Differentially private analysis of networks with covariates via a generalized $β$-model

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.