  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.
