  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.
