复杂网络上的演化博弈动力学研究
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摘要
演化博弈论提供了一种描述个体之间交互作用的通用数学框架。每个博弈者在博弈中采取一定的策略,并根据其对手的策略获取收益。在每一时步,每个博弈者在某种收益最优的原则或学习规则下更新自己的策略,所有博弈者最终达到某种演化稳定的均衡点,而不是经典博弈论所讨论的确定的静止的纳什均衡。
典型的演化博弈通常放在全混合的群体结构下讨论,即群体中的所有博弈者两两之间均存在相互作用并进行博弈。复杂网络上的演化博弈研究主要考虑结构化的群体结构,博弈者之间的相互作用关系通过一个复杂网络进行刻画。由于这种结构化的连接拓扑结构,复杂网络上的演化博弈也呈现出了更为复杂的动力学行为。
本文回顾了近年来复杂网络上的演化博弈的研究进展,着重研究了最后通牒博弈在复杂网络上的公平演化、性别对战博弈在观点动力学中的应用、以及具有单一交互能力的二分图上的雪堆博弈。
本文的主要贡献以及研究成果如下:
1.最后通牒博弈作为博弈中研究理性与非理性因素的一个典型例证,刻画了博弈者公平与自私的折衷。本文研究了在复杂网络上的公平演化机制,考虑在演化最后通牒博弈中,所有的博弈者达到收益一致意义下的公平涌现。我们发现公平涌现条件与网络拓扑结构有关,并仿真研究了网络的小世界与无标度特性对演化最后通牒博弈的公平涌现临界的影响;
2.本文提出并研究一种新的基于演化网络性别对战博弈的观点动力学模型。群体中的个体根据其不同的固定观点偏好划分为不同的类,而个体实际所持观点根据生死过程或死生过程进行更新,以模拟观点形成中的个体之间的相互说服过程。群体最终会形成统一的观点,所有个体达到一致;或共存的不同观点,分别对应“观点统治”或“观点共存”两相,而相变临界点由网络的模块度所决定。我们给出了一般的网络演化博弈策略共生稳定性条件,并加以仿真验证。仿真结果表明了网络的模块性(或社团结构特征)有利于演化策略的共生,并提供了一条研究网络模块性的新思路;
3.本文提出了一种具有可调幂律指数度分布的二部图模型,研究了其上的具有单一个体交互能力的雪堆博弈。区别于以往网络上的演化博弈动力学研究通常考虑博弈者每一时步与其网络邻居同时进行博弈,博弈者按照一定策略选择邻居与之进行(多人)博弈。在雪堆博弈的背景下,我们引入了耗时代价刻画博弈者中合作联盟带给所有博弈者的收益。仿真研究表明,在考虑耗时代价时,激励了群体中合作行为的涌现,促成合作者联盟,而网络中度分布的异质性则抑制了合作频率的提高。
Evolutionary game theory provides a versatile framework and mathemat-ical description of interactions between players or agents. In a game, the re-ward, or utility, that any player receives is a function of player’s own strategyas well as the strategies of other players, according to which the player updateshis/her strategy under a certain payoff optimal principle and/or learning rulesat each time step . As time evolves, reach is an evolutionary stable equilib-rium in a dynamical game system of players, instead of the deterministic/staticsituation(Nash equilibrium) in the classical game theory literature.
An evolutionary game are generally concerning the well-mixed popula-tion, where all the players contact (and play game) with each other. However,the evolutionary game in complex networks considers a structural population,i.e., the connections between players are described by a network of contacts(NOCs), where the randomness and complexity of topological structure de-scribing the interactions between players result in diversity and complexity ofthe game system’s asymptotic dynamic behaviours.
The dissertation surveys a wide range of related works and current situa-tion of evolutionary game dynamics on complex networks, and discusses fair-ness evolution of the Ultimatum Game on complex networks, an application ofevolutionary Battle-of-the-Sexes Game in opinion dynamics, and the SnowdriftGame on bipartite graphs with identical interactivity.
The main contributions of this dissertation are summarized as follows:
1. The ultimatum game as a typical counterexample of rationality describesthe con?ict between fairness and selfishness in the game theory literature.This dissertation investigates the fairness mechanism in the evolutionaryultimatum game among a population of players located on an network ofcontacts. We study the condition to achieve the fairness emergence, whereall players reach a consensus in their payoffs. More numerical observa-tions reveal the in?uence of small-world and scale-free features of com-plex networks on the fairness emergence in the evolutionary ultimatumgame.
2. A new evolutionary Battle-of-the-Sexes Game is proposed to model theopinion formation among a structured population on networks. The popu-lation of players is partitioned into different classes according to their un-altered opinion preferences, and their factual opinions are considered asevolutionary game strategies with two different updating rules, the’birth-death’and’death-birth’rules, to imitate the process of opinion formation.The players finally reach a consensus in the dominate opinion, or fall into(quasi-) stationary fractions of coexisting mixed opinions, which presentsa phase transition at the critical modularity of the multi-class players’par-titions on networks. In this dissertation, a broad theoretic analysis on co-existence stability of mixed strategies of the evolutionary game amongmulti-class players is given, where the analytical predictions agree wellwith numerical simulations of our model, indicating that players with acommunity- (or modular) population structure are prone to form coexist-ing opinions. It also provides a clue that the coexistence of mixed evo-lutionary strategies implies the modularity of networks under the gametheoretic framework.
3. A bipartite graph model with tunable power law exponent is proposed,on which we studied the evolutionary game dynamics under the assump-tion that each player has an identical interactivity, i.e., at each time stepplayers pick his neighbours as the game opponents in a muti-player game,instead of playing with all other connected players. With the backgroundof Snowdrift Game, the coalitionary time saving is introduced to describethe rewards brought by coalition of cooperators. Simulation results showthat coalitionary time saving and heterogeneity of network topology availto the enhancement of cooperation.
典型的演化博弈通常放在全混合的群体结构下讨论,即群体中的所有博弈者两两之间均存在相互作用并进行博弈。复杂网络上的演化博弈研究主要考虑结构化的群体结构,博弈者之间的相互作用关系通过一个复杂网络进行刻画。由于这种结构化的连接拓扑结构,复杂网络上的演化博弈也呈现出了更为复杂的动力学行为。
本文回顾了近年来复杂网络上的演化博弈的研究进展,着重研究了最后通牒博弈在复杂网络上的公平演化、性别对战博弈在观点动力学中的应用、以及具有单一交互能力的二分图上的雪堆博弈。
本文的主要贡献以及研究成果如下:
1.最后通牒博弈作为博弈中研究理性与非理性因素的一个典型例证,刻画了博弈者公平与自私的折衷。本文研究了在复杂网络上的公平演化机制,考虑在演化最后通牒博弈中,所有的博弈者达到收益一致意义下的公平涌现。我们发现公平涌现条件与网络拓扑结构有关,并仿真研究了网络的小世界与无标度特性对演化最后通牒博弈的公平涌现临界的影响;
2.本文提出并研究一种新的基于演化网络性别对战博弈的观点动力学模型。群体中的个体根据其不同的固定观点偏好划分为不同的类,而个体实际所持观点根据生死过程或死生过程进行更新,以模拟观点形成中的个体之间的相互说服过程。群体最终会形成统一的观点,所有个体达到一致;或共存的不同观点,分别对应“观点统治”或“观点共存”两相,而相变临界点由网络的模块度所决定。我们给出了一般的网络演化博弈策略共生稳定性条件,并加以仿真验证。仿真结果表明了网络的模块性(或社团结构特征)有利于演化策略的共生,并提供了一条研究网络模块性的新思路;
3.本文提出了一种具有可调幂律指数度分布的二部图模型,研究了其上的具有单一个体交互能力的雪堆博弈。区别于以往网络上的演化博弈动力学研究通常考虑博弈者每一时步与其网络邻居同时进行博弈,博弈者按照一定策略选择邻居与之进行(多人)博弈。在雪堆博弈的背景下,我们引入了耗时代价刻画博弈者中合作联盟带给所有博弈者的收益。仿真研究表明,在考虑耗时代价时,激励了群体中合作行为的涌现,促成合作者联盟,而网络中度分布的异质性则抑制了合作频率的提高。
Evolutionary game theory provides a versatile framework and mathemat-ical description of interactions between players or agents. In a game, the re-ward, or utility, that any player receives is a function of player’s own strategyas well as the strategies of other players, according to which the player updateshis/her strategy under a certain payoff optimal principle and/or learning rulesat each time step . As time evolves, reach is an evolutionary stable equilib-rium in a dynamical game system of players, instead of the deterministic/staticsituation(Nash equilibrium) in the classical game theory literature.
An evolutionary game are generally concerning the well-mixed popula-tion, where all the players contact (and play game) with each other. However,the evolutionary game in complex networks considers a structural population,i.e., the connections between players are described by a network of contacts(NOCs), where the randomness and complexity of topological structure de-scribing the interactions between players result in diversity and complexity ofthe game system’s asymptotic dynamic behaviours.
The dissertation surveys a wide range of related works and current situa-tion of evolutionary game dynamics on complex networks, and discusses fair-ness evolution of the Ultimatum Game on complex networks, an application ofevolutionary Battle-of-the-Sexes Game in opinion dynamics, and the SnowdriftGame on bipartite graphs with identical interactivity.
The main contributions of this dissertation are summarized as follows:
1. The ultimatum game as a typical counterexample of rationality describesthe con?ict between fairness and selfishness in the game theory literature.This dissertation investigates the fairness mechanism in the evolutionaryultimatum game among a population of players located on an network ofcontacts. We study the condition to achieve the fairness emergence, whereall players reach a consensus in their payoffs. More numerical observa-tions reveal the in?uence of small-world and scale-free features of com-plex networks on the fairness emergence in the evolutionary ultimatumgame.
2. A new evolutionary Battle-of-the-Sexes Game is proposed to model theopinion formation among a structured population on networks. The popu-lation of players is partitioned into different classes according to their un-altered opinion preferences, and their factual opinions are considered asevolutionary game strategies with two different updating rules, the’birth-death’and’death-birth’rules, to imitate the process of opinion formation.The players finally reach a consensus in the dominate opinion, or fall into(quasi-) stationary fractions of coexisting mixed opinions, which presentsa phase transition at the critical modularity of the multi-class players’par-titions on networks. In this dissertation, a broad theoretic analysis on co-existence stability of mixed strategies of the evolutionary game amongmulti-class players is given, where the analytical predictions agree wellwith numerical simulations of our model, indicating that players with acommunity- (or modular) population structure are prone to form coexist-ing opinions. It also provides a clue that the coexistence of mixed evo-lutionary strategies implies the modularity of networks under the gametheoretic framework.
3. A bipartite graph model with tunable power law exponent is proposed,on which we studied the evolutionary game dynamics under the assump-tion that each player has an identical interactivity, i.e., at each time stepplayers pick his neighbours as the game opponents in a muti-player game,instead of playing with all other connected players. With the backgroundof Snowdrift Game, the coalitionary time saving is introduced to describethe rewards brought by coalition of cooperators. Simulation results showthat coalitionary time saving and heterogeneity of network topology availto the enhancement of cooperation.
引文
[1] 先行一方具有必胜策略。具体操作如下,先行方将第一枚硬币放于圆桌中心,以后每一步放置硬币的位置为上一步对手放置位置关于圆桌中心的对称点。很明显,当对手可放置硬币时,己方也必然可放置,故对手一定以失败告终.
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[20] Bagrow J.P. and Bollt E.M. Local method for detecting communities. Physical ReviewE, 72(4):046108, 2005.
[21] Reichardt J. and Bornholdt S. When are networks truly modular? Physica D, 224(1-2):20–26, 2006.
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[6] Newman M.E.J. The structure and function of complex networks. Society for Industrialand Applied Mathematics Review, 45(2):167–256.
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[1] Berelson B. Voting: A study of opinion formation in a presidential campaign. Univer-sity Of Chicago Press, 1954.
[2] Galam S. Real space renormalization group and totalitarian paradox of majority rulevoting. Physica A, 285(1-2):66–76, 2000.
[3] Sznajd K. and Sznajd J. Opinion evolotion in closed community. International Journalof Modern Physics C, 11(6):1157–1165, 2000.
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[6] Krapivsky P.L. and Redner S. Dynamics of majority rule in two-state interacting spinsystems. Physical Review Letters, 90(23):238701, Jun 2003.
[7] Mobilia M. and Redner S. Majority versus minority dynamics: Phase transition in aninteracting two-state spin system. Physical Review E, 68(4):046106, Oct 2003.
[8] Holme P. and Newman M.E.J. Nonequilibrium phase transition in the coevolution ofnetworks and opinions. Physical Review E, 74(5):056108, 2006.
[9] Lambiotte R. Ausloos M. and Holyst J.A. Majority model on a network with commu-nities. Physical Review E, 75(3):030101, 2007.
[10] Horn R.A. and Johnson C.R. Matrix analysis. Cambridge University Press, 1985.
[11] Hofbauer J. and Sigmund K. Evolutionary games and population dynamics. 1998.
[12] Ohtsuki H. and Nowak M.A. The replicator equation on graphs. Journal of TheoreticalBiology, 243(1):86–97, 2006.
[13] Ohtsuki H. Hauert C. Lieberman E. et al. A simple rule for the evolution of cooperationon graphs and social networks. Nature, 441(7092):502–505, 2006.
[14] Darwin C. The origin of species. Signet Classic, 2003.
[15] Ohtsuki H. Nowak M.A. and Pacheco J.M. Breaking the symmetry between interac-tion and replacement in evolutionary dynamics on graphs. Physical Review Letters,98(10):108106, 2007.
[16] Girvan M. and Newman M.E.J. Community structure in social and biological networks.Proceedings of the National Academy of Sciences, 99(12):7821, 2002.
[17] Zhou H.J. Distance, dissimilarity index, and network community structure. PhysicalReview E, 67(6):061901, Jun 2003.
[18] Fortunato S. Latora V. and Marchiori M. Method to find community structures basedon information centrality. Physical Review E, 70(5):056104, 2004.
[19] Clauset A. Finding local community structure in networks. Physical Review E,72(2):026132, 2005.
[20] Bagrow J.P. and Bollt E.M. Local method for detecting communities. Physical ReviewE, 72(4):046108, 2005.
[21] Reichardt J. and Bornholdt S. When are networks truly modular? Physica D, 224(1-2):20–26, 2006.
[22] Boccaletti S. Ivanchenko M. Latora V. et al. Detecting complex network modularity bydynamical clustering. Physical Review E, 75(4):045102, 2007.
[23] Newman M.E.J. and Girvan M. Finding and evaluating community structure in net-works. Physical Review E, 69(2):026113, 2004.
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[26] Zachary W.W. An information ?ow model for con?ict and fission in small groups.Journal of Anthropological Research, 33(4):452–473, 1977.
[27] Palla G. Derenyi I. Farkas I. et al. Uncovering the overlapping community structure ofcomplex networks in nature and society. Nature, 435(7043):814–8, 2005.
[1] Newman M.E.J. Scientific collaboration networks. I. Network construction and funda-mental results. Physical Review E, 64(1):16131, 2001.
[2] Faloutsos M. Faloutsos P. and Faloutsos C. On power-law relationships of the internettopology. Computer Communication Review, 29(4):251–262, 1999.
[3] Krapivsky P.L. and Redner S. A statistical physics perspective on web growth. Com-puter Networks, 39(3):261–276, 2002.
[4] Newman M.E.J. Forrest S. and Balthrop J. Email networks and the spread of computerviruses. Physical Review E, 66(3):035101, Sep 2002.
[5] Baraba′si A.L. and Albert R. Emergence of scaling in random networks. Science,286(5439):509, 1999.
[6] Newman M.E.J. The structure and function of complex networks. Society for Industrialand Applied Mathematics Review, 45(2):167–256.
[7] Boccaletti S. Latora V. Moreno Y. et al. Complex networks: Structure and dynamics.Physics Reports, 424(4-5):175–308, 2006.
[8] Han J.D. Bertin N. Hao T. et al. Evidence for dynamically organized modularity in theyeast protein–protein interaction network. Nature, 430(6995):88–93, 2004.
[9] Zhou T. Liu J.G. Bai W.J. et al. Behaviors of susceptible-infected epidemics on scale-free networks with identical infectivity. Physical Review E, 74(5):56109, 2006.
[10] Zheng D.F. Yin H.P. Chan C.H. et al. Cooperative behavior in a model of evolutionarysnowdrift games with N-person interactions. Europhysics Letters, 80(1):18002, 2007.
[11] Hauert C. and Szabo′ G. Game theory and physics. American Journal of Physics,73:405, 2005.
[1] Erds P. and Re′nyi A. On the evolution of random graphs. Publications of the Mathemat-ical Institute of the Hungarian Academy of Science, 5:17–61, 1960.
[2] Watts D.J. and Strogatz S.H. Collective dynamics of’small-world’networks. Nature,393(6684):409–10, 1998.
[3] Baraba′si A.L. and Albert R. Emergence of scaling in random networks. Science,286(5439):509, 1999.