Analysis For The Topology Of Complex Networks: In Theory And Simulation

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.