Although the asymptotic properties of the parameter estimator have been derived in the $p_{0}$ model for directed graphs with the differentially private bi-degree sequence, asymptotic theory in general models is still lacking. In this paper, we release the bi-degree sequence of directed graphs via the discrete Laplace mechanism, which satisfies differential privacy. We use the moment method to estimate the unknown model parameter. We establish a unified asymptotic result, in which consistency and asymptotic normality of the differentially private estimator holds. We apply the unified theoretical result to the Probit model. Simulations and a real data demonstrate our theoretical findings.

翻译：尽管在具有差分隐私双度序列的有向图$p_{0}$模型中，参数估计量的渐近性质已得到推导，但一般模型中的渐近理论仍存在空白。本文通过离散拉普拉斯机制发布有向图的双度序列，该机制满足差分隐私。我们采用矩估计方法估计未知模型参数，建立了统一的渐近结果，使差分隐私估计量的一致性及渐近正态性得以成立。我们将该统一理论结果应用于Probit模型。仿真实验与真实数据验证了我们的理论发现。