Asymptotics In The Weighted P₀ Model Based On Differential Privacy

With the development of Internet technology,edge-weighted directed networks are widespread in daily life and work,and many scholars have further studied its statistical properties.However,the degree sequence of network graph often carries a lot of sensitive information,so the protection of privacy data has become an im-portant issue of statistical analysis.Therefore,under the premise of protecting data by differential privacy mechanism,this paper establishes a p0 model for weighted directed random graphs,and further studies its parameter properties.This paper mainly studies the problem of statistical inference in the weighted p0 network model under the differential privacy criterion,the results include:Firstly,since the object of privacy protection is the bi-degree sequence,which is also an exclusively sufficient statistic of the weighted p0 model,so a Laplace mecha-nism satisfying differential privacy is used to release the weighted directed bi-degree sequence with noise.Secondly,based on the differential privacy bi-degree sequence,the moment es-timation equation in the weighted p0 model is established to obtain the differential privacy estimator of parameter.Thirdly,we prove the consistency of the difference privacy estimator,which is expressed as follows:let q-1 be the maximum weight of the edge in the network graph,and θ be the parameter in p0 model.Let ε be the privacy parameter,and κ be the parameter of sub-exponential distribution.If(1+1(q-1)2/ε)e12||θ*||∞=Op(n1/2/(log n)1/2)where e is privacy parameter.Then as n goes to infinity,with probability approach-ing one,the estimate θ of θ exists and satisfies||θ-θ*||∞=Op(log n1/2/n1/2(1+k)e6|θ*||∞)=op(1).Further,if θ exists,it is unique.Fourthly,we prove the asymptotic normality of the differential privacy estima-tor,which is expressed as follows:assume that A～Pθ*and(1+4/(q-1)2/ε)2e18||θ*||∞=Op((n/log n)1/2).If(1+g(q-1)2/ε)2e18||θ*||∞=Op((n/log n)1/2)and e2||θ*||∞=Op(n1/2),then for any fixed k≥1,as n→∞,the first k elements of the vector θ-θ*is asymptotically multivariate normal distribution,where The parameter mean is 0,and the submatrix of S*in the upper left corner of k×k constitutes the covariance matrix of the parameter,where the matrix S*is the θ in the matrix S is replaced with the real value θ*to get.Fifthly,the theoretical results are demonstrated by numerical simulation and two practical numerical examples.