Although the theoretical properties in the $p_0$ model based on a differentially private bi-degree sequence have been derived, it is still lack of a unified theory for a general class of directed network models with the $p_{0}$ model as a special case. We use the popular Laplace data releasing method to output the bi-degree sequence of directed networks, which satisfies the private standard--differential privacy. The method of moment is used to estimate unknown parameters. We prove that the differentially private estimator is uniformly consistent and asymptotically normal under some conditions. Our results are illustrated by the Probit model. We carry out simulation studies to illustrate theoretical results and provide a real data analysis.

翻译：尽管基于差分隐私双度序列的$p_0$模型的理论性质已被推导，但对于以$p_0$模型为特例的一般有向网络模型族仍缺乏统一理论。我们采用广泛使用的拉普拉斯数据发布方法输出有向网络的双度序列，该方法满足隐私标准——差分隐私。采用矩估计法估计未知参数。我们证明，在特定条件下，差分隐私估计量具有一致性和渐近正态性。通过Probit模型验证理论结果，并开展仿真实验与真实数据分析以支撑理论结论。