Differential privacy statistical inference for a directed graph network model with covariates

Network data typically contain sensitive relational information, where direct release or sharing may lead to non-negligible privacy violations without proper statistical safeguards. While differential privacy has emerged as a powerful framework for privacy-preserving network data analysis, theoretical understanding remains limited particularly for models incorporating both network structure and nodal attributes. This paper bridges this gap by investigating a directed $\beta$-model with covariates under differential privacy constraints. Our model accounts for both node-level heterogeneity (via $2n$-dimensional degree parameters $\theta$ ) and covariate-driven homogeneity (via a $p$-dimensional parameter $\gamma$). To protect privacy, we introduce a joint Laplace mechanism for releasing network statistics while satisfying differential privacy constraints. Leveraging moment-based estimation techniques, we estimate the parameters of both degree heterogeneity and homogeneity and derive the consistency and asymptotic normality of the differentially private estimators as the network size tends to infinity. Our theoretical findings are validated through numerical simulations and real-world case studies, demonstrating the validity of our theoretical results.