and these defectors with middle degrees would have to learn the strategy of peripheral cooperators owninghigher payoffs. As a result, the difference of ρc between different degrees is quickly lessened and the cooperation level ofthe whole population is greatly promoted.In combination with all the analysis, now let us make further discussion on the effect of varying β on the evolution

of cooperation. For β > 0, increasing β leads to individuals allocating less and less funds for ones with a certain degreedifference, which makes defectors gain negligible payoffs. Thus defection would be infeasible for the choice of individuals’behavior and the flourishing cooperation emerges. For β < 0, decreasing β makes the payoffs of middle-degree defectorstaper off relatively. These defectors would turn to cooperators through the replicator dynamics. As a result, cooperationflourisheswith decreasing β . In this case, high-level cooperation emerges via feeding each other between the largest-degreenodes and the lowest-degree ones instead of between clustered hubs. However, for moderate β , a significant proportion ofpayoffs from the PGGs are nibbled away by defectors contributing nothing to their co-players, leading to the scenario ofcooperation crisis.Finally, let us take into account some possible extensions of our original model in which heterogeneous allocation

favors the emergence of cooperation. In Refs. [49–51], normalizing individuals’ payoffs greatly diminishes the positive roleof network heterogeneity in promoting cooperation. In order to further probe the effect of heterogeneous allocation oncooperation, here, we investigate the case that the payoffs of individuals are normalized by their degrees. Accordingly, thestrategy updating rule is as follows:

Wsi→sj =1

1+ exp[(Fi − Fj)/κ], (4)

where Fi (Fj) denotes the fitness of individual i (j), which is equal to Pi/ki (Pj/kj). The corresponding results are reported inFig. 6. One can see that there still exists a valley of cooperation in the moderate values of β , which to a degree indicates theuniversality of cooperation crisis. Besides, we have also investigated the effects of other typical modes of strategy updating,such as Santos rule [6] and Best-take-over [52], on the evolution of cooperation. the results indicate that these too muchheterogeneous allocation can promote cooperation best and there also exists a moderate range of parameter unfavoring theemergence of cooperation. In a word, our proposedmechanism of funds allocation is robust against various decisionmakingof individual for the evolution of cooperation in the heterogeneous scale-free networks.

4. Conclusions

We have presented a model based upon the heterogeneous allocation of funds in the context of the evolutionary PGG,focusing on the effect of different allocatingmodes on the evolution of cooperation, in which participation is compulsory forall the related PGGs. Heterogeneity of allocation was induced by the difference of phenotypic degrees of paired individuals,characterized by the link weight. We found that individuals either strengthening or weakening the allocation of funds

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for ones with different degrees, both can favor the emergence of cooperation. However, there exists moderate modes ofheterogeneous allocation resulting in the worst cooperation. When the weight factor β gets above zero, the free-ridingdefectors would gain less and less payoffs with increasing β and individuals of different degrees demonstrate almost thesame microscopic characteristic. While β gets below zero, with decreasing β , the payoffs of middle-degree individualsrelatively decrease compared to the payoffs of the others. Thus these middle-degree ones imitate the cooperative strategyof neighbors with better performance which leads to the enhanced cooperation. Our approach of heterogeneous allocationmay be helpful to better understand the intricate dynamics for the evolution of cooperation in realistic systems.

Acknowledgements

This work was partially supported by National Natural Science Foundation of China (NSFC) under grant No. 10476019,No. 60674050, No. 60736022, and No. 60528007, National 973 Program under Grant No. 2002CB312200, National 863Program under Grant No. 2006AA04Z258 and 11-5 project under Grant No. A2120061303.

References

[1] A.M. Colman, Game Theory and its Applications in the Social and Biological Sciences, Butterworth-Heinemann, Oxford, 1995.[2] L.A. Dugatkin, Cooperation among Animals: An Evolutionary Perspective, Oxford University Press, Oxford, 1997.[3] J.M. Smith, Evolution and the Theory of Games, Cambridge University Press, Cambridge, 1982.[4] J. Hofbauer, K. Sigmund, Evolutionary Games and Population Dynamics, Cambridge University Press, Cambridge, 1998.[5] H. Gintis, Game Theory Evolving, Princeton University Press, Princeton, NJ, 2000.[6] F.C. Santos, J.M. Pacheco, Phys. Rev. Lett. 95 (2005) 098104.[7] M.G. Zimmermann, V.M. Eguíluz, Phys. Rev. E 72 (2005) 056118.[8] J. Vukov, G. Szabó, Phys. Rev. E 71 (2005) 036133.[9] J. Vukov, G. Szabó, A. Szolnoki, Phys. Rev. E 73 (2006) 067103.[10] A. Traulsen, M.A. Nowak, J.M. Pacheco, Phys. Rev. E 74 (2006) 011909.[11] Z.-X. Wu, X.-J. Xu, Z.-G. Huang, S.-J. Wang, Y.-H. Wang, Phys. Rev. E 74 (2006) 021107.[12] X.-J. Chen, L. Wang, Phys. Rev. E 77 (2008) 017103.[13] R. Axelrod, W.D. Hamilton, Science 211 (1981) 1390.[14] W.-X. Wang, J. Ren, G. Chen, B.-H. Wang, Phys. Rev. E 74 (2006) 056113.[15] K.H. Lee, C.H. Chan, P.M. Hui, D.-F. Zheng, Physica A 387 (2008) 5602.[16] G. Hardin, Science 162 (1968) 1243.[17] E. Fehr, S. Gächter, Nature 415 (2002) 137.[18] K.G. Binmore, Game Theory and the Social Contract, MIT Press, Cambridge, 1994.[19] J.H. Kagel, A.E. Roth, The Handbook of Experimental Economics, Princeton University Press, Princeton, 1997.[20] P. Kollock, Annu. Rev. Sociol. 24 (1998) 183.[21] C. Hauert, S. De Monte, J. Hofbauer, K. Sigmund, J. Theoret. Biol. 218 (2002) 187.[22] C. Hauert, G. Szabó, Complexity 8 (2003) 31.[23] M. Doebeli, C. Hauert, T. Killingback, Science 306 (2004) 859.[24] C. Hauert, S. De Monte, J. Hofbauer, K. Sigmund, Science 296 (2002) 1129.[25] G. Szabó, C. Hauert, Phys. Rev. Lett. 89 (2002) 118101.[26] T. Wu, F. Fu, L. Wang, Phys. Rev. E 80 (2009) 026121.[27] F.C. Santos, M.D. Santos, J.M. Pacheco, Nature 454 (2008) 213.[28] H.-X. Yang, W.-X. Wang, Z.-X. Wu, Y.-C. Lai, B.-H. Wang, Phys. Rev. E 79 (2009) 056107.[29] A. Szolnoki, G. Szabó, Europhys. Lett. 77 (2007) 30004.[30] J.-Y. Guan, Z.-X. Wu, Y.-H. Wang, Phys. Rev. E 76 (2007) 056101.[31] H.-F. Zhang, H.-X. Yang, W.-B. Du, B.-H. Wang, X.-B Cao, Physica A 389 (2010) 1099.[32] X.-B Cao, W.-B. Du, Z.-H. Rong, Physica A 389 (2010) 1273.[33] M. Perc, A. Szolnoki, Phys. Rev. E 77 (2008) 011904.[34] W.-B. Du, H.-R. Zhang, M.-B. Hu, Physica A 387 (2008) 3796.[35] R.L. Riolo, M.D. Cohen, R. Axelrod, Nature (London) 414 (2001) 441.[36] V.A.A. Jansen, M. van Baalen, Nature (London) 440 (2006) 663.[37] A. Traulsen, M.A. Nowak, PLoS ONE 2 (2007) e270.[38] T. Antal, H. Ohtsuki, J. Wakeley, P.D. Taylor, M.A. Nowak, Proc. Natl. Acad. Sci. USA 106 (2009) 8597.[39] X.-J. Chen, L. Wang, Phys. Rev. E 80 (2009) 046109.[40] X.-J. Chen, F. Fu, L. Wang, Phys. Rev. E 80 (2009) 051104.[41] F. Fu, C. Hauert, M.A. Nowak, L. Wang, Phys. Rev. E 78 (2008) 026117.[42] M.A. Nowak, K. Sigmund, Nature (London) 437 (2005) 1291.[43] A.L. Barabási, R. Albert, Science 286 (1999) 509.[44] G. Szabó, J. Vukov, Phys. Rev. E 69 (2004) 036107.[45] G. Szabó, J. Vukov, A. Szolnoki, Phys. Rev. E 72 (2005) 047107.[46] M. Perc, Phys. Rev. E 75 (2007) 022101.[47] L. Gammaitoni, P. Hänggi, P. Jung, F. Marchesoni, Rev. Modern Phys. 70 (1998) 223.[48] J. Gómez-Gardeñes, J. Poncela, L.M. Floría, Y. Moreno, J. Theoret. Biol. 253 (2008) 296.[49] M. Tomassini, E. Pestelacci, L. Luthi, Internat J. Modern Phys. C 18 (2007) 1173.[50] Z.-X. Wu, J.-Y. Guan, X.-J. Xu, Y.-H. Wang, Physica A 379 (2007) 672.[51] A. Szolnoki, M. Perc, Z. Danku, Physica A 387 (2008) 2075.[52] G. Abramson, M. Kuperman, Phys. Rev. E 63 (2001) 030901(R).