复杂网络拓扑特征的理论研究及仿真分析
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摘要
复杂网络能够描述自然和社会中的非常广泛的系统,近年来已成为国际上一个十分引人瞩目的新兴研究领域.本论文主要从概率论、图论和统计物理角度出发,系统研究了复杂网络的一些重要拓扑特征.
本文的主要研究工作如下：
第一章中,介绍了本论文的研究背景、预备知识及本文所作的主要工作.
第二章中,从概率论角度出发,讨论了复杂网络稳态度分布的存在性及相关问题.根据复杂网络的定义和性质建立网络马氏链,进而利用马氏链理论中首达概率的方法和技巧为增长网络稳态度分布的存在性提供严格证明.
文中首先分析了一类允许重连的经典模型——DMS模型,为该网络稳态度分布的存在性提供了严格证明.另外,文中进一步探讨了一类一般的网络模型——修正Cooper-Frieze模型.我们证明了该模型稳态度分布的存在性,并通过数值仿真,将该模型的度分布及聚集性与BA模型进行了对比分析.
第三章中,主要研究了群体择优模型的一些重要拓扑特征及相应的仿真分析.群体择优模型为复杂网络研究引进了一种新的择优思想——群体择优思想.文中着重讨论了两类群体择优模型-无权群体择优模型和加权群体择优模型的拓扑特征及同步分析.
首先,对无权群体择优模型,我们利用率方程方法着重分析了其度分布、度相关性和聚集性.并且在文中我们为该模型提供了广泛的仿真分析,且得出仿真结果与解析结果完全吻合.其次,对加权群体择优模型,我们主要分析了其度分布、点权及边权分布,研究了群体择优思想在演化过程中对加权网络拓扑结构的影响,并将其结果与BBV模型作了比较分析.另外,我们还进一步对群体择优模型作了同步分析,并讨论了网络在随机故障和恶意攻击情形下的同步能力.
在文章最后两章中,主要利用复杂网络的研究方法,对社会网络及自然网络作了一些应用分析.
其中在第四章中,我们研究了财富网络模型的拓扑结构及财富分布.根据社会个体和组织之间经济关系的特点,通过考虑财富重分和择优机制等因素对网络结构的影响,我们建立了对应的财富网络模型.文中着重考虑了两类财富网络模型,其中主要对模型的度分布及其他一些重要拓扑特征作了理论研究和仿真分析.
在第五章中,我们研究了一类生物网络模型的拓扑结构及功能.通过引进新的择优思想和演化机制,我们提出了一类新的蛋白质结构域相互作用网络模型.同时,主要研究了该类蛋白质结构域相互作用网络的一些重要拓扑特征,如度分布、聚集性以及最短路径长度等等.
Complex network has become a very hot field of research in recent years, which can describe very broad systems both in nature and society. In this paper, we systematically study the topological characteristic of complex networks by using Markov chain theory, knowledge of Graph theory and Statistical Physics. The paper is organized as follows.
In chapter 1, we introduce the research background of this paper, some basic con-cepts and theories, and our main work in this paper.
In chapter 2, from the point of Probability theory, we study the existence of steady-state degree distribution of complex networks. Based on the definition and properties of complex network, we first construct network Markov chain. Then by using the first passage probability in probability theory, we provide a strict proof for the existence of the degree distribution of growing networks. In this chapter, we firstly consider a classical model, the DMS model, and provide a strict proof for the existence of the degree distribution.
Besides, we further study a more general model, the modified Cooper-Frieze model. We prove the existence of the degree distribution, a comparison with the BA model on degree distribution and clustering is also provided.
In chapter 3, we discuss some important topological characteristics of Group models, and the corresponding simulations are also provided. In the Group models, it introduces a new concept to realize the preferential attachment in the research of complex network, and it is a holistic approach. In this chapter, we discuss two kinds of Group models, un-weighted and weighted.
For the first kind of model, we consider the two-node correlation and three-node correlation by using the rate equation. And all analytical solutions are successfully contrasted with computer simulations.
What's more, we mainly research the degree distribution, weight distribution for the weighted Group model. The influence of group preference mechanism on weighted network is also studied. Besides, we study the synchronization phenomenon for both these two models, and discuss the influence of random removal of nodes and special removal of the most highly connected nodes on the synchronizability of the two Group models.
Chapter 4 and 5 are mainly the application of complex networks in real social and natural networks.
In chapter 4, we consider the topological structure in wealth network. Wealth is the main target of the economic activities of individuals and agents, and is also one of the most powerful motives for human relationship in a society. Based on the economic relations between the agents and organizations, we propose a general wealth model, which permits local preferential attachment and local redistribution of wealth. We research the degree distribution and wealth distribution, and simulations for some important properties are also provided.
And in chapter 5, the analysis of the topological structure and function for biology network is provided. By introducing of new ideas and evolving mechanism, we propose a new kind of the protein domain network. We mainly consider the topological properties of the protein domain network, such as degree distribution, clustering and the shortest path problem.
本文的主要研究工作如下：
第一章中,介绍了本论文的研究背景、预备知识及本文所作的主要工作.
第二章中,从概率论角度出发,讨论了复杂网络稳态度分布的存在性及相关问题.根据复杂网络的定义和性质建立网络马氏链,进而利用马氏链理论中首达概率的方法和技巧为增长网络稳态度分布的存在性提供严格证明.
文中首先分析了一类允许重连的经典模型——DMS模型,为该网络稳态度分布的存在性提供了严格证明.另外,文中进一步探讨了一类一般的网络模型——修正Cooper-Frieze模型.我们证明了该模型稳态度分布的存在性,并通过数值仿真,将该模型的度分布及聚集性与BA模型进行了对比分析.
第三章中,主要研究了群体择优模型的一些重要拓扑特征及相应的仿真分析.群体择优模型为复杂网络研究引进了一种新的择优思想——群体择优思想.文中着重讨论了两类群体择优模型-无权群体择优模型和加权群体择优模型的拓扑特征及同步分析.
首先,对无权群体择优模型,我们利用率方程方法着重分析了其度分布、度相关性和聚集性.并且在文中我们为该模型提供了广泛的仿真分析,且得出仿真结果与解析结果完全吻合.其次,对加权群体择优模型,我们主要分析了其度分布、点权及边权分布,研究了群体择优思想在演化过程中对加权网络拓扑结构的影响,并将其结果与BBV模型作了比较分析.另外,我们还进一步对群体择优模型作了同步分析,并讨论了网络在随机故障和恶意攻击情形下的同步能力.
在文章最后两章中,主要利用复杂网络的研究方法,对社会网络及自然网络作了一些应用分析.
其中在第四章中,我们研究了财富网络模型的拓扑结构及财富分布.根据社会个体和组织之间经济关系的特点,通过考虑财富重分和择优机制等因素对网络结构的影响,我们建立了对应的财富网络模型.文中着重考虑了两类财富网络模型,其中主要对模型的度分布及其他一些重要拓扑特征作了理论研究和仿真分析.
在第五章中,我们研究了一类生物网络模型的拓扑结构及功能.通过引进新的择优思想和演化机制,我们提出了一类新的蛋白质结构域相互作用网络模型.同时,主要研究了该类蛋白质结构域相互作用网络的一些重要拓扑特征,如度分布、聚集性以及最短路径长度等等.
Complex network has become a very hot field of research in recent years, which can describe very broad systems both in nature and society. In this paper, we systematically study the topological characteristic of complex networks by using Markov chain theory, knowledge of Graph theory and Statistical Physics. The paper is organized as follows.
In chapter 1, we introduce the research background of this paper, some basic con-cepts and theories, and our main work in this paper.
In chapter 2, from the point of Probability theory, we study the existence of steady-state degree distribution of complex networks. Based on the definition and properties of complex network, we first construct network Markov chain. Then by using the first passage probability in probability theory, we provide a strict proof for the existence of the degree distribution of growing networks. In this chapter, we firstly consider a classical model, the DMS model, and provide a strict proof for the existence of the degree distribution.
Besides, we further study a more general model, the modified Cooper-Frieze model. We prove the existence of the degree distribution, a comparison with the BA model on degree distribution and clustering is also provided.
In chapter 3, we discuss some important topological characteristics of Group models, and the corresponding simulations are also provided. In the Group models, it introduces a new concept to realize the preferential attachment in the research of complex network, and it is a holistic approach. In this chapter, we discuss two kinds of Group models, un-weighted and weighted.
For the first kind of model, we consider the two-node correlation and three-node correlation by using the rate equation. And all analytical solutions are successfully contrasted with computer simulations.
What's more, we mainly research the degree distribution, weight distribution for the weighted Group model. The influence of group preference mechanism on weighted network is also studied. Besides, we study the synchronization phenomenon for both these two models, and discuss the influence of random removal of nodes and special removal of the most highly connected nodes on the synchronizability of the two Group models.
Chapter 4 and 5 are mainly the application of complex networks in real social and natural networks.
In chapter 4, we consider the topological structure in wealth network. Wealth is the main target of the economic activities of individuals and agents, and is also one of the most powerful motives for human relationship in a society. Based on the economic relations between the agents and organizations, we propose a general wealth model, which permits local preferential attachment and local redistribution of wealth. We research the degree distribution and wealth distribution, and simulations for some important properties are also provided.
And in chapter 5, the analysis of the topological structure and function for biology network is provided. By introducing of new ideas and evolving mechanism, we propose a new kind of the protein domain network. We mainly consider the topological properties of the protein domain network, such as degree distribution, clustering and the shortest path problem.
引文
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