Limit Laws In The Generalized Random Graphs With Random Vertex Weights

Recently, the research of complex networks has become a new hotspot in-volving to many kinds of networks in nature and society science. The research greatly promotes the development of science.For about40years since1960s, ran-dom graph has been the fundamental theory of the research of complex networks. Among these kinds of random graph, the generalized random graphs with vertex weights is a good model for complex networks,because it has the properties of scale-free which is the one of the fundamental properties of complex networks. The vertex weight is deterministic or random. Van Der Hofstad R focused on the limit theorems of the number of edges and vertex degree in the situation of deterministic weights and he also paid a little attention to the situation of random weights.On the basis of his result, this text has mainly carried on discussion of the situation of random weights and got some limit theorems of the number of edges and vertex degree. First,we considered the situation that the vertex weight is i.i.d and the expectation exists, and then we got a limit theorem of the num-ber of edges.Then we considered the situation that the vertex weight is i.i.d and the distribution belongs to the domain of attraction of a stable law with index a∈(0, l)(the expectation doesn’t exist), and then we got two limit theorems of the number of edges and the vertex degree respectively.