基于复杂网络的演化博弈及一致性动力学研究
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摘要
复杂网络广泛存在于自然界和人类社会之中,对复杂网络的研究已成为对复杂系统及复杂性科学的研究中最具有挑战的前沿性课题之一。将任何复杂系统都可抽象成相互作用的个体组成的网络,不同领域的科学家积极合作试图从理论上和实证上深入探索复杂网络的奥秘,揭示人类社会甚至是生物系统中相互联系的本质与内在规律,为人们建立完善的网络模型提供坚实的理论依据与保障。然而,对复杂网络研究的终极目标是研究网络上的各种动力学过程是如何受到网络拓扑结构、动力学机制等因素的影响,研究微观的相互作用在系统整体上产生的宏观现象。根据国内外的研究动态及发展趋势,本文对复杂网络上演化博弈动力学及基于命名博弈的一致性动力学做了细致的研究,深入探讨了自然界中维持合作现象并且促进合作涌现、以及提高一致性收敛效率的一些潜在动力学机制。本文的主要工作如下:
1、囚徒困境博弈模型是社会博弈的一个典型案例,受到科研人员的广泛关注。以往的研究工作中,初始时刻都假定拥有合作策略和背叛策略的个体在系统中随机分布,合作行为与背叛行为在均等的情形下受网络结构、博弈模型参数及动力学机制的影响互相演化竞争。本文研究了不同的初始策略分布情形下,HK聚类网络对演化合作动力学的影响。初始时,合作者和背叛者以一定的比例分别占据网络中不同度等级的节点,分两种不同的情形对囚徒困境博弈进行研究。研究结果证实网络中大度数节点对系统演化具有主导作用,定性地决定着合作行为的演化趋势,网络结构对合作行为的传播与稳定具有一定的差异性。通过计算不同度等级的个体在演化稳定阶段的合作比例对呈现的结果进行了详细的分析,计算了初始合作比例的临界值。
2、研究了公共品博弈模型中异质的投资和分配对演化合作行为的影响。由于社会关系存在多样性,基于连接数的差异性,个体为不同的邻居配置不同的资金量,并且差异的多样性可通过其连接度的指数函数进行调节,表现为权重网络模型。通过数值仿真,发现这种异质性的资金配置影响着合作行为的演化。对不同度等级的个体在演化稳定阶段的合作比例及平均收益进行了分析,并对节点的连接权重进行了理论分析和数值模拟,两者比较吻合。验证了在不同的适应性计算方式及策略更新规则下这种异质配置模式对合作行为影响的鲁棒性特征。
3、在网络上对演化博弈的研究,放宽了原始演化博弈论中个体之间不发生重复博弈的限制,物种具有某种抽象记忆,对遗传策略的优劣具有感知功能。因此,本文研究了基于历史收益的公共品博弈模型。个体拥有两种类型收益,分别通过扮演合作者和背叛者的角色得到,两种类型的收益异步地衰减。增强的历史收益影响具有促进合作行为涌现的能力,刻画了不同历史收益参数影响下演化稳定阶段合作者生存的空间斑图,并对这种斑图的演化形成过程进行了仿真程序跟踪。分析了稳态时边界上的合作者和背叛者的平均适应性关系,并对不同策略选择强度下合作行为和背叛行为生存的临界值进行了研究。
4、命名博弈研究简单的局部交互结合一个自组织的过程促使系统演化到全局一致性状态的动力学行为。考虑到流行的词汇具有较大的概率被用来交流,而生僻的词汇则容易被遗忘并且从系统中消失。因此,本文提出了词汇与其权重协同演化的模型。在由无标度网络或小世界网络构成的交流系统中,每个词汇都有一个权重,权重与该词汇的流通频率相关,并且也随时间进行演化。采用倒置的命名博弈方式,研究了一致性的收敛时间,得到了最佳的权重因子参数。对词汇从系统消失时的总权重的计算方式进行构建,研究了不同词汇的数量、平均协议成功率及个体的总记忆长度随时间的演化过程,得到了收敛时间和最大记忆长度与网络尺度的函数关系。
5、在观点动力学模型中,个体的观点被看成离散的或连续的变量,经由规定的协议,个体之间的观点进行交互,从多样性的状态演化到全局统一的形式。本文研究了基于Majority-rule(即服从多数原则)的观点动力学模型。初始时刻,以命名博弈的方式产生大量观点,但仅允许个体的记忆库中保留一个观点,并且个体对此观点具有一定的坚信程度。个体之间以概率地进行成对交互或者以服从多数的方式来更新自己的观点。研究发现,个体越趋向于采纳多数人的观点,全局一致性的速度越快,并对不同的模型参数对动力学行为的影响作了细致的研究。
Complex networks exist in nature and human society widely. To study complex networks has become one of advanced subjects of the greatest challenges in the fields of complex systems and complexity science. Based on abstracting a complex system as a network consisting of many interacting individuals, scientists in different fields working together try to exploit the profound mysteries of complex networks theoretically and empirically, and to reveal the essence and inherent regularity of interaction on human society and biological system, aiming to provide solid theoretical basis for constructing perfect network models. Nevertheless, the ultimate aim for studying compelx networks is to investigate how the network topology and dynamical mechanism affect the dynamics process, and to probe macroscopic phenomena in the system induced by microscopic interaction. According to the latest trends and development strategy at home and abroad, in this paper, we have done through going and painstaking research on evolutionary game dynamics and consensus dynamics based on naming game in complex networks, explored in greater depth some underlying dynamics mechanism in persisting and promoting cooperation, as well as in improving convergence efficiency for the consensus of the system. The main innovation points of this paper are as follows:
1. The effect of HK clusterd scale-free networks on the evolution of cooperation, in the case of different initial distributions, is investigated. It is found that, on the one hand, cooperarion can be enhanced with the increasing clusting coefficeient when only the most connected nodes are occupied by cooperators initially. And this enhanced cooperation is robust with respect to the increasing number of initial cooperators. On the other hand, if cooperators just occupy the lowest-degree nodes at the beginning, then the higher the value of the clustering coefficient, the more unfavorable the environment of cooperators to survive for the increment of temptation to defect. In this case, there are a lot of cooperators in the beginning of evolution. Bsides, we investigate the Snowdrift game in BA scale-free networks and find that, the magnitude of the fitness degermines qualitatively the dissemination trend of cooperative behavior in complex networks.
2. We investigate the effects of heterogeneous investment and distribution on the evolution of cooperation in the context of the public goods games. To do this, we develop a simple model in which each individual allocates differing funds to his direct neighbors based upon 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. By numerical simulations, it is found that allocating both too much and too little funds to diverse neighbors can remarkably improve the cooperation level. However, there exists a worst mode of funds allocation leading to the most unfavorable cooperation induced by the moderate values of the parameter.
3. The effect of memory on the evolution of cooperation on a square lattice is investigated. The fitness of individuals are characterized by two types of payoffs being obtained by acting as cooperators and defectors, respectively, both of which are the linear combination of the current payoffs and the cumulative historical payoffs. Simulation results show that cooperation is promoted by an increasing memory effect over a wide range of the multiplication factor. Defectors can just survive through forming narrower clusters to exploit cooperators more widely. For each decaying factor of historical payoffs, there exist two threshold values of the multiplication factor, below/above which cooperators/defectors would vanish completely from the system.
4. We propose a coevolutionary version to investigate the naming game, a model recently introduced to describe how shared vocabulary can emerge and persist spontaneously in communication systems. We base our model on the fact that more popular names have more opportunities to be selected by agents and then spread in the population. A name’s popularity is concerned with its communication frequency, characterized by its weight coevolving with the name. A tunable parameter governs the influence of name weight. We implement thismodified version on both scale-free networks and small-world networks, in which interactions proceed between paired agents by means of the reverse naming game. It is found that there exists an optimal value of the parameter that induces the fastest convergence of the population. This illustration indicates that a moderately strong influence of evolving name weight favors the rapid achievement of final consensus, but very strong influences inhibit the convergence process. The rank-distribution of the final accumulated weights of names qualitatively explains this nontrivial phenomenon. Investigations of some pertinent quantities are also provided, including the time evolution of the number of different names and the success rate, as well as the total memory of agents for different parameter values, which are helpful for better understanding the coevolutionary dynamics. Finally, we explore the scaling behavior in the convergence time and conclude a smaller scaling parameter compared to the previous naming gamemodels.
5. We propose a simple model to investigate the evolutionary dynamics of opinions on well-mixed populations.We assume that each individual has an inherent propensity to maintain his own word (opinion) about an object whereas other individuals would affect his decision when they communicate. On the one hand, individuals learn the opinion of another one with a probability pertaining to their propensities. On the other hand, the focal individual would adopt the word held by the majority in a randomly selected group. We have numerically explored how dynamical behavior evolves as a result of combination of these two competing update patterns. A parameter governs the time scale ratio at which the two update patterns separately progress.We find that an increasing tendency to adopt the opinion held by the majority results in a rapid extinction of most opinions, thus more easily induces the system to a global consensus. Large initial probabilities denoting propensity are found to be unfavorable for the achievement of the consensus. Interestingly, simulation results indicate that the convergence time is negligibly affected by the number of initial distinct words when this number exceeds a certain value. Results from our model may offer an insight into better understanding the intricate dynamics of opinions.
1、囚徒困境博弈模型是社会博弈的一个典型案例,受到科研人员的广泛关注。以往的研究工作中,初始时刻都假定拥有合作策略和背叛策略的个体在系统中随机分布,合作行为与背叛行为在均等的情形下受网络结构、博弈模型参数及动力学机制的影响互相演化竞争。本文研究了不同的初始策略分布情形下,HK聚类网络对演化合作动力学的影响。初始时,合作者和背叛者以一定的比例分别占据网络中不同度等级的节点,分两种不同的情形对囚徒困境博弈进行研究。研究结果证实网络中大度数节点对系统演化具有主导作用,定性地决定着合作行为的演化趋势,网络结构对合作行为的传播与稳定具有一定的差异性。通过计算不同度等级的个体在演化稳定阶段的合作比例对呈现的结果进行了详细的分析,计算了初始合作比例的临界值。
2、研究了公共品博弈模型中异质的投资和分配对演化合作行为的影响。由于社会关系存在多样性,基于连接数的差异性,个体为不同的邻居配置不同的资金量,并且差异的多样性可通过其连接度的指数函数进行调节,表现为权重网络模型。通过数值仿真,发现这种异质性的资金配置影响着合作行为的演化。对不同度等级的个体在演化稳定阶段的合作比例及平均收益进行了分析,并对节点的连接权重进行了理论分析和数值模拟,两者比较吻合。验证了在不同的适应性计算方式及策略更新规则下这种异质配置模式对合作行为影响的鲁棒性特征。
3、在网络上对演化博弈的研究,放宽了原始演化博弈论中个体之间不发生重复博弈的限制,物种具有某种抽象记忆,对遗传策略的优劣具有感知功能。因此,本文研究了基于历史收益的公共品博弈模型。个体拥有两种类型收益,分别通过扮演合作者和背叛者的角色得到,两种类型的收益异步地衰减。增强的历史收益影响具有促进合作行为涌现的能力,刻画了不同历史收益参数影响下演化稳定阶段合作者生存的空间斑图,并对这种斑图的演化形成过程进行了仿真程序跟踪。分析了稳态时边界上的合作者和背叛者的平均适应性关系,并对不同策略选择强度下合作行为和背叛行为生存的临界值进行了研究。
4、命名博弈研究简单的局部交互结合一个自组织的过程促使系统演化到全局一致性状态的动力学行为。考虑到流行的词汇具有较大的概率被用来交流,而生僻的词汇则容易被遗忘并且从系统中消失。因此,本文提出了词汇与其权重协同演化的模型。在由无标度网络或小世界网络构成的交流系统中,每个词汇都有一个权重,权重与该词汇的流通频率相关,并且也随时间进行演化。采用倒置的命名博弈方式,研究了一致性的收敛时间,得到了最佳的权重因子参数。对词汇从系统消失时的总权重的计算方式进行构建,研究了不同词汇的数量、平均协议成功率及个体的总记忆长度随时间的演化过程,得到了收敛时间和最大记忆长度与网络尺度的函数关系。
5、在观点动力学模型中,个体的观点被看成离散的或连续的变量,经由规定的协议,个体之间的观点进行交互,从多样性的状态演化到全局统一的形式。本文研究了基于Majority-rule(即服从多数原则)的观点动力学模型。初始时刻,以命名博弈的方式产生大量观点,但仅允许个体的记忆库中保留一个观点,并且个体对此观点具有一定的坚信程度。个体之间以概率地进行成对交互或者以服从多数的方式来更新自己的观点。研究发现,个体越趋向于采纳多数人的观点,全局一致性的速度越快,并对不同的模型参数对动力学行为的影响作了细致的研究。
Complex networks exist in nature and human society widely. To study complex networks has become one of advanced subjects of the greatest challenges in the fields of complex systems and complexity science. Based on abstracting a complex system as a network consisting of many interacting individuals, scientists in different fields working together try to exploit the profound mysteries of complex networks theoretically and empirically, and to reveal the essence and inherent regularity of interaction on human society and biological system, aiming to provide solid theoretical basis for constructing perfect network models. Nevertheless, the ultimate aim for studying compelx networks is to investigate how the network topology and dynamical mechanism affect the dynamics process, and to probe macroscopic phenomena in the system induced by microscopic interaction. According to the latest trends and development strategy at home and abroad, in this paper, we have done through going and painstaking research on evolutionary game dynamics and consensus dynamics based on naming game in complex networks, explored in greater depth some underlying dynamics mechanism in persisting and promoting cooperation, as well as in improving convergence efficiency for the consensus of the system. The main innovation points of this paper are as follows:
1. The effect of HK clusterd scale-free networks on the evolution of cooperation, in the case of different initial distributions, is investigated. It is found that, on the one hand, cooperarion can be enhanced with the increasing clusting coefficeient when only the most connected nodes are occupied by cooperators initially. And this enhanced cooperation is robust with respect to the increasing number of initial cooperators. On the other hand, if cooperators just occupy the lowest-degree nodes at the beginning, then the higher the value of the clustering coefficient, the more unfavorable the environment of cooperators to survive for the increment of temptation to defect. In this case, there are a lot of cooperators in the beginning of evolution. Bsides, we investigate the Snowdrift game in BA scale-free networks and find that, the magnitude of the fitness degermines qualitatively the dissemination trend of cooperative behavior in complex networks.
2. We investigate the effects of heterogeneous investment and distribution on the evolution of cooperation in the context of the public goods games. To do this, we develop a simple model in which each individual allocates differing funds to his direct neighbors based upon 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. By numerical simulations, it is found that allocating both too much and too little funds to diverse neighbors can remarkably improve the cooperation level. However, there exists a worst mode of funds allocation leading to the most unfavorable cooperation induced by the moderate values of the parameter.
3. The effect of memory on the evolution of cooperation on a square lattice is investigated. The fitness of individuals are characterized by two types of payoffs being obtained by acting as cooperators and defectors, respectively, both of which are the linear combination of the current payoffs and the cumulative historical payoffs. Simulation results show that cooperation is promoted by an increasing memory effect over a wide range of the multiplication factor. Defectors can just survive through forming narrower clusters to exploit cooperators more widely. For each decaying factor of historical payoffs, there exist two threshold values of the multiplication factor, below/above which cooperators/defectors would vanish completely from the system.
4. We propose a coevolutionary version to investigate the naming game, a model recently introduced to describe how shared vocabulary can emerge and persist spontaneously in communication systems. We base our model on the fact that more popular names have more opportunities to be selected by agents and then spread in the population. A name’s popularity is concerned with its communication frequency, characterized by its weight coevolving with the name. A tunable parameter governs the influence of name weight. We implement thismodified version on both scale-free networks and small-world networks, in which interactions proceed between paired agents by means of the reverse naming game. It is found that there exists an optimal value of the parameter that induces the fastest convergence of the population. This illustration indicates that a moderately strong influence of evolving name weight favors the rapid achievement of final consensus, but very strong influences inhibit the convergence process. The rank-distribution of the final accumulated weights of names qualitatively explains this nontrivial phenomenon. Investigations of some pertinent quantities are also provided, including the time evolution of the number of different names and the success rate, as well as the total memory of agents for different parameter values, which are helpful for better understanding the coevolutionary dynamics. Finally, we explore the scaling behavior in the convergence time and conclude a smaller scaling parameter compared to the previous naming gamemodels.
5. We propose a simple model to investigate the evolutionary dynamics of opinions on well-mixed populations.We assume that each individual has an inherent propensity to maintain his own word (opinion) about an object whereas other individuals would affect his decision when they communicate. On the one hand, individuals learn the opinion of another one with a probability pertaining to their propensities. On the other hand, the focal individual would adopt the word held by the majority in a randomly selected group. We have numerically explored how dynamical behavior evolves as a result of combination of these two competing update patterns. A parameter governs the time scale ratio at which the two update patterns separately progress.We find that an increasing tendency to adopt the opinion held by the majority results in a rapid extinction of most opinions, thus more easily induces the system to a global consensus. Large initial probabilities denoting propensity are found to be unfavorable for the achievement of the consensus. Interestingly, simulation results indicate that the convergence time is negligibly affected by the number of initial distinct words when this number exceeds a certain value. Results from our model may offer an insight into better understanding the intricate dynamics of opinions.
引文
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