Asymptotic In Undirected Random Graph Models With A Noisy Degree Sequence

In the case of differential privacy under Laplace mechanism,the asymptotic properties of parameter estimator have been derived in some special models such as-model,but under a general noisy mechanism the results are lacking.In this paper,we release the degree sequences of undirected weighted networks under a general noisy mechanism with the discrete Laplace mechanism as a special case.We establish a unified asymptotic result including the consistency and asymptotically normality of the parameter estimator.This paper mainly studies the statistical inference problem of undirected random graph model under general noise mechanism,the main result includes:Firstly,based on the general noise mechanism,the moment estimation equation is established to obtain the estimator of parameter estimation.Secondly,we prove the consistency of estimators,which is expressed as followes:Assume that ?is the parameter of model,? is the parameter of sub-exponential,and ?i(i=1,2,3)is the parameter of function.Supposed that max i=1,…n ki?k,which k=op(n1/2),and(?)(?) Then as n goes to infinity,with probability approaching one,the estimate ?exists and satisfies:(?)Thirdly,we prove the asymptotic normality of the differential privacy estimator,which is expressed as followes:If(?)M2/m3=o(n),the for any fixed k1,as n??,the vector consisting of the first k elements(B-1)1/2(?-?)is asymptotic standard multivariate normal,where(B-1)1/2=diag(v1112,…,vnn1/2).Fourthly,we apply it to the ?-model,Chung and Lu(2004)log-linear model,maximum entropy models with discrete weights.Fifthly,we give the numerical simulation in maximum entropy models with discrete weights.