Evolutionary games on complex network

Wen-Xu Wang1,2 and Bing-Hong Wang1,3∗1Department of Modern Physics, University of Science and Technology of China, Hefei 230026, China

2Department of Electronic Engineering, City University of Hong Kong, Hong Kong SAR, China3Shanghai Academy of System Science, Shanghai, 200093 China

(Dated: February 22, 2007)

Cooperation is ubiquitous in the real world, ranging from biological systems to economic and social systems.Evolutionary game theory has been considered an important approach to characterizing and understanding theemergence of cooperative behavior in systems consisting of selfish individuals. In this paper, we review someof our works about dynamics of evolutionary games over complex networks, including the effects of networkstructures on the emergence and persistence of cooperation, resonance type phenomena in evolutionary games,and the memory-based evolution of game dynamics.

PACS numbers: 87.23.Kg, 02.50.Le, 87.23.Ge, 89.65.-s, 89.75.Fb

I. INTRODUCTION

Game theory provides a useful framework for describingthe evolution of systems consisting of selfish individuals [1–3]. The prisoner’s dilemma game (PDG) as a metaphor forinvestigating the evolution of cooperation has drawn consider-able attention [4, 5]. In the PDG, two players simultaneouslychoose whether to cooperate or defect. Mutual cooperationresults in payoffR for both players, whereas mutual defectionleads to payoffP gained both. If one cooperates while theother defects, the defector gains the highest payoffT , whilethe cooperator bears a costS. This thus gives a simply rankof four payoff values:T > R > P > S. One can see thatin the PDG, it is best to defect regardless of the co-player’sdecision to gain the highest payoffT . However, besides thewidely observed selfish behavior, many natural species andhuman being show the altruism that individuals bear cost tobenefit others. These observation brings difficulties in evalu-ating the fitness payoffs for different behavioral patterns, evenchallenge the rank of payoffs in the PDG. Since it is not suit-able to consider the PDG as the sole model to discuss cooper-ative behavior, the snowdrift game (SG) has been proposed aspossible alternative to the PDG, as pointed out in Ref [6]. Themain difference between the PDG and the SG is in the order ofP andS, asT > R > S > P in the SG. This game, equiva-lent to the hawk-dove game, is also of much biological interest[7, 8]. However, the original PDG and SG cannot satisfyinglyreproduce the widely observed cooperative behavior in natureand society. This thus motivates numerous extensions of theoriginal model to better mimic the evolution of cooperation inthe real world [9–12].

Since the spatial structure is introduced into the evolution-ary games by Nowak and May [13], there has been a contin-uous effort on exploring effects of spatial structures on thecooperation [6, 14, 15]. It has been found that the spatialstructure promotes evolution of cooperation in the PDG [13],while in contrast often inhibits cooperative behavior in the SG[6]. In recent years, extensive studies indicate that many real

∗Electronic address:bhwang@ustc.edu.cn

networks are far different from regular lattices, instead, showsmall-world and scale-free topological properties. Hence, itis naturally to consider evolutionary games on networks withthese kinds of properties [16–21]. An interesting result foundby Santos and Pacheco is that “Scale-free networks provide aunifying framework for the emergence of cooperation” [22].

In this paper, we review some of our works in the field ofevolutionary games. By means of some simple models, wehave studied how an important topological structural feature,the average degree, affect the cooperative behavior [23]. Wefound there exists the highest cooperation level induced byan optimal value of average degree for different types of net-works. Besides, we investigate the randomness effect on thecooperative behavior by introducing both topological and dy-namical randomness [24]. We found a resonance type phe-nomena reflected by the existence of highest level of coopera-tion in the case of appropriate randomness. Moreover, we pro-pose a memory-based snowdrift game over complex networks[25]. Some very interesting behavior are observed, such asthe nonmonotonous behavior of frequency of cooperation as afunction of payoff parameter, spatial pattern transition and soon.

II. THE PRISONER’S DILEMMA GAME ANDSNOWDRIFT GAME

In The individuals can follow only two simple strategies:C(cooperate) orD (defect), described by

s =(

10

)or

(01

)(1)

respectively. Each individual plays the PDG with its “neigh-bors” defined by their spatial relationships. The total incomeof the player at the sitex can be expressed as

Mx =∑

y∈Ωx

sTx · P · sy (2)

wheresx andsy denote the strategy of nodex andy. The sumruns over all the neighboring sites ofx (this set is indicated byΩx).

2

For the PDG and SG, the payoff matrix has rescaled formsrespectively, as follows

PPDG =(

1 0b 0

)PSG =

(1 1− r

1 + r 0

)(3)

where1 < b < 2 and0 < r < 1.

III. EFFECTS OF AVERAGE DEGREE ONCOOPERATION

In this section, we focus on the influence of the average de-gree of complex networks on the evolution of the networkedPDG. Firstly, we construct networks using three typical mod-els - the BA, NW and ER Models. The details of these net-work models can be seen in Refs. [26], [27] and [28], respec-tively.

Each site of the network is occupied with an individual. Anindividual can be either a cooperator or a defector. All pairsof connected individuals play the game simultaneously andgain benefits according to the payoff parameters mentioned inthe last section. During the evolutionary process, each playeris allowed to learn from one of its neighbors and update itsstrategy at each round. The probability of a nodei selectingone of its neighborsj is Πi→j = kj/

∑l kl, where the sum

runs over the set of neighbor nodes ofi. The assumption ofΠtakes into account the fact that individuals with more interac-tions usually cause more attraction in society. In other words,well-known persons will have more influences than the oth-ers. Whereafter, the nodei will adopt the selected neighbor’sstrategy with a probability determined by the normalized pay-off difference between them [29], i.e.,

W =1

1 + exp[(Ei/ki − Ej/kj)β], (4)

whereki andkj respectively represent the degrees of nodeiand j. Ei andEj respectively represent the total payoff ofnodei andj, andβ characterizes the noise introduced to per-mit irrational choices. Here,β is set to50. According to theevolutionism,W reflects the rule of natural selection based onrelative fitness.

One of the key quantities for characterizing the cooperativebehavior is the density of cooperatorsρc, which is defined asthe fraction of cooperators in the whole population. We studyρc as a function of the average degree〈k〉 for three types ofnetworks. As shown in Fig. 1, One can find thatρc exhibitsa nonmonotonous behavior with a peak at some specific val-ues of〈k〉 for the BA network. Simulation results for the NWand ER networks can be seen in Ref. [23]. The commonnontrivial behavior shared by the scale-free, small-world net-works indicate that the average degree is a crucial feature forthe networked evolutionary game. Such nonmonotonous be-havior has been well explain in Ref. by means of consideringthe situation in two limits of〈k〉.

We further concern the relationship between the average de-gree〈k〉 and the average payoff〈M〉 in the whole population

FIG. 1: (color online). Cooperator densityρc vs average degree〈k〉on the BA networks for different values of the parameterb . Simu-lation were carried out for network sizeN = 5000. b ranges from1.05 to 1.45 with a 0.05 interval. The upward curve corresponds tosmall value ofb.

FIG. 2: (color online). The average payoff〈M〉 vs the average de-gree〈k〉 for different values of parameterb in the cases of the BAnetworks. Symbols are the simulation results and curves are the cor-respondent theoretical predictions. The network sizeN is 5000.

for different types of networks (BA, NW, ER Networks). Theaverage payoff is defined as

〈M〉 =1N

N∑

i=1

Ei, (5)

whereEi is the total income of individuali. Also we foundsimilar non-monotonous phenomena exhibited in the depen-dence ofρc on 〈k〉. Simulation results of〈M〉 versus〈k〉 fordifferent values ofb with adopting the BA network are re-ported in Fig. 3. In the following, we provide theoretical pre-dictions for the average payoff of individuals〈M〉 by assum-ing cooperators are distributed uniformly among the network.〈M〉 can be expressed as

〈M〉 = 〈k〉 × ρc × ((1− ρc)× b + ρc). (6)

Since the density of cooperatorsρc cannot be reproducedby the mean-field approach, except for the well-mixed

3

FIG. 3: (Color online). (a) The frequencies of cooperatorsρc vs thetemptation to defectb for p = 0, 0.2, 1, respectively, withT = 0.08.(b) ρc as a function of the topological randomnessp with variousvalues of the temptation to defectb for T = 0.08. The lines are usedto guide eyes.

cases(fully connected networks),ρc used in Eq. (6) for cal-culating〈M〉 is obtained by simulations, as shown in Fig. 1.As shown in Fig. 2. In the case of BA networks, analyticalresults are in very good agreement with numerical ones.

we have clarified the effect of average connectivity on thecooperate behavior. Interestingly, the average connectivityplays a non-trivial role in the cooperation, i.e, there exists theoptimum average connectivity resulting in the highest cooper-ation level.

IV. RESONANCE TYPE PHENOMENON INEVOLUTIONARY GAMES

It is well known that intrinsically noisy and disordered pro-cesses can generate surprising phenomena, such as stochas-tic resonance [30], coherence resonance [31], ordering spa-tiotemporal chaos by disorder [32], disorder-enhanced syn-chronization [33], ordering chaos by randomness [34]etc. Inthis section, we investigate the effect of both the topologicalrandomness in individual relationships and the dynamical ran-domness in decision makings on the dynamics of evolutionarygames. Our work is inspired Perc, who recently studied PDGby introducing the random disorder [35].

In order to explore the topological randomness, we considera homogeneous small-world network (HSWN) [36]. Startingfrom a undirected regular graph with fixed connectivityz andsizeN , two-step circular procedure is introduced: (i) choosetwo different edges randomly, which have not been used yetin step (ii) and (ii) swap the ends of the two edges. Here,duplicate connections and disconnected graphs are avoided.The annealed randomness is characterized by the parameterp,which denotes the fraction of swapped edges in the network.(Details can be seen in Ref. [24]). In contrast to the Watts-Strogatz (WS) model [37], this network has small-world ef-fect together with keeping the degree of each individual un-

FIG. 4: (Color online). (a) The frequencies of cooperatorsρc vs theparameter space(b, p) for T = 0.08. This figure illustrates two re-gions forb: I is the resonant region where there is an optimal amountof topological randomnessp enhancingρc; II is the harmful regionwhere the topological randomnessp decreases the level of the coop-erationρc. In fact, there exists a region III beyond the shown rangeof b in this figure, where the cooperators vanish and there is nop canpersist or enhance the cooperation. We call it the absorbing region.(b) The phase diagram of the three regions ofρc in the parameterspace(b, T ). It illustrates that the resonant region decreases as thedynamical randomnessT increases.

changed, so that the pure topological randomness can be in-vestigated by avoiding any associated heterogeneity of degreedistribution [36]. At each time step, each play selects a neigh-bor at random, and updates its strategy according to Eq. (4).

Figure 3(a) shows the frequencies of cooperatorsρc on theHSWN as a function ofb for different values of the topolog-ical randomnessP with T = 0.08. In the equilibrium state,ρc is independent of the initial state and decreases monoton-ically asb increases. One can find that whenb < 1.04, co-operators dominate defectors significantly on the regular ringgraph (p = 0) and the more randomness of the network, theworse the cooperation is. While forb > 1.04, the coopera-tor is nearly extinct in the case ofp = 0 andp = 1, whichcorrespond to the complete regular network and the completerandom network, respectively. However, the cooperator cansurvive aroundp = 0.2, i.e., intermediate topological ran-domness. he dependence ofρc on the topological randomnessp is presented in Fig. 2 (b). It illustrates that there is a clearmaximumρc aroundp = 0.2, where cooperation can be revi-talized and maintained for substantially large values ofb. Thisphenomenon reveals that there exists somewhat resonant be-haviors reflected by the optimal cooperation level at interme-diate topological randomness, similar to the effects of noiseand disorder in nonlinear systems.

To quantify the ability of topological randomnessp to fa-cilitate and maintain cooperation for variousb more precisely,we studyρc depending onb and p together, as shown inFig.4(a). One can find that whenb < 1.04, ρc is a monoton-ically decreasing function ofp. We call it the harmful region(denoted byII in Fig.4(a)) because the topological random-nessp always decreasesρc. While for b > 1.04 (the regionis denoted byI in Fig.4(a)), there exists an optimal level of

4

FIG. 5: (Color online) The frequencies of cooperatorsρc vs the pa-rameter space(T, p) for fixed b = 1.08.

p around0.2, resulting in the maximum value ofρc, whichresembles a resonance-like manner in the evolutionary game.Thus, we callI the resonant region. In fact, there exists aregion III beyond the shown range ofb in Fig.4(a), wherethe cooperators vanish and there is nop can persist or en-hance the cooperation. We call it the absorbing region. Be-sides the topological randomnessp, we have studied the ef-fect of the dynamical randomnessT . Figure 3(b) illustratesthe phase diagram of the three regions ofρc in the parameterspace(b, T ). It shows clearly that as the dynamical random-nessT increases, the resonant region reduces,i.e. the areaof the range ofb where the optimalp can promote the co-operation decreases, indicating the constructive effect of theoptimal topological randomness is restricted by the higher dy-namical randomness.

To investigate the combined effect of both the topologicalrandomness and the dynamical randomness on the evolution-ary dynamics, we fixb = 1.08, and calculateρc in dependenceon variousp andT , as shown in Fig.5. It is found that thereexists a clear “optimal island” in the parameter space(T, p)whereρc reaches the highest value, indicating that the coop-eration can be promoted by both the appropriate topologicaland the dynamical randomness. In other words, the resonanceinduced by the dynamical randomness can be enhanced by thetopological randomness, just as the noise-induced temporaland spatiotemporal order can be greatly enhanced by an ap-propriately pronounced small-world connectivity of coupledelements [35].

we have studied the effects of both the topological ran-domness and the dynamical randomness on the PDG andfound that there exists an optimal amount of randomness,leading to the highest level of cooperation. The mechanismof randomness promoting cooperation resembles an interest-ing resonance-like phenomenon, wherein the randomness-induced prevalence of the cooperation evokes the positive roleof the topological and dynamical randomness in the system.

V. MEMORY-BASED SNOWDRIFT GAME

Among the previous work, the effects of individuals’ mem-ory have not received much attention in the study of evolu-tionary games on networks. We argue that individuals usu-ally make decisions based on the knowledge of past recordsin nature and society, and the historical memory would playa key role in an evolutionary game. Therefore, in the presentwork, we propose a memory-based snowdrift game (MBSG),in which players update their strategies based on their past ex-perience. Our work is partially inspired by Challet and Zhang[38], who presented a so-called minority game, in whichagents make decisions exclusively according to the commoninformation stored in their memories.

The rules of the MBSG are described as follows. ConsiderthatN players are placed on the nodes of a certain network.In every round, all pairs of connected players play the gamesimultaneously. The total payoff of each player is the sumover all its encounters. After a round is over, each player willhave the strategy information (C or D) of its neighbors. Sub-sequently, each player knows its best strategy in that roundby means of self-questioning, i.e., each player adopts its anti-strategy to play a virtual game with all its neighbors, and cal-culates the virtual total payoff. Comparing the virtual payoffwith the actual payoff, each player can get its optimal strategycorresponding to the highest payoff and then record it into itsmemory. Taking into account the bounded rationality of play-ers, we assume that players are quite limited in their analyzingpower and can only retain the lastM bits of the past strategyinformation. At the start of the next generation, the probabil-ity of making a decision (choosing C or D) for each playerdepends on the ratio of the numbers of C and D stored in itsmemory, i.e.,PC = NC

NC+ND= NC

M andPD = 1−PC , whereNC andND are the numbers of C and D, respectively. Then,all players update their memories simultaneously. Repeat theabove process and the system evolves.

A. MBSG on lattices

First, we investigate the MBSG on two-dimensional squarelattices of four and eight neighbors with periodic boundaryconditions, as shown in Fig. 5 (a) and (b). In these two fig-ures, four common features should be noted: (i)fC has a stepstructure, and the number of steps corresponds to the num-ber of neighbors on the lattice, i.e.,4 steps for the4-neighborlattice and8 steps for the8-neighbor lattice; (ii) the two fig-ures have180-rotational symmetry about the point (0.5, 0.5);(iii) the memory lengthM has no influences on the dividingpoint rc between any two cooperation levels, but has strongeffects on the value offC in each level; (iv) for a large pay-off parameterr, the system still behaves in a high cooperationlevel, contrary to the results reported in [6]. It indicates thatalthough selfish individuals make decisions based on the bestchoices stored in their memories to maximize their own bene-fits, the cooperative behavior can emerge in the population inspite of the highest payoff of D.

The effects of memory lengthM on fC in the4-neighbor

5

FIG. 6: (Color online). The frequency of cooperationfC as a func-tion of the payoff parameterr for two-dimensional (a)4-neighborand (b)8-neighbor lattices, respectively. (4), (©) and (¤) are forM = 2, 7 and30, respectively. Each data point is obtained by av-eraging over40 different initial states andfC for each simulation isobtained by averaging from MC time stept = 5000 to t = 10000,where the system has reached a steady state. The top inset of (a) isfC as a function of memory lengthM for 2 different cooperationlevels. The bottom inset of (a) is a time series offC for r = 0.4 inthe case ofM = 1. Since forM = 1, fC as a function oft displaysa big oscillation, we do not compute thefC over a period of MC timesteps. The inset of (b) isfC depending onM for 4 cooperation levelsin the range of0 < r < 0.5. The network size isN = 10000.

lattice are shown in the insets of Fig. 6. One can find thatfC is a monotonous function ofM for both levels and the de-creasing velocity offC in the 1st level is faster than that inthe2nd one. In contrast, in the8-neighbor lattice,fC exhibitssome non-monotonous behaviors asM increases. As shownin the bottom inset of Fig. 6(b),fC peaks in the1st level cor-responding toM = 23, andfC is an increasing function ofMin the2nd level. A maximum value offC exists in the3rd and4th levels whenM is chosen to be5, as shown in the top insetof Fig. 1(b). Thus, memory lengthM plays a very complexrole in fC reflected by the remarkably different behaviors in4 cooperation levels. A typical example withM = 1 for twotypes of lattices is shown in the bottom inset of Fig. 1 (a). Abig oscillation offC is observed. The dividing points of dif-ferent levels can be obtained via local stability analysis. Thedetails can be seen in Ref. [25].

Next, we reports the spital pattern for each cooperationlevel for two types of lattices. As shown in Figs. 7 and 8, thereare different patterns for different cooperation levels. The pat-tern formation can be explained in terms of steady local pat-terns. In Fig. 7(c), we show the steady local patterns existingin the1st cooperation level. From the payoff ratio by choosingC and D of individual A, i.e.,WC : WD, the3rd local patternis the most stable one with the highest payoff ratio. In paral-lel, the4th local pattern is the counterpart of the3nd one, sothat it is also very stable. Hence, the pattern in Fig. 7(a) has achessboard-like background together with C lines composedof the 1st and2nd local patterns. Similarly, the chessboard-

A

D C W W :

(c) (d) 0 : 3

5 . 2 : 5 . 3

75 . 3 : 75 . 3 5 : 4

5 . 1 : 5 . 2 3 : 3

5 . 4 : 5 . 3 0 : 2 6 : 2

(a) (b)

FIG. 7: Typical spatial patterns in two distinct payoff parameterranges: (a)0 < r < 0.25, (b) 0.25 < r < 0.5. The C is in blackand the D is in white. A50× 50 portion of the full100× 100 latticewith 4 neighbors is illustrated. (c) and (d) are the relevant stable lo-cal patterns of (a) and (b).WC andWD are the payoffs of the centerindividual A by choosing C and D with fixing strategies of neighborsfor each local pattern.r = 0.25 in (c) and0.5 in (d).

(a) (b)

FIG. 8: (Color online). Typical spatial patterns in two distinct payoffparameter ranges: (a)0.25 < r < 0.375; (b) 0.375 < r < 0.5. Thecolor coding is the same as Fig. 2. A50 × 50 portion of the full100× 100 lattice with8 neighbors is illustrated.

like background in Fig. 7(b) is also attributed to the strongeststability of the4th and5th local patterns, and the probabil-ity of the occurrence of other local patterns is correlated withtheir payoff ratios.

B. MBSG on scale-free networks

Going beyond two-dimensional lattices, we also investigatethe MBSG on scale-free (SF) networks, since such structuralproperty is ubiquitous in natural and social systems. Figure 5shows the simulation results on the Barabasi-Albert networks[26], which are constructed by the preferential attachmentmechanism. Each data point is obtained by averaging over30different network realizations with20 different initial states ofeach realization. Figures 9 (a1) and (a2) displayfC dependingon r on BA networks in the cases of average degree〈k〉 = 4and〈k〉 = 8 for different memory lengthsM . There are somecommon features in these two figures: (i) in sharp, contrast

6

FIG. 9: (Color online).fC as a function ofr in BA networks with(a1) average degree〈k〉 = 4 and (a2)〈k〉 = 8 for differentM . Atime series offC for M = 1 is shown in the inset of (a2). (b1) and(b2) arefC as a function ofM in the case of〈k〉 = 4 and〈k〉 = 8for a special range ofr. (c1) and (c2) are average degrees〈ks〉 of Cand D players depending onr in the case ofM = 7 for 〈k〉 = 4 and〈k〉 = 8, respectively. The network size is10000. Each data pointis obtained by averaging over30 different network realizations with20 different initial state of each realization.fC for each simulationis obtained by averaging from MC time stept = 5000 to t = 10000,where the system has reached a steady state.

to the cases on lattices,fC is a non-monotonous function ofr with a peak at a specific value ofr. This interesting phe-nomenon indicates that properly encouraging selfish behav-iors can optimally enhance the cooperation on SF networks;(ii) it is the same as the cases on lattices that the continuity offC is broken by some sudden decreases. The number of con-tinuous sections corresponds to the average degree〈k〉; (iii)two figures have a180-rotational symmetry about the point(0.5, 0.5); (iv) the memory lengthM does not influence thevalues ofr, at which sudden decreases occur, as well as thetrend offC , but affects the values offC in each continuoussection. Then, we investigate the effect ofM on fC in detail.Due to the inverse symmetry offC about point(0.5, 0.5), ourstudy focus on the range of0 < r < 0.5. We found that inboth SF networks, there exists a unique continuous section, inwhichM plays different roles infC . For the case of〈k〉 = 4,the special range is fromr = 0.34 to 0.49, as shown in Fig.9(a1). In this regionfC as a function ofM is displayed inFig. 9(b1). One can find that forr = 0.42, fC is independentof M . For 0.34 < r < 0.42, fC is a decreasing function ofM ; while for 0.42 < r < 0.49, fC becomes an increasingfunction of M . Similar phenomena are observed in the SF

FIG. 10: (Color online). Distributions of strategies in BA networks.Cooperators and defectors are denoted by gray bars and black bars,respectively. Each bar adds up to a total fraction of1 per degree, thegray and black fractions being directly proportional to the relativepercentage of the respective strategy for each degree of connectivityk. (a) is for the case of〈k〉 = 4 with r = 0.1 and (b) is for thecase of〈k〉 = 4 with r = 0.49, at whichfC peaks. (c) shows thecase of〈k〉 = 8 with r = 0.05 and (d) displays the case of〈k〉 = 8with r = 0.16, which corresponds to the maximum value offC . Allthe simulations are obtained for network sizeN = 1000 in order tomake figures clearly visible.

network with〈k〉 = 8, as exhibited in Fig. 5 (b2).r = 0.45is the dividing point, and forr < 0.45 and r > 0.45, fC

shows decreasing and increasing behaviors respectively asMincreases. In the case ofM = 1, the system has big oscil-lations as shown in the inset of Fig. 5 (a2). Similar to thecases on lattices, the behavior of large proportion of individu-als’ strategy switches that induces the big oscillation offC inthe SF network.

In order to give an explanation for the non-monotonous be-haviors reported in Figs. 9(a1) and (a2), we study the averagedegree〈ks〉 of cooperators and defectors depending onr. InFigs. 9(c1) and (c2),〈ks〉 of D vs r shows almost the sametrend as that offC in Figs. 9(a1) and (a2), also the samesudden decreasing points at specific values ofr. Whenr isaugmented from0, large-degree nodes are gradually occupiedby D, reflected by the enhancement of D’s〈ks〉. The detaileddescription of the occupation of nodes with given degree canbe seen in Fig. 10. One can clearly find that on the4-neighborlattice, in the case of low value offC (Fig. 10(a)), almostall high degree nodes are occupied by cooperators and mostlow degree nodes are occupied by defectors; while at the peakvalue of fC (Fig. 10(b)), cooperators on most high degreenodes are replaced by defectors and on low degree nodes co-operators take the majority. Similarly, asfC increases in the8-neighbor lattices, defectors gradually occupy those high de-gree nodes, together with most very low degree nodes takenby cooperators (Fig. 10(c) and (d)). Moreover, note that inSF networks, large-degree nodes take the minority and most

7

neighbors of small-degree nodes are those large-degree ones,so that when more and more large-degree nodes are taken byD, more and more small-degree nodes have to choose C togain payoff1 − r from each D neighbor. Thus, it is the pas-sive decision making of small-degree nodes which take themajority in the whole populations that leads to the increase offC . However, for very larger, poor benefit of C results in thereduction offC . Therefore,fC peaks at a specific value ofron SF networks. In addition, it is worthwhile to note that inthe case of highfC , the occupation of large degree nodes inthe MBSG on SF networks is different from recently reportedresults in Ref. [39]. The authors found that all (few) highdegree nodes are occupied by cooperators, whereas defectorsonly manage to survive on nodes of moderate degree. Whilein our work, defectors take over almost all high degree nodes,which induces a high level of cooperation.

we have studied the memory-based snowdrift game on net-works, including lattices and scale-free networks. Transitionsof spatial patterns are observed on lattices, together with thestep structure of the frequency of cooperation versus the pay-off parameter. The memory length of individuals plays dif-ferent roles at each cooperation level. In particular, non-monotonous behavior are found on SF networks, which canbe explained by the study of the occupation of nodes withgive degree. Interestingly, in contrast to previously reportedresults, in the memory-based snowdrift game, the fact of highdegree nodes taken over by defectors leads to a high cooper-ation level on SF networks. Furthermore, similar to the caseson lattices, the average degrees of SF networks is still a signif-icant structural property for determining cooperative behav-ior. The memory effect on cooperative behavior investigatedin our work may draw some attention from scientific commu-nities in the study of evolutionary games.

VI. CONCLUSION

We have reviewed three of our work in the field of evo-lutionary games on complex networks. We have studied

the network effect, in particular the average degree, on thecooperative behavior. It is observed that the average degreeplays a universal role on the prisoner’s dilemma game overscale-free, small-world and random networks, that is thedensity of cooperator peaks at some specific values of theaverage degree. We as well have studied both the topologicaland dynamical randomness effects on the cooperation level.We found a resonance type phenomenon in the prisoner’sdilemma game, which is reflected by the existence of theoptimal amount of randomness, leading to the highest levelof cooperation. At last, we have investigate a new kind ofevolutionary games, i.e., memory-based snowdrift game, withrespect to the influence of historical experience of individualson the decision making. Interestingly, we found in the frame-work of this new game with adopting scale-free networks,properly encouraging selfish behaviors can optimally enhancecooperative behavior. We as well found the phase transitionof spatial patterns over regular lattices. Our work may shedsome new light on the study of evolutionary games overcomplex networks.

VII. ACKNOWLEDGEMENT

This work was supported by the National Basic Re-search Program of China (973 Project No.2006CB705500),by the National Natural Science Foundation of China (GrantNos.10635040, 10532060, and 10472116), by the PresidentFunding of Chinese Academy of Science, and by the Spe-cialized Research Fund for the Doctoral Program of HigherEducation of China.

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