Inference In The Directed Probit Network Model

The data of the relationship network(graph)is one of the most common da-ta types in the real world.Nodes represent individuals,and edges represent the relationship between two individuals.Many network graph models,including ex-ponential random graphs,have been proposed successively,but the generation of network data is not limited to exponential distribution family.Using the cumu-lative distribution function of standard normal distribution to model the edge of the directed network,we propose Probit model for directed networks,and explores the parameter estimation of the relationship network data from the framework of non-exponential random graph model.This paper mainly discusses the problem of statistical inference in the Probit model for directed networks,the results include:Firstly,degree sequence is the basic feature of network model,which reflects the basic information of network structure.Thus,in this paper,the bi-degree sequence is selected as the statistics.Then,the moment estimation equation is established based on the bi-degree sequence to obtain the estimators of the parameters in probit model.Secondly,we prove the consistency of moment estimators,and the specific ex-pression is as follows:if adjacency matrix A?P?*,where ?*? R2n-1 and P?.denotes the probability distribution on A under the parameter ?*in a Probit mod-el,and(?),where 0<?<1/24 is a constant.Then as n goes to infinity,with probability approaching one,the moment estimator ? exists;and if ?exists,it is unique.Thirdly,we prove the asymptotically normal distribution of moment estima-tors,which are described in detail as follows:if A?P?*,(?),where 0<?<1/36 is a constant.For any fixed k? 1,as n??,the vector consisting of the first k elements of ???*is asymptotically multivariate normal.Finally,the the-oretical results are demonstrated by numerical simulation and three real numerical examples.