How to achieve the tradeoff between privacy and utility is one of fundamental problems in private data analysis.In this paper, we give a rigourous differential privacy analysis of networks in the appearance of covariates via a generalized $\beta$-model, which has an $n$-dimensional degree parameter $\beta$ and a $p$-dimensional homophily parameter $\gamma$.Under $(k_n, \epsilon_n)$-edge differential privacy, we use the popular Laplace mechanism to release the network statistics.The method of moments is used to estimate the unknown model parameters. We establish the conditions guaranteeing consistency of the differentially private estimators $\widehat{\beta}$ and $\widehat{\gamma}$ as the number of nodes $n$ goes to infinity, which reveal an interesting tradeoff between a privacy parameter and model parameters. The consistency is shown by applying a two-stage Newton's method to obtain the upper bound of the error between $(\widehat{\beta},\widehat{\gamma})$ and its true value $(\beta, \gamma)$ in terms of the $\ell_\infty$ distance, which has a convergence rate of rough order $1/n^{1/2}$ for $\widehat{\beta}$ and $1/n$ for $\widehat{\gamma}$, respectively. Further, we derive the asymptotic normalities of $\widehat{\beta}$ and $\widehat{\gamma}$, whose asymptotic variances are the same as those of the non-private estimators under some conditions. Our paper sheds light on how to explore asymptotic theory under differential privacy in a principled manner; these principled methods should be applicable to a class of network models with covariates beyond the generalized $\beta$-model. Numerical studies and a real data analysis demonstrate our theoretical findings.

翻译：如何在隐私保护与数据效用之间取得平衡是隐私数据分析的基本问题之一。本文通过广义β模型对存在协变量的网络进行了严格的差分隐私分析，该模型包含n维度参数β和p维同质性参数γ。在(k_n, ε_n)-边差分隐私约束下，我们采用广泛应用的拉普拉斯机制发布网络统计量，并通过矩估计方法估计未知模型参数。我们建立了保证差分隐私估计量̂β和̂γ相合性的条件（当节点数n趋于无穷时），这揭示了隐私参数与模型参数之间有趣的权衡关系。通过应用两阶段牛顿法获得(̂β, ̂γ)与其真实值(β, γ)之间ℓ∞距离误差的上界来证明相合性，其中̂β的收敛速度约为1/n^{1/2}量级，而̂γ约为1/n量级。进一步，我们推导了̂β和̂γ的渐近正态性，在特定条件下其渐近方差与非隐私估计量相同。本文系统阐明了如何在差分隐私框架下探索渐近理论，这些方法论应适用于广义β模型之外的一类含协变量的网络模型。数值模拟与真实数据分析验证了我们的理论发现。