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Although the asymptotic properties of the parameter estimator have beenderived in the $p_{0}$ model for directed graphs with the differentiallyprivate bi-degree sequence, asymptotic theory in general models is stilllacking. In this paper, we release the bi-degree sequence of directed graphsvia the discrete Laplace mechanism, which satisfies differential privacy. Weuse the moment method to estimate the unknown model parameter. We establish aunified asymptotic result, in which consistency and asymptotic normality of thedifferentially private estimator holds. We apply the unified theoretical resultto the Probit model. Simulations and a real data demonstrate our theoreticalfindings.

Asymptotic theory in a class of directed random graph models with a differentially private bi-degree sequence