Study On Complex Network Models And Its Application

Networks exist in every aspect of nature and society, the most commons are the ecosystems, the World Wide Web, human relationship networks and transportation networks, etc. The importance and urgency of understanding real networks made complex networks become a hot research area in recent years. In order to explore the behavior and function of real networks, one needs to make a scrutiny into the topological structure of real networks. The main task of this paper is to build up network models that can simulate the evolving behavior of real networks, and to work out rigorous methods for the statistical properties of networks. In this paper, rigorous methods for degree distribution are proposed from the standpoint of probability and statistics respectively. Meanwhile, some new models are presented, the statistical parameters are analyzed and simulations are also taken to verify the theoretical derivations. The main results are as follows:1,Rigorous derivations of the degree distribution of the LCD model and the B-O model are given from the probability aspect. The frequently used methods of solving degree distribution are the mean-field method, the rate equation method, the master equation method and so on. While use these methods, one needs to make some assumptions, and obtains only approximate expressions of degree distribution. Although, the LCD model and the B-O model were rigorously derived with the martingale method, the calculating processes were somewhat complicated. Moreover, the martingale method can only analyze networks with multiple edges and loops. As we know, from the aspect of probability, the degree distributions of vertices are series of Markov chains. Thus, with the relationship of degree probability and first passage probability of Markov chains, this paper gives a rigorous proof to the existence of degree distribution, and obtains the exact analytic expressions of degree distribution. Moreover, a simplified method is proposed, which also gives rigorous analysis to the degree distribution. Finally, simulations of the LCD model and the B-O model are taken. The numerical simulations and theoretical analyses show that both the two models are power-law. The degree exponent of the LCD model is 3, while that of the B-O model depends on the attractiveness.2,A new method based on a limit theorem is proposed to solve degree distribution from the point of statistics. The average number of vertices with degree k at time t is analyzed firstly. Then, the relationship of degree distributions between k and k-1 is derived with the Stolz theorem. With this relationship, the existence of the degree distribution is thus proved and the exact expression of the degree distribution is derived. This method is simple and can be widely extended to other models, such as models with multiple edges and loops. Furthermore, this method can also be easily applied to solve vertex strength distribution and edge weight distribution of weighted networks.3,Some network models with edge iterations are proposed. Vertices are correlated in most real networks, and it is difficult to describe a real system if the correlations among vertices are ignored in the evolving processes. To give expressions to correlations of vertices in the preferential process, some models based on edge iterations are given. Firstly, the network model with one edge iterated is studied. Analyses show that the model has power-law degree distribution, high clustering coefficient and the two vertex degree correlations are logarithmically increased with the network size. Secondly, the network model with one edge iterated is extended. Network models with multiple edges iterated and weighted network models with edge iterations are suggested. With the limit theorem method, we give rigorous and detailed analyses to the topological characteristics of networks. Results show that the vertex degree distribution, the vertex strength distribution and the edge weight distribution are all power-law for the weighted networks, and the models display high clustering coefficient, which possesses the essential feature of real networks.4,The deterministic model, the random model and the merging model are proposed based on collaboration networks. At first, the model is evolved with fixed number of people for each project. With the limit theorem method, the clique degree distribution and the degree distribution are studied, both with scale-free property and the degree exponents depend on the number of people involved in each project. Besides, it is find that the degree correlations grow exponentially with network size, and the model shows high clustering coefficient. However, the number of people involved is usually different from one project to another, we assume the number of people involved in a project to be a random variable, and propose the random model. The clique degree distribution and the degree distribution are similar to that of the deterministic model, but the degree exponents are related to the parameters of the random variable. Meanwhile, based on the fact that the vertices of new projects often coincide with vertices of old projects, a merging model with vertices correlated to each other is presented. The simulation results and theoretical derivations indicate that the model is scale-free, the degree exponent depends on the proportion of vertices coincided between new and old projects. In addition, the two degree correlations are related to the network size, and the model owns high clustering coefficient.