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1、河北工业大学 硕士学位论文 复杂网络的统计描述与网络建模的研究 姓名：王双进 申请学位级别：硕士 专业：理论物理 指导教师：何文辰 20070601 河北工业大学硕士学位论文 i 复杂网络的统计描述与网络建模的研究复杂网络的统计描述与网络建模的研究 摘摘 要要 目前国际上已经形成了复杂性问题和复杂网络的研究热潮。自然界中存在的大量的复 杂系统如 Internet 网、计算机网络、神经系统、社会关系网络等都可以通过复杂网络加以 描述。研究这些复杂网络是为了弄清产生它们的内在机制，了解它们的演化规律，进而找 到这些网络之上的复杂行为与网络结构的关系，以便积累对支配复杂系统的自然规律的认 识。本文采
2、取从复杂网络建模入手，用统计物理和随机图论等工具对复杂网络的统计特征 与性质、结构和功能等宏观规律进行研究。 第一章首先介绍了复杂网络的研究背景与发展过程以及研究现状。所做工作集中在下 面几章。 第二章先阐述了复杂网络的度分布、平均路径长度和集聚系数等几个统计指标，再介 绍 ER 模型、WS 模型等几种早期的网络模型以及流行病的两种模型，最后描述了因特网、 生物系统中的网络和科研合作网络等几个现实中的网络。 第三章对随机图论内容进行了讨论。对于网络的研究，最早是从数学家开始的，其基 本的理论就是图论，图论是研究网络共性的有力工具。本章分别从子图、图的演化、度分 布、集聚系数和图谱等角度来分析复
3、杂网络的相关性质。 第四章对渗流模型及理论进行了研究。渗流模型是从与随机图论相似但又与其不同的 角度临界概率角度，来分析复杂网络的结构和性质的变化。本章着重介绍了渗流模型 及理论的几个重要性质、Cayley 树上的渗流和集团结构。 第五章从统计力学的角度分析和考察了无标度网络的形成机理与基本特征，介绍了无 标度网络最常用的动力学模型BA 模型。 着重分析了基于连续性理论的动力学方法 （包 括平均场和速率方程）和基于概率理论的主方程方法在无标度网络研究中的应用及其相互 关系。接下来提出了增长网络的形成机理，着重介绍由线性增长与择优连接组成的 BA 模 型，并基于马尔可夫链理论尝试了数值计算方法。
4、另外，用这种数值计算方法研究了一个 具有对数增长的加速增长模型。 关键词关键词：复杂网络 网络模型 统计特性 网络的动力学行为 随机图论 渗流理论 复杂网络的统计描述与网络建模的研究 ii THE STATISTICAL DESCRIPTION OF COMPEX NETWORKS AND THE RESEARCHES ON ESTABLISHING NETWORK MODELS ABSTRACT At present the researches on the complex question and the complex networks are on the upsurge in th
5、e international. The massive complex systems, such as Internet, the computer network, the nervous system, the social relate network and so on, exist in the nature. These complex systems are all described through the complex networks. The aim of the studies on these networks is to investigate their m
6、echanisms, understand their developing rules and find the relationship between their structure and the complex functioning processes on them, so as to accumulate the knowledge about the natural disciplinarians dominating the complex systems. This thesis starts from the establishment of network model
7、s, takes the statistical physics and random-graph theory as tools, researches the macroscopic rules of the statistical properties, the structure and the function of complex networks. The first chapter introduces the background and the developing process as well as the research status quo of complex
8、networks. Most of our works concentrate on the following several chapters. The second chapter firstly elaborates several statistical targets of complex networks, such as degree distribution, average path length and clustering coefficient etc., then introduces several kinds of early network models,su
9、ch as the ER model and the WS model etc.,introduces two kinds of the epidemic disease models as well, finally describes several real networks,such as the Internet, the living system network and the scientific research cooperation network and so on. The third chapter discusses the content of random-g
10、raph theory.Regarding the network researches, the earliest studies are started from the mathematician.Its basic theory is the graph theory which is a powerful tool of discussion on network general character. This chapter respectively analyzes the relative properties of complex networks, such as subg
11、raphs, graph evolution,degree distribution,clustering coefficient, graph spectra and so on. The fourth chapter studies the percolation model and the relative theory. The percolation theory starts from the critical probabilitys angle which is similar but different to random-graph theory, analyzes the
12、 structure of complex networks and the changes of properties. This chapter emphatically introduces several important properties of percolation model and the relative theory, percolation on a Cayley and cluster structure. The fifth chapter analyzes and inspectes the formation mechanism and the basic
13、河北工业大学硕士学位论文 iii characteristics of scale-free networks from the statistical mechanics angle, then introduces BA model which is most commonly used dynamics model in scale-free networks. We emphatically analyze the dynamic method (including Mean-field and Rate-equation) based on the Continuum theory
14、and Master-equation method based on the probability theory in the scale-free networks, research their application and reciprocity of scale-free networks. Then we propose the formation mechanism of growth networks, introduce emphatically the BA model which is composed of the linear growth and prefere
15、ntial attachment, and attempt the value computational method based on Markov-chains theory. Moreover, we study the accelerating growth model which has the logarithm growth by the value computational method. KEY WORDS: complex networks, network models, statistical properties, network dynamical behavi
16、or, random-graph theory, percolation theory 河北工业大学硕士学位论文 1 第一章 第一章 复杂网络的研究背景与现状复杂网络的研究背景与现状 1.1 复杂网络的研究现状简介 1.1 复杂网络的研究现状简介 自1735年数学家欧拉对著名的哥尼斯堡七桥问题的解答开创了图论的研究以来。 两百多年来， 对网 络理论的研究已经取得了重大进展。 上世纪兴起的复杂性科学及其复杂性研究已经成为最受关注的新兴 交叉学科之一，引起了国内外关注，并且与非线性科学及其混沌动力学的复杂性研究交错在一起，在国 际上纷纷形成了非线性科学和复杂性问题研究的热潮。 许多杰出的科学家为发展复杂性科学进行了不懈 的努力。然而迄今为止，复杂性科学尚未取得重要突破。数学家F.A.Doria教授表达了多年来从事复杂性 科学研究的科学家的共同感受,他说:“We go from complexity to perplexity(我们从复杂感到了困惑)”。也 如当代最富有
