Markov Chain Application On Topological Structure Of Complex Networks

Recently, complex networks attract more and more attentions from various fields. In short, complex networks are very complicated networks and present complex topological structure and dynamics action. In our life and scientific research, many natural and man-made networks are complex networks. For example, Internet network, collaboration network, biology network etc. Therefore, it is necessary to profound research properties of complex networks, for better to design fact networks. The main task of this paper is to bulid up network models that can simulate the evolving behavior of real networks, and to work out methods for the statistical properties of network. Rigorous methods for degree distribution are proposed by using Probability theory, Graph theory and Statistical Physics. Meanwhile, some new models are presented, and systematically study the topological characteristic of these complex networks. These studies are very important both in theory and in practical applications.The paper is organized as follows:In chapter 1, we introduce the background, status quo and significance in researching of complex networks, and our main work in this paper.In chapter 2, first, some important characteristic parameters in complex networks are introduced, such as degree disturbution, correlation, clustering coefficient, average path length and assortative coefficient and so on. Second, we recommend several methods in studying complex networks. For example, mean-field method, rate equation method, master equation method, Markov chain method and so on.In chapter 3, we mainly research the degree disturbution, degree correlation and clustering coefficient for attachment number is random varible's generalized collaboration networks. We divide the generalized collaboration networks into 3 kinds accord attachment type:random attachment network, preferential attachment network and mixed attachment network. Moreover, the degree distribution of preferential attachment network and mixed attachment network is scale-free network.Finally, we investigate a mixed attachmen evolving network model, incorporating the mixed attachment, and the removals of links. Based on Markov chain theory, paper provides a rigorous proof for the existence of the steady-state degree distribution of the network and gets its corresponding exact formulas and show that the model can generate scale-free network.