Asymptotics In Undirected Random Graphs With Covariate

In this paper,on the basis of Graham(2017),we study the consistency of maximum likelihood estimation of degree sequence parameter ? and the asymptotic normality of ?(?*)under the condition that the covariable parameter is true,and the consistency of maximum likelihood estimation of degree sequence parameter ?has been studied by Graham et al.on the basis of undirected network graph model with covariates.However,it is difficult to obtain the true value ?*of covariable parameter in real life,so it is necessary to study the asymptotic theory of ?(?*),which replaces the real value ?*of covariable parameter with its estimation ?.In this paper,we considered when the number of nodes in undirected graph with covariates tends to infinity,the central limit theorem of parameter maximum likelihood estimation is established,and the research results are proved by numerical simulation and real data.The results of this paper are as follows:??.Assume that ?*and A?P?*,?*,where P?*,?*denotes the probability distribution P(aij=1)=exp(ZijT?+?i+?j)/1+exp(Zij?+?i+?j)on A under the true paraneters ?*and?*.If ||?*||??<L,then for any fixed r>1,as n ??,the vector {v111/2(?1-?1*),...,vrr1/2(?r-?r*)} converges in distribution to the r-dimensional standardized multivariate normal distribution.