复杂网络上的博弈演化
摘要
近十年来,复杂网络结构和网络上各种动力学模型的研究成为非线性物理和统计物理的新热点。因为博弈模型贴近生活,各种博弈模型在复杂网络上演化问题已经是复杂网络研究中重要的一个研究内容。和传统的物理系统相比,博弈的演化问题展示出了更丰富物理现象。
本文首先对复杂网络的研究现状和经典的博弈理论进行了介绍。其中先介绍了度分布、聚集系数等可以描述网络的结构的参数;然后对规则网络、随机网络、小世界网络和无标度网络这几种主要的复杂网络形式进行了介绍。在这一章中还对纳什均衡理论和一些经典的博弈模型进行了介绍。
在第二章中,我们介绍了如何在复杂网络上研究博弈的演化问题。我们把各种文献中的这一类模型分成了四个独立的模块,并对这四个模块进行了介绍。这样可以方便读者以后阅读相关的文献。由于复杂网络所展示出的网络结构不同于一般的现实空间,所以统计物理中一些常用的分析方法如平均场方法在这个领域中并不能很好的适用,这样就需要一些特殊的技巧。在这一章中还介绍了一种广义平均场方法,这种方法可以比较好的分析规则网络中的博弈演化模型。
接下来的一章中我们介绍了一种带有自愿行为的囚徒困境博弈在两个网络中分别进行演化的模型。这个模型研究了博弈在多个网络之间相互作用的形式。通过设定一个影响因子,可以发现两个网络上的博弈演化会出现丰富的同步行为。
在第四章中研究了记忆效应在博弈模型中的作用。模拟结果表明,记忆效应的存在可以促进网络中的合作态密度,它是社会生活中合作态能够普遍存在的原因之一。
第五章介绍了一种复杂网络和博弈相互作用的模型。这种模型体现了社会人际关系网络形成的基本原因。在这一章的模型中定义当博弈参与者模仿了其邻居策略的时候就要同时改变其可调整长程连接。参与者以合作态密度为概率选择随机调整长程连接,否则偏好连接到网络中收益大的参与者上。在这个模型中还定义了一个α来控制偏好连接的强度。模拟结果表明,α可以促进合作态密度的增大。当α大于一定阈值的时候,在部分诱惑参数下的合作态密度会被显著的增强。同时可以发现,网络结构也发生了质的变化。网络中会出现度值非常大的中心点,这个中心点总是选择合作态的策略,并且和中心点的网络邻居也都选择合作态的策略。这也正是合作态密度显著增加的原因。
在本文的最后一章中,我们对这个领域的研究进行了总结和展望。
In the past ten years,complex network and the dynamics on it becomepopular topics of statistic physics and nonlinear physics.Evolutionarygames,an important dynamics in complex networks,have also show morecomplicated behaviors.
At the beginning of this thesis,we review some network structureparameters like degree distribution,clustering coefficient,and some typicalnetwork structures like regular network,random network,small-worldnetwork,and scale-free network.Then we present some game models andexplain the concept of Nash equilibrium.
In the second section,we introduce how to study the evolutionarygame on graph.We depart this topic to four independent modules and introducethem separately.These introductions are help for readers to readother papers.The traditional physical system can be reproduced qualitativelywell by mean-field approximation.This appropriate description of anevolutionary game requires more sophisticated technique.We introduce ageneral mean-field approximation that can analyze the evolutionary gameson regular network.
Then we study the synchronization between the prisoner's dilemmagames with voluntary participation in two Newman-Watts small-world networks.It is found that there are three kinds of synchronizations:partialphase synchronization,total phase synchronization,and complete synchronization,for varied coupling factors.Besides,two games can reach completesynchronization for the large enough coupling factor.We also discussthe effect of the coupling factor on the amplitude of oscillation of cooperatordensity.
In the forth section,we study the effect of memory on the evolutionofthe prisoner's dilemma game in square lattice networks.Based on extensivesimulations,it shows that the density of cooperators is enhancedby an increasing memory effect for most parameters.However,we also observethat the density of cooperators decrease with an increased memoryeffect in the case of a large memory and moderate temptation.It is interestingto note that memory makes cooperators immune from temptation.The strength of protection reaches its maximal value only for a moderatememory effect.
In the fifth section,we build a coevolution model of prisoner's dilemmagame and network structure to study the dynamic interaction in the realworld.Different from other game and network structure coevolution models,players rewire their network connection according to other players'payoffs.We use a parameterαto control the effect of payoff in the processof rewiring.Based on the asynchronous update rule and Monte Carlosimulation,it is found that,when players prefer to rewire their links to thericher,the cooperation density will increase.The reason of it is analyzed.
At last,we make a conclusion and outlook of this thesis.
本文首先对复杂网络的研究现状和经典的博弈理论进行了介绍。其中先介绍了度分布、聚集系数等可以描述网络的结构的参数;然后对规则网络、随机网络、小世界网络和无标度网络这几种主要的复杂网络形式进行了介绍。在这一章中还对纳什均衡理论和一些经典的博弈模型进行了介绍。
在第二章中,我们介绍了如何在复杂网络上研究博弈的演化问题。我们把各种文献中的这一类模型分成了四个独立的模块,并对这四个模块进行了介绍。这样可以方便读者以后阅读相关的文献。由于复杂网络所展示出的网络结构不同于一般的现实空间,所以统计物理中一些常用的分析方法如平均场方法在这个领域中并不能很好的适用,这样就需要一些特殊的技巧。在这一章中还介绍了一种广义平均场方法,这种方法可以比较好的分析规则网络中的博弈演化模型。
接下来的一章中我们介绍了一种带有自愿行为的囚徒困境博弈在两个网络中分别进行演化的模型。这个模型研究了博弈在多个网络之间相互作用的形式。通过设定一个影响因子,可以发现两个网络上的博弈演化会出现丰富的同步行为。
在第四章中研究了记忆效应在博弈模型中的作用。模拟结果表明,记忆效应的存在可以促进网络中的合作态密度,它是社会生活中合作态能够普遍存在的原因之一。
第五章介绍了一种复杂网络和博弈相互作用的模型。这种模型体现了社会人际关系网络形成的基本原因。在这一章的模型中定义当博弈参与者模仿了其邻居策略的时候就要同时改变其可调整长程连接。参与者以合作态密度为概率选择随机调整长程连接,否则偏好连接到网络中收益大的参与者上。在这个模型中还定义了一个α来控制偏好连接的强度。模拟结果表明,α可以促进合作态密度的增大。当α大于一定阈值的时候,在部分诱惑参数下的合作态密度会被显著的增强。同时可以发现,网络结构也发生了质的变化。网络中会出现度值非常大的中心点,这个中心点总是选择合作态的策略,并且和中心点的网络邻居也都选择合作态的策略。这也正是合作态密度显著增加的原因。
在本文的最后一章中,我们对这个领域的研究进行了总结和展望。
In the past ten years,complex network and the dynamics on it becomepopular topics of statistic physics and nonlinear physics.Evolutionarygames,an important dynamics in complex networks,have also show morecomplicated behaviors.
At the beginning of this thesis,we review some network structureparameters like degree distribution,clustering coefficient,and some typicalnetwork structures like regular network,random network,small-worldnetwork,and scale-free network.Then we present some game models andexplain the concept of Nash equilibrium.
In the second section,we introduce how to study the evolutionarygame on graph.We depart this topic to four independent modules and introducethem separately.These introductions are help for readers to readother papers.The traditional physical system can be reproduced qualitativelywell by mean-field approximation.This appropriate description of anevolutionary game requires more sophisticated technique.We introduce ageneral mean-field approximation that can analyze the evolutionary gameson regular network.
Then we study the synchronization between the prisoner's dilemmagames with voluntary participation in two Newman-Watts small-world networks.It is found that there are three kinds of synchronizations:partialphase synchronization,total phase synchronization,and complete synchronization,for varied coupling factors.Besides,two games can reach completesynchronization for the large enough coupling factor.We also discussthe effect of the coupling factor on the amplitude of oscillation of cooperatordensity.
In the forth section,we study the effect of memory on the evolutionofthe prisoner's dilemma game in square lattice networks.Based on extensivesimulations,it shows that the density of cooperators is enhancedby an increasing memory effect for most parameters.However,we also observethat the density of cooperators decrease with an increased memoryeffect in the case of a large memory and moderate temptation.It is interestingto note that memory makes cooperators immune from temptation.The strength of protection reaches its maximal value only for a moderatememory effect.
In the fifth section,we build a coevolution model of prisoner's dilemmagame and network structure to study the dynamic interaction in the realworld.Different from other game and network structure coevolution models,players rewire their network connection according to other players'payoffs.We use a parameterαto control the effect of payoff in the processof rewiring.Based on the asynchronous update rule and Monte Carlosimulation,it is found that,when players prefer to rewire their links to thericher,the cooperation density will increase.The reason of it is analyzed.
At last,we make a conclusion and outlook of this thesis.
引文
[1]汪小帆，李翔，陈关荣，复杂网络理论及其应用，清华大学出版社，(2006)．
[2]吴金闪，狄增如，从统计物理学看复杂网络研究，物理学进展24， 18(2004)．
[3]A．R(?)ka， and B．Albert-L(?)szl(?)， Statistical mechanics of complex networks， Rev．Mod．Phys 74， 47(2002)．
[4]E．P．Wigner， Characteristic vectors of bordered matrices of infinite dimensions， Ann．Math．62， 548(1955)．
[5]S．Brin and L．Page， Proceedings of the 7th International World Wide Web Conference (Elsevier， Amsterdam， 1998)．
[6]M．Girvan， M．E．J．Newman， Community structure in social and biological networks， Proc．Natl．Acad．Sci．USA 99， 7821(2002)．
[7]I．K．Goh， E．Oh， H．Jeong， B．Kahng， and D．Kim， Classification of scale-free networks，Proc．Natl．Acad．Sci．USA 99， 12583(2002)．
[8]P．Erd(o|¨)s， and A．R(?)nyi， On random graphs， Publ．Math．6， 290(1959)．
[9]P．Erd(o|¨)s， and A．R(?)nyi， On the evolution of random graphs， Publ．Math．Inst．Hung．Acad5，17(1960)．
[10]P．Erd(o|¨)s， and A．R(?)nyi， On the evolution of random graphs， Bull．Inst．Int．Stat 38，343(1961)．
[11]D．J．Watts， and S．H．Strogatz， Collective dynamics of small-world Networks， Nature 393，440(1998)．
[12]M．E．J．Newman， C．Moore， and D．J．Watts，Mean-Field Solution of the Small-World Network Model， Phys．Rev．Lett， 84， 3201(2000)．
[13]G．Abramson and M．Kuperman， Social game in a social network， Phys．Rev．E 63，030901(R)(2001)．
[14]B．J．Kim， A．T usina， P．Holme， and P．Minnhagen， J．S．Chung， and M．Y．Choi， Dynamic instablilities induced by asymmetric influence： Prisoners' dilemma game in small-world networks， Phys．Rev．E 66， 021907(2001)．
[15]A．-L．Barabasi， and R．Albert， Emergence of Scaling in random networks， Science 286，509(1999)．
[16]B．Bollobas， and O．Riordan， The diameter of a scale-free random graph， University of Memphis (2002) ．
[17]P．L．Krapivsky， S．Redner， and F．Leyvraz， Connectivity of Growing Random Networks，Phys．Rev．Lett．85， 4629(2000)．
[18]施锡铨，博弈论，上海财经大学出版社，(2005)．
[19] J. von Nermann, and O. Morgenstern, Theory of Game and Economic Behaviour (Princeton University Press, Princeton).
[20] J. Nash, Proc. Nat. Acad, Sci. USA 36, 48(1950).
[21] R. Axelrod, The Evolution of Cooperation, New York (1984).
[22] M. A. Nowak, and R. M. May, Evolutionary games and spatial chaos, Nature 359,826(1992).
[23] G. Szabo, and G. Fath, Evolutionary games on graphs 446, 97 (2007).
[24] G. Szabo, and C. Toke, Evolutionary prisoner's dilemma game on a square lattice, 58,69(1998).
[25] H. X. Yang, B. H. Wang, W. X. Wang, and Z. H. Rong, Spatial games based on pursuing the highest average payoff, Chin. Phys. Lett. 28, 3504(2008).
[26] C. Hauter, S. De Monte, J. Hofbauer, and K. Sigmund, Volunteering as red queen mechanism for cooperation in public goods games, Science 296, 1129(2002).
[27] C. Hauert, and G. Szabo, Phase transitions and volunteering in spatial public goods games,Phys. Rev. Lett. 89, 118101(2002).
[28] C. Hauert, and G. Szabo, Game theory and physics, Am. J. Phys. 73, 405(2004).
[29] M. Tomassini, L. Luthi, and M. Giacobini, Hawks and Doves on small-world networks,Phys. Rev. E 73, 016132(2006).
[30] W. Li, X. Zhang, and G. Hu, How scale-free networks and large-scale collective cooperation emerge in complex homogeneous social systems, Phys. Rev. E 76, 045102(R) (2007).
[31] Z. X. Wu, X. J. Xu, Z. G. Huang, S. J. Wang, and Y. H. Wang Evolutionary prisoner's dilemma game with dynamic preferential selection, Phys. Rev. E 74, 021107(2006).
[32] G. Szabo, J. Vukov, and A. Szolnoki, Phase diagrams for an evolutionary prisoner's dilemma game on two-dimensional lattices, Phys. Rev. E 72, 047107(2005).
[33] Zhi-Xi Wu, Xin-Jian Xu, Yong Chen, and Ying-Hai Wang, Spatial prisoner's dilemma game with volunteering in Newman-Watts small-world networks, Phys. Rev. E 71, 037103(2005).
[34] F. C. Santos and J. M. Pacheco, Scale-free networks provide a unifying framework for the emergence of cooperation, Phys. Rev. Lett. 95, 098104 (2005);
[35] B. A. Huberman, and N. S. Glance, Evolutionary games and computer simulations, Proc.Natl. Acad. Sci. USA 90, 7716(1993).
[36] G. Szabo and G. Odor, Extended mean-field study of a stochastic cellular automaton, Phys.Rev. E 49, 2764(1994).
[37] G. Szabo and A. Szolnoki, Generalized mean-field study of a driven lattice gas, Phys. Rev.E 53, 2196(1996).
[38] R. Donangelo, M. H. Jensen, I. Simonsen, and K. Sneppen, Synchronization model for stock market asymmetry, J. Stat. Mech.: Theory Exp. L11001(2006).
[39]D．G．Myers， Social Psychology (McGraw-Hill， New York， 1993)．
[40]Y．Chen， S．-M．Qin， L．Yu， and S．Zhang， Emergence of synchronization induced by the interplay between two prisoner' s dilemma games with volunteering in small-world networks，Phys．Rev．E 77， 032103(2008)．
[41]J．A．Freund， L．Schimancky-Geier， and P．H(a|¨)nngi， Frequency and phase synchronization in stochastic systems， Chaos 13， 225(2003)．
[42]Y．Kuramoto， and I．Nishikava， Statistical macrodynamics of large dynamical systems．Case of a phase transition in oscillator communities， J．Stat．Phys．49， 569(1987)．
[43]W．-X．Wang， J．Ren， G．R．Chen， and B．-H．Wang， Memory-based snowdrift game on networks， Phys．Rev．E b f 74， 056113(2006)．
[44]S．-M．Qin， Y．Chen， X．-Y．Zhao， and J．Shi， Effect of memory on the prisoner's dilemma game in a square lattice， Phys．Rev．E 78， 041129(2008)．
[45]http：//www．youtube．com/watch?v=oVhxQNyqFkY http：//www．youtube．com/watch?v=n f vuNTx f 888 http：/www．youtube．com/watch?v=6FSIdge0W-U http：//www．youtube．corn/watch?v=2oHU5oT21_M http：//www．youtube．corn/watch?v=ul4MVS4FrRs http：//www．youtube．com/watch?v=edsW jmt3oFA．
[46]J．Vukov， G．Szab(?)， and A．Szolnoki， Evolutionary prisoner's dilemma game on Newman-Watts networks， Phys．Rev．E 77， 026109(2008)．
[47]T．M．Liggett， Interacting Particle Systems (Springer， New York， 1985)．
[48]J．L．Cardy and U．C．T(a|¨)uber， Theory of branching and annihilating random walks， Phys．Rev．Lett．77， 4780 (1996)．
[49]P．Grassberger， On phase transitions in Schl(o|¨)gl's second model， Z．Phys．B 47， 365(1982)；[50]T．Reichenbach， M．Mobilia， and E．Frey， Mobility promotes and jeopardizes biodiversity in rock-paper-sissors games， Nature (London) 448， 1046 (2007)．
[51]F．C．Santos， J．M．Pacheco， and T．Lenaerts， Evolutionary dynamics of social dilemmas in structured heterogeneous populations， Proc．Natl．Acad．Sci．U．S．A．103， 3490 (2006)．
[52]M．G．Zimmermann， V．M．Egu(?)luz， and M．San Miguel， Coevolution of dynamical states and interactions in dynamic networks， Phys．Rev．E 69， 065102(R) (2004)．
[53]S．-M．Qin， G．-Y．Zhang， and Y．Chen， Coevolution of game and network structure： The temptation increases the cooperator density， e-print arXiv： 0901．2759．
[54]A．Szolnokil and M．Perc， Coevolution of teaching activity promotes cooperation， New J．Phys．10， 043036 (2008)．
[55]J．M．Pacheco， A．Traulsen， and M．A．Nowak， Coevolution of Strategy and Structure in Complex Networks with Dynamical Linking， Phys．Rev．Lett．97， 258103 (2006)．
[56] M. G. Zimmermann and V. M. Egu(?)luz, Cooperation, social networks, and the emergence of leadership in a prisoner's dilemma with adaptive local interactions, Phys. Rev. E 72, 056118 (2005).
[2]吴金闪，狄增如，从统计物理学看复杂网络研究，物理学进展24， 18(2004)．
[3]A．R(?)ka， and B．Albert-L(?)szl(?)， Statistical mechanics of complex networks， Rev．Mod．Phys 74， 47(2002)．
[4]E．P．Wigner， Characteristic vectors of bordered matrices of infinite dimensions， Ann．Math．62， 548(1955)．
[5]S．Brin and L．Page， Proceedings of the 7th International World Wide Web Conference (Elsevier， Amsterdam， 1998)．
[6]M．Girvan， M．E．J．Newman， Community structure in social and biological networks， Proc．Natl．Acad．Sci．USA 99， 7821(2002)．
[7]I．K．Goh， E．Oh， H．Jeong， B．Kahng， and D．Kim， Classification of scale-free networks，Proc．Natl．Acad．Sci．USA 99， 12583(2002)．
[8]P．Erd(o|¨)s， and A．R(?)nyi， On random graphs， Publ．Math．6， 290(1959)．
[9]P．Erd(o|¨)s， and A．R(?)nyi， On the evolution of random graphs， Publ．Math．Inst．Hung．Acad5，17(1960)．
[10]P．Erd(o|¨)s， and A．R(?)nyi， On the evolution of random graphs， Bull．Inst．Int．Stat 38，343(1961)．
[11]D．J．Watts， and S．H．Strogatz， Collective dynamics of small-world Networks， Nature 393，440(1998)．
[12]M．E．J．Newman， C．Moore， and D．J．Watts，Mean-Field Solution of the Small-World Network Model， Phys．Rev．Lett， 84， 3201(2000)．
[13]G．Abramson and M．Kuperman， Social game in a social network， Phys．Rev．E 63，030901(R)(2001)．
[14]B．J．Kim， A．T usina， P．Holme， and P．Minnhagen， J．S．Chung， and M．Y．Choi， Dynamic instablilities induced by asymmetric influence： Prisoners' dilemma game in small-world networks， Phys．Rev．E 66， 021907(2001)．
[15]A．-L．Barabasi， and R．Albert， Emergence of Scaling in random networks， Science 286，509(1999)．
[16]B．Bollobas， and O．Riordan， The diameter of a scale-free random graph， University of Memphis (2002) ．
[17]P．L．Krapivsky， S．Redner， and F．Leyvraz， Connectivity of Growing Random Networks，Phys．Rev．Lett．85， 4629(2000)．
[18]施锡铨，博弈论，上海财经大学出版社，(2005)．
[19] J. von Nermann, and O. Morgenstern, Theory of Game and Economic Behaviour (Princeton University Press, Princeton).
[20] J. Nash, Proc. Nat. Acad, Sci. USA 36, 48(1950).
[21] R. Axelrod, The Evolution of Cooperation, New York (1984).
[22] M. A. Nowak, and R. M. May, Evolutionary games and spatial chaos, Nature 359,826(1992).
[23] G. Szabo, and G. Fath, Evolutionary games on graphs 446, 97 (2007).
[24] G. Szabo, and C. Toke, Evolutionary prisoner's dilemma game on a square lattice, 58,69(1998).
[25] H. X. Yang, B. H. Wang, W. X. Wang, and Z. H. Rong, Spatial games based on pursuing the highest average payoff, Chin. Phys. Lett. 28, 3504(2008).
[26] C. Hauter, S. De Monte, J. Hofbauer, and K. Sigmund, Volunteering as red queen mechanism for cooperation in public goods games, Science 296, 1129(2002).
[27] C. Hauert, and G. Szabo, Phase transitions and volunteering in spatial public goods games,Phys. Rev. Lett. 89, 118101(2002).
[28] C. Hauert, and G. Szabo, Game theory and physics, Am. J. Phys. 73, 405(2004).
[29] M. Tomassini, L. Luthi, and M. Giacobini, Hawks and Doves on small-world networks,Phys. Rev. E 73, 016132(2006).
[30] W. Li, X. Zhang, and G. Hu, How scale-free networks and large-scale collective cooperation emerge in complex homogeneous social systems, Phys. Rev. E 76, 045102(R) (2007).
[31] Z. X. Wu, X. J. Xu, Z. G. Huang, S. J. Wang, and Y. H. Wang Evolutionary prisoner's dilemma game with dynamic preferential selection, Phys. Rev. E 74, 021107(2006).
[32] G. Szabo, J. Vukov, and A. Szolnoki, Phase diagrams for an evolutionary prisoner's dilemma game on two-dimensional lattices, Phys. Rev. E 72, 047107(2005).
[33] Zhi-Xi Wu, Xin-Jian Xu, Yong Chen, and Ying-Hai Wang, Spatial prisoner's dilemma game with volunteering in Newman-Watts small-world networks, Phys. Rev. E 71, 037103(2005).
[34] F. C. Santos and J. M. Pacheco, Scale-free networks provide a unifying framework for the emergence of cooperation, Phys. Rev. Lett. 95, 098104 (2005);
[35] B. A. Huberman, and N. S. Glance, Evolutionary games and computer simulations, Proc.Natl. Acad. Sci. USA 90, 7716(1993).
[36] G. Szabo and G. Odor, Extended mean-field study of a stochastic cellular automaton, Phys.Rev. E 49, 2764(1994).
[37] G. Szabo and A. Szolnoki, Generalized mean-field study of a driven lattice gas, Phys. Rev.E 53, 2196(1996).
[38] R. Donangelo, M. H. Jensen, I. Simonsen, and K. Sneppen, Synchronization model for stock market asymmetry, J. Stat. Mech.: Theory Exp. L11001(2006).
[39]D．G．Myers， Social Psychology (McGraw-Hill， New York， 1993)．
[40]Y．Chen， S．-M．Qin， L．Yu， and S．Zhang， Emergence of synchronization induced by the interplay between two prisoner' s dilemma games with volunteering in small-world networks，Phys．Rev．E 77， 032103(2008)．
[41]J．A．Freund， L．Schimancky-Geier， and P．H(a|¨)nngi， Frequency and phase synchronization in stochastic systems， Chaos 13， 225(2003)．
[42]Y．Kuramoto， and I．Nishikava， Statistical macrodynamics of large dynamical systems．Case of a phase transition in oscillator communities， J．Stat．Phys．49， 569(1987)．
[43]W．-X．Wang， J．Ren， G．R．Chen， and B．-H．Wang， Memory-based snowdrift game on networks， Phys．Rev．E b f 74， 056113(2006)．
[44]S．-M．Qin， Y．Chen， X．-Y．Zhao， and J．Shi， Effect of memory on the prisoner's dilemma game in a square lattice， Phys．Rev．E 78， 041129(2008)．
[45]http：//www．youtube．com/watch?v=oVhxQNyqFkY http：//www．youtube．com/watch?v=n f vuNTx f 888 http：/www．youtube．com/watch?v=6FSIdge0W-U http：//www．youtube．corn/watch?v=2oHU5oT21_M http：//www．youtube．corn/watch?v=ul4MVS4FrRs http：//www．youtube．com/watch?v=edsW jmt3oFA．
[46]J．Vukov， G．Szab(?)， and A．Szolnoki， Evolutionary prisoner's dilemma game on Newman-Watts networks， Phys．Rev．E 77， 026109(2008)．
[47]T．M．Liggett， Interacting Particle Systems (Springer， New York， 1985)．
[48]J．L．Cardy and U．C．T(a|¨)uber， Theory of branching and annihilating random walks， Phys．Rev．Lett．77， 4780 (1996)．
[49]P．Grassberger， On phase transitions in Schl(o|¨)gl's second model， Z．Phys．B 47， 365(1982)；[50]T．Reichenbach， M．Mobilia， and E．Frey， Mobility promotes and jeopardizes biodiversity in rock-paper-sissors games， Nature (London) 448， 1046 (2007)．
[51]F．C．Santos， J．M．Pacheco， and T．Lenaerts， Evolutionary dynamics of social dilemmas in structured heterogeneous populations， Proc．Natl．Acad．Sci．U．S．A．103， 3490 (2006)．
[52]M．G．Zimmermann， V．M．Egu(?)luz， and M．San Miguel， Coevolution of dynamical states and interactions in dynamic networks， Phys．Rev．E 69， 065102(R) (2004)．
[53]S．-M．Qin， G．-Y．Zhang， and Y．Chen， Coevolution of game and network structure： The temptation increases the cooperator density， e-print arXiv： 0901．2759．
[54]A．Szolnokil and M．Perc， Coevolution of teaching activity promotes cooperation， New J．Phys．10， 043036 (2008)．
[55]J．M．Pacheco， A．Traulsen， and M．A．Nowak， Coevolution of Strategy and Structure in Complex Networks with Dynamical Linking， Phys．Rev．Lett．97， 258103 (2006)．
[56] M. G. Zimmermann and V. M. Egu(?)luz, Cooperation, social networks, and the emergence of leadership in a prisoner's dilemma with adaptive local interactions, Phys. Rev. E 72, 056118 (2005).