The Complex Network Theory And The Application To The Study Of Biological Network

In this paper, in order to explore further the mechanism responsible for complex networks, we extend the binomial random graph model and get a general model on which the connected probability between each pair of vertices follows by geometric-like distribution. With the theory of stochastic analysis, we analyze the geometric properties of this model and derive analytical expressions for the distribution of the degree P(k) and the clustering coefficient of the extended model. The part of the work is an extension to the study of the static statistical characteristic of the classical random graph model. Furthermore, recently study appears that the weight is very important in anglicizing the characteristic of the networks. So in the second part, in order to explore mechanism responsible for scale-free phenomena in weighted networks, we present a generalized model of weighted networks in which the system growth incorporated eight operations: weights'growth, weights'decrease, intrinsic strength growth and decrease, the addition of new links between the existing nodes, the rewiring and deleting links. In particular, the model yields scale-free behaviors for the weight, strength and degree distributions, meanwhile we get the analytical expressions of the strength and degree distributions through analytical method. Furthermore, we investigate the strength and degree distributions of the uniformly selected model, and find the preferential attachment is necessary to the emergence of scale-free phenomena in weighted networks. Finally, we found the analytical results are consistent with the numerical simulations. Due to the biological network is an effective method to studying the phenomenon of biology from the system and the global perspective. In final, we firstly construct a biological network to reflect the interaction of the proteins in the cells. Then using the theory of the graph-valued Markov process and derived process of random network processes we get the recurrence equation of the node degree distribution in the model. Base on the equation, we obtain the conclusion that the degree distribution following the power law. Meanwhile,we verify the reasonable of the model by the numerical simulation.