Although a lot of approaches are developed to release network data with a differentially privacy guarantee, inference using noisy data in many network models is still unknown or not properly explored. In this paper, we release the bi-degree sequences of directed networks using the Laplace mechanism and use the $p_0$ model for inferring the degree parameters. The $p_0$ model is an exponential random graph model with the bi-degree sequence as its exclusively sufficient statistic. We show that the estimator of the parameter without the denoised process is asymptotically consistent and normally distributed. This is contrast sharply with some known results that valid inference such as the existence and consistency of the estimator needs the denoised process. Along the way, a new phenomenon is revealed in which an additional variance factor appears in the asymptotic variance of the estimator when the noise becomes large. Further, we propose an efficient algorithm for finding the closet point lying in the set of all graphical bi-degree sequences under the global $L_1$ optimization problem. Numerical studies demonstrate our theoretical findings.

中文翻译：

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具有差分私有双度序列的有向网络

尽管开发了许多方法来发布具有差异隐私保证的网络数据，但在许多网络模型中使用噪声数据进行推理仍然未知或没有得到适当的探索。在本文中，我们使用拉普拉斯机制发布有向网络的双度序列，并使用 $p_0$ 模型来推断度参数。$p_0$ 模型是一个指数随机图模型，以双度序列作为其唯一的充分统计量。我们表明，没有去噪过程的参数估计量是渐近一致且正态分布的。这与一些已知结果形成鲜明对比，即估计量的存在性和一致性等有效推理需要去噪过程。一路上，揭示了一种新现象，即当噪声变大时，估计量的渐近方差中会出现一个额外的方差因子。此外，我们提出了一种有效的算法，用于在全局 $L_1$ 优化问题下找到位于所有图形双度序列集合中的壁橱点。数值研究证明了我们的理论发现。