The real network has two characteristics: heterogeneity and homogeneity. A directed network model with covariates is proposed to analyze these two features, and the asymptotic theory of parameter Maximum likelihood estimators(MLEs) is established. However, in many practical cases, network data often carries a lot of sensitive information. How to achieve the trade-off between privacy and utility has become an important issue in network data analysis. In this paper, we study a directed $\beta$-model with covariates under differential privacy mechanism. It includes $2n$-dimensional node degree parameters $\boldsymbol{\theta}$ and a $p$-dimensional homogeneity parameter $\boldsymbol{\gamma}$ that describes the covariate effect. We use the discrete Laplace mechanism to release noise for the bi-degree sequences. Based on moment equations, we estimate the parameters of both degree heterogeneity and homogeneity in the model, and derive the consistency and asymptotic normality of the differentially private estimators as the number of nodes tends to infinity. Numerical simulations and case studies are provided to demonstrate the validity of our theoretical results.

翻译：真实网络具有异质性和同质性两个特征。本文提出了一种带协变量的有向网络模型来分析这两个特征，并建立了参数极大似然估计的渐近理论。然而，在许多实际案例中，网络数据往往包含大量敏感信息。如何在隐私与效用之间实现权衡已成为网络数据分析中的一个重要问题。本文研究差分隐私机制下带协变量的有向β模型。该模型包含2n维节点度参数θ和一个描述协变量效应的p维同质性参数γ。我们使用离散拉普拉斯机制对双度序列添加噪声。基于矩方程，我们估计了模型中节点度异质性参数和同质性参数，并推导了当节点数趋于无穷时差分隐私估计量的一致性和渐近正态性。数值模拟和案例研究验证了我们理论结果的有效性。