A Possion Network Model With Covariates

In recent years,network data are common in a wide variety of field.Most of literatures focus on the analysis of binary edges(e.g.,appear or disappear),while most of edges are weighted.Many networks can be described by two significant features.One of the important characteristics is the node heterogeneity which nodes exhibit different degrees of interaction.For example,different people have different circles of friends.Another important feature is homogeneity which nodes sharing the same features tend to associate with each other.For example,people of the same age are more likely to be friends.In this paper,we study the Poisson distribution model in undirected graph,whose edge value is infinite discrete.Our model captures degree heterogeneity via node-specific parametrization,which is measured by the degree vetor ?,and ho-mogeneity by incorporating covariates,which is the isomorphic coefficient ?.This paper mainly studies the statistical inference of degree parameters and homogeneity parameters in Poisson distribution.The study results are as follow:First,under the Poisson distribution model,the maximum likelihood estimator of model parameters is studied.Second,under certain conditions,the consistency of MLE is proved,? and ?exists and satisfies||?-?||?=op(1),and||?-?||?=op(1).Third,under certain conditions,the asymptotic normality of the MLE ? and? is proved.Fourth,numerical simulation demonstrate our theoretical findings and one data analyse cinfirm the usefulness of our model.