复杂系统中合作涌现的几种机制
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摘要
复杂系统研究关注由个体之间的微观相互作用所导致的自组织、涌现等宏观整体现象。近二十年来复杂系统研究的一大进展是对其粗粒化描述——复杂网络的研究。因而网络无处不在,遍及自然界和人类社会,具有代表性的复杂网络有因特网、万维网、食物网、人际关系网、科研合作网络、电力网、蛋白质相互作用网络、基因调控网络等。不同于传统的图论,复杂网络研究着眼于网络大规模的统计特性,统计物理学家在其中做出了许多重要贡献。人们发现现实的复杂网络往往具有短距离、幂律度分布、高簇系数等普适特征；并根据这些统计特征构造了网络模型,其中著名的比如小世界网络、无尺度网络等。在对各种网络模型充分研究之后,需要还原其节点上的动力学自由度。复杂网络研究的最终目标是理解网络结构如何影响其上的动力学过程,并反过来研究其上动力学过程如何驱动网络结构演化。
生物系统和社会经济是典型的复杂系统。构成这些体系的基本单元不再是传统物理上的粒子,而是各种复杂有机体；这样,体系中个体之间的相互作用不直接表现为通常的四种基本力,体系的演化也不直接由经典或量子的物理定律描述。一个富有挑战性的课题是如何用数学来定量的研究生物和社会体系。事实上,现有的一个定量研究生物进化、社会经济演化的理论框架是演化博弈论。最近的演化博弈论研究从充分混合的体系转向具有空间结构的种群体系,并与复杂网络的研究结合了起来,被称为网络演化博弈或空间演化博弈。空间博弈模型与晶格统计模型有许多相似之处,而目前数学在描述自然界多体问题最成功的理论是统计物理学,因而后者值得生物学和社会学借鉴。近年来统计物理学家在这类复杂体系的研究中做了大量工作。统计物理中的重要概念和理论框架诸如相变,临界性和普适性,都能应用其中。统计物理中的一些方法比如平均场近似、主方程法也可以应用其中。反过来,对这些系统的多学科交叉研究,也对统计物理学提出了新的挑战。
我们系统研究了网络上演化博弈模型中的一些合作涌现机制,本文的主要工作如下。
1.社会分工与合作的涌现：合作的一个前提是分工,通过分工和专业化,个体才能发挥特长进而提高生产效率。分工现象在动物和人类社会中无处不在,正是分工使得合作有意义。我们通过将总人口划分成两种工作类型,研究了分工对合作的影响。我们假定当两个个体类型不同时他们具有更高的工作效率,即将收益矩阵用一个因子放大；当两者类型相同时,收益矩阵不变。模拟结果表明,类型的划分与空间网格结构相结合能够诱导出几个社会等级,这种社会等级的出现有利于合作。并且,当类型比例参数和收益矩阵缩放参数取适当值时,最有利于合作的涌现。
2.演化博弈模型中的生命过程：多数演化博弈动力学只包含了生-灭过程,然而包括出生、成长、衰老、疾病、死亡等整个生命过程对生物系统的描述都很重要。在前人所做的对于不同的相互作用和策略更新时间尺度的研究的基础上,我们提出可以将时间尺度因子与个体适应度相关联。这样可以在演化博弈论中更加完整地描述和研究个体的生命过程。结果表明,时间尺度与个体适应度的关联可以进一步提高合作水平。并且,时间尺度取适当值时合作水平最高。
3.中间策略的影响：空间博弈模型和统计物理上的晶格统计模型——比如伊辛模型——相似。在统计物理上,另一个晶格模型——q态Potts模型是伊辛模型的一个重要推广,其中两个自旋值被推广到q个。受此启发,我们将通常仅包含合作-背叛二元策略的空间博弈模型推广到q种策略,其中包含了分数值混合策略。我们将分数值策略解释为中间策略,即既不完全合作也不完全背叛。结果表明,中间策略能够在一个更大的叛诱惑参数范围背内保持合作水平,这是因为,一些中间策略可以认为是既削弱了纯背叛者的剥削又保持一定的空间互惠效应的优化的策略。
Complex system studies concentrate on the integrated macroscopic phenomena of self-organization and emergence caused by microscopic interactions. In the last two decades one major progress in the study of complex systems is on their coarse-grained description-the complex networks. Thus networks are ubiquitous in nature and hu-man society. Most typical complex networks include the Internet, World Wide Web, social networks, scientific collaboration networks, power grids, protein-protein inter-action networks, genetic regulatory networks, etc. Unlike traditional graph theory, the studies on complex networks emphasize on their large scale statistical properties, in which statistical physicists have made significant contributions. People have found that complex networks in real lives obey several universal properties such as short distance, power-law degree distributions and high cluster coefficients. They have built many mathematical models according to those properties, among which well known are the small world networks and scale free networks. Opun sufficient investigations on the net-work models one still needs to recover the dynamical freedoms attached to the nodes, as the ultimate goals of complex network studies are trying to understand how network structures affect the dynamic processes and conversely how dynamic processes drive the evolution of the network structures.
Bio-systems and Social economy are typical complex systems. The units that com-prise these systems are no longer traditional physical particles, but are various compli-cated organisms, so that the interactions among the agents don't appear as the four kind of fundamental forces and the evolution of these systems are not directly governed by classical or quantum laws. One challenging topic is how to study the society and bio-systems in a mathematical and quantitative approach. In fact, one existing framework on the evolution of bio-systems and social economical systems is the evolutionary game theory. Recently evolutionary game studies have turned from well-mixed populations to structured population and have closely connected with researches on complex networks, which has been called evolutionary games on graphs or spatial evolutionary games. So far the most successful application of mathematics in many body systems is in statisti-cal mechanics, while spacial game models resemble lattice statistical models in several ways, thus the later could serve as a guid for the former. Indeed, in recent years statisti-cal physicists have made a large amount of contributions in these problems. Important concepts in statistical mechanics such as phase transition, criticality and universality have all been adopted. Many famous methods in statistical mechanics such as the mean field approximation and master equation method have been extended to these problems as well. Conversely, cross-disciplinary investigations about these systems have raised many new challenges and opportunities to statistical mechanics.
We have systematically investigated the mechanisms of the emergence of coopera-tion in several evolutionary game models on networks. Our chief works are as follows:
1. Social division of work and the emergence of cooperation:one prerequisite of cooperation is division of work. Through division of work and specialization indi-viduals could make full use of their special talents and increase the productivity. The phenomena of division of work is ubiquitous in animals as well as human societies. It is division of work that makes cooperation meaningful. We divided the whole population into two different work types and investigated its impact on cooperation. We assume that two individuals have higher productivity if their types are different, that is their payoff matrix are amplified by a multiplicative factor; while the payoff matrix remains unchanged, if their are of a same type. Simulation results show that division into dif-ferent types together with the spacial lattice structure can induce several social ranks, which is beneficial for cooperation. Besides, we found that moderate values for both the type fraction parameter and multiplicative factor are more favorable for cooperation.
2. Life processes in evolutionary game models:most evolutionary dynamics in-clude merely birth-death process, though the whole life process including birth, growth, aging, disease, death and so on are all very important for the description of bio-systems. Based on previous investigations about different interaction and strategy update time scales, we proposed that the time scale factor should be correlated with individuals' fitness, so that the description and investigation on individuals'life processes become more complete. Simulation results show that the correlation between time scale and individuals'fitness can further increase the cooperation level. Besides, moderate time scale values lead to the highest cooperation level.
3. Impact of intermediate strategies:spatial game models resemble some lattice statistical models, such as the Ising model. In statistical mechanics, there is another model——the q state Potts model which is an important generalization of Ising model, and in which two spin values is generalized to q values. Inspired by that, we general-ized the usual cooperation-defection binary strategy spatial game to a q-strategy model, where fractional valued mixed strategies are allowed. We interpret the fractional strate-gies as intermediate strategies, namely, neither completely cooperate nor defect. We showed that intermediate strategies can sustain cooperation in a broader range of the temptation to defect parameter, because some intermediate strategies are optimal strate-gies in that they can on the one hand reduce the exploitation by the pure defectors and on the other hand keep a certain spatial reciprocity effect.
生物系统和社会经济是典型的复杂系统。构成这些体系的基本单元不再是传统物理上的粒子,而是各种复杂有机体；这样,体系中个体之间的相互作用不直接表现为通常的四种基本力,体系的演化也不直接由经典或量子的物理定律描述。一个富有挑战性的课题是如何用数学来定量的研究生物和社会体系。事实上,现有的一个定量研究生物进化、社会经济演化的理论框架是演化博弈论。最近的演化博弈论研究从充分混合的体系转向具有空间结构的种群体系,并与复杂网络的研究结合了起来,被称为网络演化博弈或空间演化博弈。空间博弈模型与晶格统计模型有许多相似之处,而目前数学在描述自然界多体问题最成功的理论是统计物理学,因而后者值得生物学和社会学借鉴。近年来统计物理学家在这类复杂体系的研究中做了大量工作。统计物理中的重要概念和理论框架诸如相变,临界性和普适性,都能应用其中。统计物理中的一些方法比如平均场近似、主方程法也可以应用其中。反过来,对这些系统的多学科交叉研究,也对统计物理学提出了新的挑战。
我们系统研究了网络上演化博弈模型中的一些合作涌现机制,本文的主要工作如下。
1.社会分工与合作的涌现：合作的一个前提是分工,通过分工和专业化,个体才能发挥特长进而提高生产效率。分工现象在动物和人类社会中无处不在,正是分工使得合作有意义。我们通过将总人口划分成两种工作类型,研究了分工对合作的影响。我们假定当两个个体类型不同时他们具有更高的工作效率,即将收益矩阵用一个因子放大；当两者类型相同时,收益矩阵不变。模拟结果表明,类型的划分与空间网格结构相结合能够诱导出几个社会等级,这种社会等级的出现有利于合作。并且,当类型比例参数和收益矩阵缩放参数取适当值时,最有利于合作的涌现。
2.演化博弈模型中的生命过程：多数演化博弈动力学只包含了生-灭过程,然而包括出生、成长、衰老、疾病、死亡等整个生命过程对生物系统的描述都很重要。在前人所做的对于不同的相互作用和策略更新时间尺度的研究的基础上,我们提出可以将时间尺度因子与个体适应度相关联。这样可以在演化博弈论中更加完整地描述和研究个体的生命过程。结果表明,时间尺度与个体适应度的关联可以进一步提高合作水平。并且,时间尺度取适当值时合作水平最高。
3.中间策略的影响：空间博弈模型和统计物理上的晶格统计模型——比如伊辛模型——相似。在统计物理上,另一个晶格模型——q态Potts模型是伊辛模型的一个重要推广,其中两个自旋值被推广到q个。受此启发,我们将通常仅包含合作-背叛二元策略的空间博弈模型推广到q种策略,其中包含了分数值混合策略。我们将分数值策略解释为中间策略,即既不完全合作也不完全背叛。结果表明,中间策略能够在一个更大的叛诱惑参数范围背内保持合作水平,这是因为,一些中间策略可以认为是既削弱了纯背叛者的剥削又保持一定的空间互惠效应的优化的策略。
Complex system studies concentrate on the integrated macroscopic phenomena of self-organization and emergence caused by microscopic interactions. In the last two decades one major progress in the study of complex systems is on their coarse-grained description-the complex networks. Thus networks are ubiquitous in nature and hu-man society. Most typical complex networks include the Internet, World Wide Web, social networks, scientific collaboration networks, power grids, protein-protein inter-action networks, genetic regulatory networks, etc. Unlike traditional graph theory, the studies on complex networks emphasize on their large scale statistical properties, in which statistical physicists have made significant contributions. People have found that complex networks in real lives obey several universal properties such as short distance, power-law degree distributions and high cluster coefficients. They have built many mathematical models according to those properties, among which well known are the small world networks and scale free networks. Opun sufficient investigations on the net-work models one still needs to recover the dynamical freedoms attached to the nodes, as the ultimate goals of complex network studies are trying to understand how network structures affect the dynamic processes and conversely how dynamic processes drive the evolution of the network structures.
Bio-systems and Social economy are typical complex systems. The units that com-prise these systems are no longer traditional physical particles, but are various compli-cated organisms, so that the interactions among the agents don't appear as the four kind of fundamental forces and the evolution of these systems are not directly governed by classical or quantum laws. One challenging topic is how to study the society and bio-systems in a mathematical and quantitative approach. In fact, one existing framework on the evolution of bio-systems and social economical systems is the evolutionary game theory. Recently evolutionary game studies have turned from well-mixed populations to structured population and have closely connected with researches on complex networks, which has been called evolutionary games on graphs or spatial evolutionary games. So far the most successful application of mathematics in many body systems is in statisti-cal mechanics, while spacial game models resemble lattice statistical models in several ways, thus the later could serve as a guid for the former. Indeed, in recent years statisti-cal physicists have made a large amount of contributions in these problems. Important concepts in statistical mechanics such as phase transition, criticality and universality have all been adopted. Many famous methods in statistical mechanics such as the mean field approximation and master equation method have been extended to these problems as well. Conversely, cross-disciplinary investigations about these systems have raised many new challenges and opportunities to statistical mechanics.
We have systematically investigated the mechanisms of the emergence of coopera-tion in several evolutionary game models on networks. Our chief works are as follows:
1. Social division of work and the emergence of cooperation:one prerequisite of cooperation is division of work. Through division of work and specialization indi-viduals could make full use of their special talents and increase the productivity. The phenomena of division of work is ubiquitous in animals as well as human societies. It is division of work that makes cooperation meaningful. We divided the whole population into two different work types and investigated its impact on cooperation. We assume that two individuals have higher productivity if their types are different, that is their payoff matrix are amplified by a multiplicative factor; while the payoff matrix remains unchanged, if their are of a same type. Simulation results show that division into dif-ferent types together with the spacial lattice structure can induce several social ranks, which is beneficial for cooperation. Besides, we found that moderate values for both the type fraction parameter and multiplicative factor are more favorable for cooperation.
2. Life processes in evolutionary game models:most evolutionary dynamics in-clude merely birth-death process, though the whole life process including birth, growth, aging, disease, death and so on are all very important for the description of bio-systems. Based on previous investigations about different interaction and strategy update time scales, we proposed that the time scale factor should be correlated with individuals' fitness, so that the description and investigation on individuals'life processes become more complete. Simulation results show that the correlation between time scale and individuals'fitness can further increase the cooperation level. Besides, moderate time scale values lead to the highest cooperation level.
3. Impact of intermediate strategies:spatial game models resemble some lattice statistical models, such as the Ising model. In statistical mechanics, there is another model——the q state Potts model which is an important generalization of Ising model, and in which two spin values is generalized to q values. Inspired by that, we general-ized the usual cooperation-defection binary strategy spatial game to a q-strategy model, where fractional valued mixed strategies are allowed. We interpret the fractional strate-gies as intermediate strategies, namely, neither completely cooperate nor defect. We showed that intermediate strategies can sustain cooperation in a broader range of the temptation to defect parameter, because some intermediate strategies are optimal strate-gies in that they can on the one hand reduce the exploitation by the pure defectors and on the other hand keep a certain spatial reciprocity effect.
引文
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