Markov Chain Application On Some Collboration Networks

Many natural and man-made networks are complex networks. For example, Internet network, WWW network, collaboration network, biology network etc. These networks have affinity in our life, therefore it is necessary to profound research and comprehend topology structure runmechanism, dynamics action, synchronization ability, anti-amming ability of complex nelworks, for bebetter to design and manage these 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 rigorous methods for the statistical properties of network. Rigorous methods for degree distribution are proposed by using Markov chain theory, knowledge of Graph theory and Statistical Physics. Meanwhile, some new models are presented, and systematically study the topological characteristic of these complex networks.The paper is organized as follows:In chapter 1, we introduce the background and evolvement in researching of complex networks, and our main work in this paper.In chapter 2, first, we introduce some important characteristic parameters in complex networks, such as degree disturbution, degree correlation, clustering coefficient and average path length and so on. Second, we present three well-known network models. For instance, random graph, small world network, and scale-free network. Third, we recommend several methods in studying complex networks. For example, mean-field method, rate equation method, master equation method, martingale method and Markov chain method.In chapter 3, we discuss a evolving model of collaboration networks, where the act-size is fixed. Based on the first-passage probability of Markov chain theory, this chapter provides a rigorous proof for the existence of a limiting degree distribution of this model and proves that the degree distribution obeys the power-law, further, we obtain the clustering coefficient, average path length and degree correlation function, and we found the network model is small world network.In chapter 4, we mainly research the degree disturbution, degree correlation clustering coefficient for generalized collaboration networks. Especially, we divide the generalized collaboration networks into 3 kinds by 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, and the exponent is adjustable.Finally, we investigate a special evolving network model, incorporating the additions of new nodes, new links, 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 generated by this model and gets its corresponding exact formulas and show that the model can generate scale-free evolving network.