Limit Laws For The Subgraph Counts In The Random Dot Product Graphs

In recent years,complex networks have attracted more and more attention from researchers in various fields of science.The objects of study include various networks in the natural and social sciences,which play a crucial role in the development of science.When facing complex network,a new and dynamic large-scale relational data,the theory of random graph and the complex network theory derived from it have attracted more and more researchers attention.It has made great contributions to the study of small-world phenomena,clustering relations and power-law behaviors in complex networks.Based on a kind of important random graph model-random dot product graph,this paper makes a theoretical study of random dot product graph in dense and sparse cases respectively.Specifically,we provide the limit theorem of the number of edges in random dot product graph,and obtains the convergence theorem in probability and the central limit theorem of edges in these two cases.In addition,we investigate the asymptotic behavior for the number of triangles in a random dot product graphs with random vertex vectors.when the number of vertices tends to infinity,we show that the asymptotic distribution of the triangle number converges to a Poisson distribution with parameter related to the second moments of vertex vectors.