一般复杂网络及经济网络的动态模型与稳定性研究
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摘要
复杂网络在诸如社会、政治、医药、经济、管理等领域的广泛应用而受到自然科学领域内研究者们越来越多的关注,成为近年来的一个研究热点。在复杂网络研究中,网络模型和网络上的传播动力学研究是其中的两个重要研究方向。而随着复杂网络的兴起,经济学家和经济物理学家们也开始对经济系统的网络结构进行研究。经济网络恰好刻画了经济系统中微观个体之间存在的相互联系与相互作用的关系结构,这种关系可以是经济个体之间的信息交流、商品或证券交易、投资关系、信用关系或者隶属控制关系等等。现实的经济系统往往存在着巨大而复杂的网络结构,然而传统的经典经济学理论却假设经济系统具有完全或者星形连接的简单网络结构,所以在以往的经济学研究中网络的结构被忽略了,这或许是造成经典经济学理论对现实的经济现象常常缺乏解释能力的原因之一。因此经济网络的研究对经济学来说就显得尤为重要,这在目前还是经济学中的一个全新的研究领域,虽然还有相当多方面的研究有待进一步的深入,但是现在的进展情况已显示出这项研究的潜力。
本论文将不确定理论、效用理论、博弈论、随机占优理论及平均场方法等理论和方法应用到一般复杂网络和经济网络的研究中,对一般复杂网络的网络模型、网络特性、动力学行为及经济网络的动态模型与稳定性等方面进行了研究。通过对网络结构和网络特性的研究,我们可以利用理论结果来解释一些相应的复杂系统所呈现的特性;通过对网络动力学行为的研究,一方面我们可以获取对网络结构产生影响的机制,进而解释复杂系统所呈现出的动力学现象,并将这种理论研究成果应用到建模过程中,设计出具有更好特性的网络模型;另一方面,可以利用理论成果来对实际网络的形成过程进行干扰,使网络结构朝利好方向发展。因此,对网络结构模型和网络特性研究的重要意义是显而易见的。本文的主要内容和创新之处概述如下:
1、建立了不确定无标度网络模型。
网络的形成过程中会存在着各种不确定性,考虑到这种不确定性对网络形成过程的影响,在网络择优机制上,运用不确定变量对网络的择优概率进行改良,再考虑到实际网络中存在的旧连线与旧节点删除现象,建立了一个基于不确定理论的网络演化模型。然后运用平均场方法,建立节点度数的动力学方程。通过讨论不确定变量的两种不同分布情况,根据连线增加和删除数目之间的不同关系,导出不确定无标度网络的度分布和幂率指数的表达方程式,并进行了实验模拟和理论结果的比较来检查模型的一般适用性。但由于对度小的分布状态时,运用平均场理论误差较大,因此对模型进行了马氏链数值分析。运用数值分析结果和模拟结果的比较,发现两者的吻合性较好。由幂率指数和分析结果表明:该模型可以自组织演化成无标度网络,又由于不确定性的普遍存在性,因此,模型具有一般的适用性。
2、网络形成过程中的随机性对网络结构及网络特性的影响。
许多现实网络结构的动态形成过程中,新连线增加的连接机制中,不仅具有网络演化模型中提到的基于现有网络结构的择优连接,还有节点之间的随机连接。在社会经济网络的形成过程中,随机性连接所占比例的不同对网络结构及网络特性有明显不同的影响,为了研究这种影响,我们假设在网络结构形成过程中基于两种连接机制:一是基于完全的随机连接,二是基于网络现有结构的连接。给这两种连接机制以及旧连线的删除机制所占概率分别设定不同的参数,通过这些参数的变化来分析网络结构和网络特征的变化。首先运用平均场方法导出网络的度分布和网络的累积度分布和度指数,通过各参数的变化分析对网络度指数的影响,进而讨论形成的网络结构的不同。其次,根据参数对动力学指数的影响,分别对网络的平均路径长度、聚类系数等网络统计特征进行分析,找出这些结构参数对网络统计特征的影响规律。
3、利用随机占优理论分析网络上的传播和免疫特性。
网络传播和免疫是网络动力学行为之一。研究网络的传播特性可以对现实网络中的疾病、流言等等传播行为进行深入了解,研究网络的免疫特性则可以对网络上的传播行为加以控制。网络结构是对网络传播的速度和面积产生重要影响的因素,因此,要了解网络结构对网络传播和免疫特性的影响。文中首先讨论网络中存在稳定传播状态的临界值问题,建立代表网络结构的度分布为变量的网络传播稳定状态方程式,根据随机占优理论的思想,以传统SIS传播模型为例,对于具有不同度分布的网络结构,可以比较得知两种网络结构上稳定传播状态的高低。利用此比较结果,不用计算网络传播的临界值,可以得知不同的网络结构对网络中疾病和信息等的传播速度及状态的影响,进而进行网络免疫,以使在疾病爆发或信息传播过程中根据网络结构的调整控制网络传播的方向和速度,对分析社会中涉及传播的问题提供一种新的方法。再者,针对前面提到的随机性网络结构,根据参数的不同,运用此方法来进行网络传播特性的分析,分析网络参数对网络传播速率和传播状态的影响,并且进一步分析,这些参数对网络免疫力的影响。
4、分析了经济网络中内生形成模型的动态扩展模型及动态稳定性和动态有效性。
经济网络中的内生形成模型恰当的刻画了社会经济系统中个体之间的关系,其中的稳定性定理和有效性定理也有利于一些社会经济现象的解释。但是,内生形成模型的静态性和社会经济系统的动态变化性存在着差距,这使一些理论分析存在偏差,基于此点考虑,本文在网络静态内生形成模型的基础上,分析了有新节点加入网络时,所形成的动态稳定网络和动态有效网络结构。因为内生形成模型是静态网络,在模型中定义的稳定性和有效性不适合动态网络,因此,文中首先给出了动态稳定性和动态有效性的定义,然后根据网络的形成过程:基于个体自私和短视的假定,由连接的效用和成本来决定连线的建立过程,导出在不同条件下的动态稳定网络结构和动态有效网络结构,并运用效用理论加以证明。为了进一步扩展模型的适用性,重新假设网络中的个体是异质个体(即个体之间的连接成本存在差异)的情况下网络结构的形成过程及形成的动态稳定网络结构,并运用效用理论加以证明。
Complex networks have been found many applications in a variety of fields including society, politics, medicine, economics and management, and attract more and more attentions from various fields of science and engineering. Evolving networks and epidemic dynamics are two important fields in complex networks. And, Economists and econophysicists have been starting to research on the network formation of economic systems from the rise of the complex networks. Economic networks describe the relationships and the influences among the economic agents precisely. The relationships can be the information changing among the economic agents, commodity transaction, dealing in securities, investment relation, credit relationship, membership and so on. It is neglected that the real economic systems always have a large and complex network formation in classic economic theory. It is supposed that the economic systems have the simple network formation like star network full connected network. It can partly explain why the economic theory can not resolve the economic phenomena sometimes. So, it is important to research on the economic network formation. It is a new field in economics and there are many fields need further study. But it shows the potentiality on economic networks through current progress report.
The theories and methods which contain uncertainty theory, utility theory, Game, stochastic dominance theory and the mean-filed method are used to the research on network formation in this paper. And the research contains network formation, the characters and the dynamics. It can be used to explain the characters of real-world networks through the results that received from the research. And by studying the dynamic properties of complex networks, on the one hand, we can learn the mechanism of network formation and build better network structure by using the results; on the other hand, we can apply these theoretical results to influence the formation process of real-world networks and let it to change advantaged. So, the importance of the research on network formation and the properties of networks are clearly self-evident.
The main contents and originalities in this paper can be summarized as follows:
1. The evolving network based on uncertainty theory is proposed.
It contains various uncertainties in the process of network formation. First, an evolving network model by using proving rules is presented. The proving contains both adding the uncertainty variable to preferential probability and deleting the old links. Then the dynamical equation is established via using mean-filed approach. The expression equations of degree distribution and the degree exponent are deduced both by discussing two kinds of distributing of uncertainty variable and by the different relationship between the numbers of added and deleted links. And then, the degree of fit between the theoretical result and simulation is compared. But it will have a higher error if the degree is lower, it is analyzed by Markov-based numerical method. It is well fit by comparing the numerical result and simulation. It is showed that the uncertainty evolving network can evolve to scale-free network and it will be used generally through the degree exponent and the theoretical result.
2. The influence of randomness on the network structure and the properties is studied.
There is not only preferential attachment but also random attachment in the process of network formation. And the different probability of the two kinds of attachment will lead to different network structure. The model which includes both random attachment and preferential attachment is presented. The parameters are set to the probability of two kinds of attachment and the probability of deleting, and the change of the network formation and the properties following the change of parameters is analyzed. Firstly, the equation of degree distribution, cumulative degree distribution and degree exponent are deduced via mean field approach. The influence of randomness on the degree distribution is discussed, and network stucture is different because of the change of degree distribution and degree exponent. Then, the influence of randomness on the properties of network which include average path length and the clustering coefficient is discussed, and the regularity of the influence is analyzed.
3. The epidemic dynamics and immunization in complex networks based on stochastic dominance theory is studied.
The epidemic dynamics and immunization are the important dynamic behaviors. It is to help learn properties of the spread of epidemic and gossips through studying the epidemic dynamics of network. Network structure is the important factor to influence the properties of epidemic dynamics and immunization. So, it is important to learn the influence. Firstly, the equation of the positive steady state is built and the positive steady state is discussed. Then, the epidemic speed and ratio between different network structures can be made comparison based on the stochastic dominance theory in SIS model. The diffusion speed and ration can be learned from the comparison results, and the immunization which can influenced epidemic dynamics the can be carried on from the results, all these conclusions can be used to change the epidemic property of real-world networks like epidemic and gossip broadcasting. So, It is supplied a new method to these problems. Based on the conclusion, the randomness model which is presented above is discussed. The changes of speed and the ratio of epidemic dynamics following the change of the parameters are analyzed. And then, the property of immunization changing by the parameters is also analyzed.
4. The dynamic economic models and the stability are analyzed.
The endogenous model which is one kind of economic networks is described the relationship between the agents in economic systems precisely. The stability and efficiency studied in the model can assist to explain some economic phenomenon. But, it is some difference from the static state of the model to the dynamic process of real-world economic systems and let the theoretical results show the shortage in some analysis. The dynamic model that contains new nodes adding is presented in this paper in order to solve this problem. The definition of the stability and efficiency which give in static model is not suit here, so, firstly, the new meaning is defined based on the dynamic formation process. Then the dynamic stability and the dynamic efficiency network formation are showed based on the rules of endogenous model and the hypotheses that the agent is selfish and myopic. And the process of the linking is decided by the benefit and the cost. The dynamic stability and the dynamic efficiency network formations are proved through utility theory. In order to expand the applied range of the results, the hypotheses is changed to that the agent is heterogeneous (the cost of agent is different). The dynamic stability network formation is showed in this kind of hypotheses and is proved through utility theory.
本论文将不确定理论、效用理论、博弈论、随机占优理论及平均场方法等理论和方法应用到一般复杂网络和经济网络的研究中,对一般复杂网络的网络模型、网络特性、动力学行为及经济网络的动态模型与稳定性等方面进行了研究。通过对网络结构和网络特性的研究,我们可以利用理论结果来解释一些相应的复杂系统所呈现的特性;通过对网络动力学行为的研究,一方面我们可以获取对网络结构产生影响的机制,进而解释复杂系统所呈现出的动力学现象,并将这种理论研究成果应用到建模过程中,设计出具有更好特性的网络模型;另一方面,可以利用理论成果来对实际网络的形成过程进行干扰,使网络结构朝利好方向发展。因此,对网络结构模型和网络特性研究的重要意义是显而易见的。本文的主要内容和创新之处概述如下:
1、建立了不确定无标度网络模型。
网络的形成过程中会存在着各种不确定性,考虑到这种不确定性对网络形成过程的影响,在网络择优机制上,运用不确定变量对网络的择优概率进行改良,再考虑到实际网络中存在的旧连线与旧节点删除现象,建立了一个基于不确定理论的网络演化模型。然后运用平均场方法,建立节点度数的动力学方程。通过讨论不确定变量的两种不同分布情况,根据连线增加和删除数目之间的不同关系,导出不确定无标度网络的度分布和幂率指数的表达方程式,并进行了实验模拟和理论结果的比较来检查模型的一般适用性。但由于对度小的分布状态时,运用平均场理论误差较大,因此对模型进行了马氏链数值分析。运用数值分析结果和模拟结果的比较,发现两者的吻合性较好。由幂率指数和分析结果表明:该模型可以自组织演化成无标度网络,又由于不确定性的普遍存在性,因此,模型具有一般的适用性。
2、网络形成过程中的随机性对网络结构及网络特性的影响。
许多现实网络结构的动态形成过程中,新连线增加的连接机制中,不仅具有网络演化模型中提到的基于现有网络结构的择优连接,还有节点之间的随机连接。在社会经济网络的形成过程中,随机性连接所占比例的不同对网络结构及网络特性有明显不同的影响,为了研究这种影响,我们假设在网络结构形成过程中基于两种连接机制:一是基于完全的随机连接,二是基于网络现有结构的连接。给这两种连接机制以及旧连线的删除机制所占概率分别设定不同的参数,通过这些参数的变化来分析网络结构和网络特征的变化。首先运用平均场方法导出网络的度分布和网络的累积度分布和度指数,通过各参数的变化分析对网络度指数的影响,进而讨论形成的网络结构的不同。其次,根据参数对动力学指数的影响,分别对网络的平均路径长度、聚类系数等网络统计特征进行分析,找出这些结构参数对网络统计特征的影响规律。
3、利用随机占优理论分析网络上的传播和免疫特性。
网络传播和免疫是网络动力学行为之一。研究网络的传播特性可以对现实网络中的疾病、流言等等传播行为进行深入了解,研究网络的免疫特性则可以对网络上的传播行为加以控制。网络结构是对网络传播的速度和面积产生重要影响的因素,因此,要了解网络结构对网络传播和免疫特性的影响。文中首先讨论网络中存在稳定传播状态的临界值问题,建立代表网络结构的度分布为变量的网络传播稳定状态方程式,根据随机占优理论的思想,以传统SIS传播模型为例,对于具有不同度分布的网络结构,可以比较得知两种网络结构上稳定传播状态的高低。利用此比较结果,不用计算网络传播的临界值,可以得知不同的网络结构对网络中疾病和信息等的传播速度及状态的影响,进而进行网络免疫,以使在疾病爆发或信息传播过程中根据网络结构的调整控制网络传播的方向和速度,对分析社会中涉及传播的问题提供一种新的方法。再者,针对前面提到的随机性网络结构,根据参数的不同,运用此方法来进行网络传播特性的分析,分析网络参数对网络传播速率和传播状态的影响,并且进一步分析,这些参数对网络免疫力的影响。
4、分析了经济网络中内生形成模型的动态扩展模型及动态稳定性和动态有效性。
经济网络中的内生形成模型恰当的刻画了社会经济系统中个体之间的关系,其中的稳定性定理和有效性定理也有利于一些社会经济现象的解释。但是,内生形成模型的静态性和社会经济系统的动态变化性存在着差距,这使一些理论分析存在偏差,基于此点考虑,本文在网络静态内生形成模型的基础上,分析了有新节点加入网络时,所形成的动态稳定网络和动态有效网络结构。因为内生形成模型是静态网络,在模型中定义的稳定性和有效性不适合动态网络,因此,文中首先给出了动态稳定性和动态有效性的定义,然后根据网络的形成过程:基于个体自私和短视的假定,由连接的效用和成本来决定连线的建立过程,导出在不同条件下的动态稳定网络结构和动态有效网络结构,并运用效用理论加以证明。为了进一步扩展模型的适用性,重新假设网络中的个体是异质个体(即个体之间的连接成本存在差异)的情况下网络结构的形成过程及形成的动态稳定网络结构,并运用效用理论加以证明。
Complex networks have been found many applications in a variety of fields including society, politics, medicine, economics and management, and attract more and more attentions from various fields of science and engineering. Evolving networks and epidemic dynamics are two important fields in complex networks. And, Economists and econophysicists have been starting to research on the network formation of economic systems from the rise of the complex networks. Economic networks describe the relationships and the influences among the economic agents precisely. The relationships can be the information changing among the economic agents, commodity transaction, dealing in securities, investment relation, credit relationship, membership and so on. It is neglected that the real economic systems always have a large and complex network formation in classic economic theory. It is supposed that the economic systems have the simple network formation like star network full connected network. It can partly explain why the economic theory can not resolve the economic phenomena sometimes. So, it is important to research on the economic network formation. It is a new field in economics and there are many fields need further study. But it shows the potentiality on economic networks through current progress report.
The theories and methods which contain uncertainty theory, utility theory, Game, stochastic dominance theory and the mean-filed method are used to the research on network formation in this paper. And the research contains network formation, the characters and the dynamics. It can be used to explain the characters of real-world networks through the results that received from the research. And by studying the dynamic properties of complex networks, on the one hand, we can learn the mechanism of network formation and build better network structure by using the results; on the other hand, we can apply these theoretical results to influence the formation process of real-world networks and let it to change advantaged. So, the importance of the research on network formation and the properties of networks are clearly self-evident.
The main contents and originalities in this paper can be summarized as follows:
1. The evolving network based on uncertainty theory is proposed.
It contains various uncertainties in the process of network formation. First, an evolving network model by using proving rules is presented. The proving contains both adding the uncertainty variable to preferential probability and deleting the old links. Then the dynamical equation is established via using mean-filed approach. The expression equations of degree distribution and the degree exponent are deduced both by discussing two kinds of distributing of uncertainty variable and by the different relationship between the numbers of added and deleted links. And then, the degree of fit between the theoretical result and simulation is compared. But it will have a higher error if the degree is lower, it is analyzed by Markov-based numerical method. It is well fit by comparing the numerical result and simulation. It is showed that the uncertainty evolving network can evolve to scale-free network and it will be used generally through the degree exponent and the theoretical result.
2. The influence of randomness on the network structure and the properties is studied.
There is not only preferential attachment but also random attachment in the process of network formation. And the different probability of the two kinds of attachment will lead to different network structure. The model which includes both random attachment and preferential attachment is presented. The parameters are set to the probability of two kinds of attachment and the probability of deleting, and the change of the network formation and the properties following the change of parameters is analyzed. Firstly, the equation of degree distribution, cumulative degree distribution and degree exponent are deduced via mean field approach. The influence of randomness on the degree distribution is discussed, and network stucture is different because of the change of degree distribution and degree exponent. Then, the influence of randomness on the properties of network which include average path length and the clustering coefficient is discussed, and the regularity of the influence is analyzed.
3. The epidemic dynamics and immunization in complex networks based on stochastic dominance theory is studied.
The epidemic dynamics and immunization are the important dynamic behaviors. It is to help learn properties of the spread of epidemic and gossips through studying the epidemic dynamics of network. Network structure is the important factor to influence the properties of epidemic dynamics and immunization. So, it is important to learn the influence. Firstly, the equation of the positive steady state is built and the positive steady state is discussed. Then, the epidemic speed and ratio between different network structures can be made comparison based on the stochastic dominance theory in SIS model. The diffusion speed and ration can be learned from the comparison results, and the immunization which can influenced epidemic dynamics the can be carried on from the results, all these conclusions can be used to change the epidemic property of real-world networks like epidemic and gossip broadcasting. So, It is supplied a new method to these problems. Based on the conclusion, the randomness model which is presented above is discussed. The changes of speed and the ratio of epidemic dynamics following the change of the parameters are analyzed. And then, the property of immunization changing by the parameters is also analyzed.
4. The dynamic economic models and the stability are analyzed.
The endogenous model which is one kind of economic networks is described the relationship between the agents in economic systems precisely. The stability and efficiency studied in the model can assist to explain some economic phenomenon. But, it is some difference from the static state of the model to the dynamic process of real-world economic systems and let the theoretical results show the shortage in some analysis. The dynamic model that contains new nodes adding is presented in this paper in order to solve this problem. The definition of the stability and efficiency which give in static model is not suit here, so, firstly, the new meaning is defined based on the dynamic formation process. Then the dynamic stability and the dynamic efficiency network formation are showed based on the rules of endogenous model and the hypotheses that the agent is selfish and myopic. And the process of the linking is decided by the benefit and the cost. The dynamic stability and the dynamic efficiency network formations are proved through utility theory. In order to expand the applied range of the results, the hypotheses is changed to that the agent is heterogeneous (the cost of agent is different). The dynamic stability network formation is showed in this kind of hypotheses and is proved through utility theory.
引文
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