Inference using noisy degrees: Differentially private $\beta$-model and synthetic graphs
研究了β模型的极大似然估计存在条件，提出了差分隐私下的参数估计方法，该估计量具有相合性和渐近正态性，可用于生成合成图并准确发布度分布，在真实图上表现优于现有算法。
The $\beta$-model of random graphs is an exponential family model with the degree sequence as a sufficient statistic. In this paper, we contribute three key results. First, we characterize conditions that lead to a quadratic time algorithm to check for the existence of MLE of the $\beta$-model, and show that the MLE never exists for the degree partition $\beta$-model. Second, motivated by privacy problems with network data, we derive a differentially private estimator of the parameters of $\beta$-model, and show it is consistent and asymptotically normally distributed—it achieves the same rate of convergence as the nonprivate estimator. We present an efficient algorithm for the private estimator that can be used to release synthetic graphs. Our techniques can also be used to release degree distributions and degree partitions accurately and privately, and to perform inference from noisy degrees arising from contexts other than privacy. We evaluate the proposed estimator on real graphs and compare it with a current algorithm for releasing degree distributions and find that it does significantly better. Finally, our paper addresses shortcomings of current approaches to a fundamental problem of how to perform valid statistical inference from data released by privacy mechanisms, and lays a foundational groundwork on how to achieve optimal and private statistical inference in a principled manner by modeling the privacy mechanism; these principles should be applicable to a class of models beyond the $\beta$-model.