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The real network has two characteristics: heterogeneity and homogeneity. Adirected network model with covariates is proposed to analyze these twofeatures, and the asymptotic theory of parameter Maximum likelihoodestimators(MLEs) is established. However, in many practical cases, network dataoften carries a lot of sensitive information. How to achieve the trade-offbetween privacy and utility has become an important issue in network dataanalysis. In this paper, we study a directed $\beta$-model with covariatesunder differential privacy mechanism. It includes $2n$-dimensional node degreeparameters $\boldsymbol{\theta}$ and a $p$-dimensional homogeneity parameter$\boldsymbol{\gamma}$ that describes the covariate effect. We use the discreteLaplace mechanism to release noise for the bi-degree sequences. Based on momentequations, we estimate the parameters of both degree heterogeneity andhomogeneity in the model, and derive the consistency and asymptotic normalityof the differentially private estimators as the number of nodes tends toinfinity. Numerical simulations and case studies are provided to demonstratethe validity of our theoretical results.

Differential privacy statistical inference for a directed graph network model with covariates