复杂网络及其上的进化博弈研究
摘要
现实世界的生物系统、生态系统、社会系统、经济系统等，都是由大量具有相互作用的个体所组成的．这些复杂系统的宏观结构属性可以用复杂网络来描述．我们在阐述复杂网络基本理论及研究概况的基础上，研究了加权结构化网络，特别是以常用的进化囚徒困境博弈模型为例，考虑不同复杂网络模型上进化博弈的动力学演化，详细研究了网络的拓扑结构对合作行为演化的影响，同时探讨了其他一些支持合作现象涌现与稳定维持的动力学机制．本文的创新工作主要如下：
一、提出一种普遍的加权结构化网络模型．基于真实复杂系统中个体具有的老化现象，我们提出了一种基于节点权重钝化机制的演化网络模型．应用主方程的解析分析方法对其度分布进行了解析分析，并做了相应的数值模拟．理论分析与模拟结果符合的相当好，其都证实所得到的网络具有非常强的结构效应(即节点间具有相当强的成团趋势)．具体的，当对网络中的节点进行目标钝化时，得到具有无标度度分布的结构化网络；而当对网络中的节点进行随机钝化时，则得到具有指数度分布的结构化网络．
二、研究了Newman-Watts小世界网络上个体自愿参加的空间囚徒困境博弈．每个博弈个体可以采取三种策略：合作、欺骗和单干。个体策略的转变既与其邻居和其自身在上一轮博弈中的收益有关，也与这些个体当时所采取的策略状态有关。为了模拟复杂系统的适应性能力，我们在博弈动力学中引入了随机的策略突变规则：当博弈个体陷入到局部共同态时，其以相应的规则进行策略转变．研究发现了丰富的动力学现象：在较弱的欺骗诱惑下，系统中的个体在小世界网络拓扑结构下都愿意参与到博弈中去；而在随机网络拓扑结构下，系统的演化出现了强烈的振荡现象．
三、通过在博弈动力学中考虑个体间非对称的影响权重，我们发现了一个新的有利于合作行为产生的机制：动态(或静态)的优先选择机制．很多现实社会群体中存在非对称的异质影响效应，因此在博弈模型中我们对任意两个相互作用的博弈个体定义了他们之间的影响权重，并且这种影响权重随着博弈过程的演化而改变．博弈个体在策略更新时，其以正比于影响权重大小的概率选择一个邻居作为参考者．研究表明，策略更新结果与影响权重的协同演化，即动态(或静态)优先选择机制的存在，使得博弈个体间的影响权重具有一个非常宽广的分布形状，这有利于相互之间具有强影响力的合作者形成稳定的紧致集团结构，从而能够有效地抵御欺骗者的入侵，继而有助于合作行为的涌现与持续．
四、为了研究群体中常常具有的异质连接属性对合作涌现的影响，我们运用三种不同的策略更新规则详细地研究了Barabási-Albert无标度网络上合作演化问题．结果表明相互作用网络基底的拓扑结构、具体的策略更新动力学规则、策略更新事件的同步性或异步性、博弈个体适合度的具体评价函数形式、欺骗诱惑量的大小，都对进化囚徒困境博弈模型的演化结果有着决定性的影响．当用个体的平均收益作为其适合度函数时，在欺骗诱惑量非常小的情况下，Barabási-Albert网络的无标度拓扑属性对于合作的形成是一个明显的抑制性因素；而当欺骗诱惑量较大时，网络的无标度拓扑属性则有利于合作者在系统中存活．
五、我们研究了双层网络上的进化囚徒困境博弈．其中底层的网络为相互作用网络，即博弈个体在其上发生相互作用；顶层网络为信息获取网络，即每轮博弈过后，博弈个体通过此网络来获得其他个体在上轮中的收益与策略状态信息，并根据与这些策略学习邻居的比较结果来决定下轮中要采取的策略．通过Monte-Carlo模拟和对近似的解析分析，我们研究了两个子模型．在第一个模型中，所有的博弈个体具有相同大小的策略学习邻居；而在第二个模型中，我们赋予博弈个体异质的信息获取能力．研究发现，相互作用网络与策略学习网络之间的差异性能够实质性地促进群体合作行为的涌现．这种差异性对合作的促进方式类似于一种“相干共振”现象，即差异性太大或太小都不利于合作行为的涌现，对合作行为促进的最优效果出现在差异性为中等程度的时候．
Coupled biological and ecological systems, social interacting species, economic agents, are typical examples of systems composed by a large number of highly interconnected dynamical units. The global properties of such complex systems can be modeled by complex networks whose nodes represent the dynamical units, and whose links stand for the interactions between them. The emergence and abundance of cooperation in these systems poses a tenacious and challenging puzzle to evolution theory. In this thesis, we explore evolutionary prisoner's dilemma games on complex networks and address how the evolution of cooperation is affected by the network topology, and also search for new mechanisms supporting the emergence and persistence of cooperation.
First, motivated by aging phenomenon of individuals of real complex systems, a weight-dependent deactivation model generating networks with high clustering coefficient is proposed to model evolving networks. We determine the degree distribution of the generated networks by master-equation approach complemented by Monte-Carlo simulation. Both analytical solutions and numerical simulations show that the generated networks possess strong structural effect. Weighted, structured scale-free networks are obtained as the deactivated vertex is target selected at each time step, and weighted, structured exponential networks are realized for the random-selected case.
In the second, a modified spatial PDG with voluntary participation in Newman-Watts small-world networks is studied. Each agent in the network is a pure strategist and can only take one of three strategies: cooperate, defect and loner; its strategical transformation is associated with both the number of current strategical states and the magnitude of average profits of the involved players; a stochastic strategy mutation is applied when it gets into the trouble of local commons. In the case of very low temptation to defect, it is found that agents are willing to participate in the game in typical small-world region and intensive collective oscillations arise in more random region.
Thirdly, we incorporate a dynamic (or static) preferential selection (DPS) mechanism into an evolutionary PDG and reveal a new mechanism for maintaining cooperation. By considering asymmetric and heterogeneous influential effects in many natural populations, we define impact weights for any pairs of neighboring individuals, which describes the influence of one player on another and evolves promptly. Based on this quantity, a DPS mechanism is introduced into the dynamics: the more influential a neighbor is, the greater probability it is picked as a reference. We find that the DPS gives rise to very large broad distributions of the impact weights, which favors the influential cooperators to form stable communities, and thereby prevents the invasion from defectors, hence contributes to the emergence and persistence of cooperation.
Fourthly, in order to investigate the influence of heterogeneous interaction neighborhood on the evolution of cooperation, we study an evolutionary PDG with players located on Barabasi-Albert scale-free networks with different update rules that determine a player's future strategy. We find the overall result that cooperation is sometimes inhibited and sometimes enhanced by the scale-free topology. The differences depend on the detailed evaluation function of the players' success, the different update rules that determine a player's future strategy, the synchronous and asynchronous events of strategy-updating, and also on the magnitude of the temptation to defect.
Finally, we study an evolutionary PDG with two layered graphs, where the lower layer is the physical infrastructure on which the interactions are taking place and the upper layer represents the connections for the strategy learning mechanism. This system is investigated by means of Monte Carlo simulations as well as an extended pair-approximation method. We consider the average density of cooperators in the stationary state for fixed interaction graph, while varying the number of edges in the learning graph. According to the Monte Carlo simulations, the cooperation is modified substantially in a way resembling a coherence-resonance-like behavior when the number of learning edges is increased. Too little learning information favors defection, but apparently so does too much information. The optimal enhancement is induced by moderate difference between the interaction and learning neighborhoods.
一、提出一种普遍的加权结构化网络模型．基于真实复杂系统中个体具有的老化现象，我们提出了一种基于节点权重钝化机制的演化网络模型．应用主方程的解析分析方法对其度分布进行了解析分析，并做了相应的数值模拟．理论分析与模拟结果符合的相当好，其都证实所得到的网络具有非常强的结构效应(即节点间具有相当强的成团趋势)．具体的，当对网络中的节点进行目标钝化时，得到具有无标度度分布的结构化网络；而当对网络中的节点进行随机钝化时，则得到具有指数度分布的结构化网络．
二、研究了Newman-Watts小世界网络上个体自愿参加的空间囚徒困境博弈．每个博弈个体可以采取三种策略：合作、欺骗和单干。个体策略的转变既与其邻居和其自身在上一轮博弈中的收益有关，也与这些个体当时所采取的策略状态有关。为了模拟复杂系统的适应性能力，我们在博弈动力学中引入了随机的策略突变规则：当博弈个体陷入到局部共同态时，其以相应的规则进行策略转变．研究发现了丰富的动力学现象：在较弱的欺骗诱惑下，系统中的个体在小世界网络拓扑结构下都愿意参与到博弈中去；而在随机网络拓扑结构下，系统的演化出现了强烈的振荡现象．
三、通过在博弈动力学中考虑个体间非对称的影响权重，我们发现了一个新的有利于合作行为产生的机制：动态(或静态)的优先选择机制．很多现实社会群体中存在非对称的异质影响效应，因此在博弈模型中我们对任意两个相互作用的博弈个体定义了他们之间的影响权重，并且这种影响权重随着博弈过程的演化而改变．博弈个体在策略更新时，其以正比于影响权重大小的概率选择一个邻居作为参考者．研究表明，策略更新结果与影响权重的协同演化，即动态(或静态)优先选择机制的存在，使得博弈个体间的影响权重具有一个非常宽广的分布形状，这有利于相互之间具有强影响力的合作者形成稳定的紧致集团结构，从而能够有效地抵御欺骗者的入侵，继而有助于合作行为的涌现与持续．
四、为了研究群体中常常具有的异质连接属性对合作涌现的影响，我们运用三种不同的策略更新规则详细地研究了Barabási-Albert无标度网络上合作演化问题．结果表明相互作用网络基底的拓扑结构、具体的策略更新动力学规则、策略更新事件的同步性或异步性、博弈个体适合度的具体评价函数形式、欺骗诱惑量的大小，都对进化囚徒困境博弈模型的演化结果有着决定性的影响．当用个体的平均收益作为其适合度函数时，在欺骗诱惑量非常小的情况下，Barabási-Albert网络的无标度拓扑属性对于合作的形成是一个明显的抑制性因素；而当欺骗诱惑量较大时，网络的无标度拓扑属性则有利于合作者在系统中存活．
五、我们研究了双层网络上的进化囚徒困境博弈．其中底层的网络为相互作用网络，即博弈个体在其上发生相互作用；顶层网络为信息获取网络，即每轮博弈过后，博弈个体通过此网络来获得其他个体在上轮中的收益与策略状态信息，并根据与这些策略学习邻居的比较结果来决定下轮中要采取的策略．通过Monte-Carlo模拟和对近似的解析分析，我们研究了两个子模型．在第一个模型中，所有的博弈个体具有相同大小的策略学习邻居；而在第二个模型中，我们赋予博弈个体异质的信息获取能力．研究发现，相互作用网络与策略学习网络之间的差异性能够实质性地促进群体合作行为的涌现．这种差异性对合作的促进方式类似于一种“相干共振”现象，即差异性太大或太小都不利于合作行为的涌现，对合作行为促进的最优效果出现在差异性为中等程度的时候．
Coupled biological and ecological systems, social interacting species, economic agents, are typical examples of systems composed by a large number of highly interconnected dynamical units. The global properties of such complex systems can be modeled by complex networks whose nodes represent the dynamical units, and whose links stand for the interactions between them. The emergence and abundance of cooperation in these systems poses a tenacious and challenging puzzle to evolution theory. In this thesis, we explore evolutionary prisoner's dilemma games on complex networks and address how the evolution of cooperation is affected by the network topology, and also search for new mechanisms supporting the emergence and persistence of cooperation.
First, motivated by aging phenomenon of individuals of real complex systems, a weight-dependent deactivation model generating networks with high clustering coefficient is proposed to model evolving networks. We determine the degree distribution of the generated networks by master-equation approach complemented by Monte-Carlo simulation. Both analytical solutions and numerical simulations show that the generated networks possess strong structural effect. Weighted, structured scale-free networks are obtained as the deactivated vertex is target selected at each time step, and weighted, structured exponential networks are realized for the random-selected case.
In the second, a modified spatial PDG with voluntary participation in Newman-Watts small-world networks is studied. Each agent in the network is a pure strategist and can only take one of three strategies: cooperate, defect and loner; its strategical transformation is associated with both the number of current strategical states and the magnitude of average profits of the involved players; a stochastic strategy mutation is applied when it gets into the trouble of local commons. In the case of very low temptation to defect, it is found that agents are willing to participate in the game in typical small-world region and intensive collective oscillations arise in more random region.
Thirdly, we incorporate a dynamic (or static) preferential selection (DPS) mechanism into an evolutionary PDG and reveal a new mechanism for maintaining cooperation. By considering asymmetric and heterogeneous influential effects in many natural populations, we define impact weights for any pairs of neighboring individuals, which describes the influence of one player on another and evolves promptly. Based on this quantity, a DPS mechanism is introduced into the dynamics: the more influential a neighbor is, the greater probability it is picked as a reference. We find that the DPS gives rise to very large broad distributions of the impact weights, which favors the influential cooperators to form stable communities, and thereby prevents the invasion from defectors, hence contributes to the emergence and persistence of cooperation.
Fourthly, in order to investigate the influence of heterogeneous interaction neighborhood on the evolution of cooperation, we study an evolutionary PDG with players located on Barabasi-Albert scale-free networks with different update rules that determine a player's future strategy. We find the overall result that cooperation is sometimes inhibited and sometimes enhanced by the scale-free topology. The differences depend on the detailed evaluation function of the players' success, the different update rules that determine a player's future strategy, the synchronous and asynchronous events of strategy-updating, and also on the magnitude of the temptation to defect.
Finally, we study an evolutionary PDG with two layered graphs, where the lower layer is the physical infrastructure on which the interactions are taking place and the upper layer represents the connections for the strategy learning mechanism. This system is investigated by means of Monte Carlo simulations as well as an extended pair-approximation method. We consider the average density of cooperators in the stationary state for fixed interaction graph, while varying the number of edges in the learning graph. According to the Monte Carlo simulations, the cooperation is modified substantially in a way resembling a coherence-resonance-like behavior when the number of learning edges is increased. Too little learning information favors defection, but apparently so does too much information. The optimal enhancement is induced by moderate difference between the interaction and learning neighborhoods.
引文
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