A standing challenge in data privacy is the trade-off between the level of privacy and the efficiency of statistical inference. Here we conduct an in-depth study of this trade-off for parameter estimation in the $\beta$-model (Chatterjee, Diaconis and Sly, 2011) for edge differentially private network data released via jittering (Karwa, Krivitsky and Slavkovi\'c, 2017). Unlike most previous approaches based on maximum likelihood estimation for this network model, we proceed via method of moments. This choice facilitates our exploration of a substantially broader range of privacy levels -- corresponding to stricter privacy -- than has been to date. Over this new range we discover our proposed estimator for the parameters exhibits an interesting phase transition, with both its convergence rate and asymptotic variance following one of three different regimes of behavior depending on the level of privacy. Because identification of the operable regime is difficult to impossible in practice, we devise a novel adaptive bootstrap procedure to construct uniform inference across different phases. In fact, leveraging this bootstrap we are able to provide for simultaneous inference of all parameters in the $\beta$-model (i.e., equal to the number of vertices), which would appear to be the first result of its kind. Numerical experiments confirm the competitive and reliable finite sample performance of the proposed inference methods, next to a comparable maximum likelihood method, as well as significant advantages in terms of computational speed and memory.
翻译:数据隐私方面的一个长期挑战是,在数据保密性水平和统计推算效率之间的权衡。在这里,我们深入研究了在美元模型(Chatterjee, Diaconis和Sly,2011年)中,对通过对口系统(Karwa, Krivitsky 和 Slavkovi\'c, 2017年)发布的边缘有差异的私人网络数据进行参数估算的这一权衡(Chatterjee, Diaconis和Sly,2011年)中,对数据保密性水平与统计推算效率之间的权衡。与以往基于对网络模型最大可能性估计的多数方法不同,我们通过时间方法进行。这一选择有助于我们探索比以往更为广泛的隐私水平 -- -- 相对更为严格的隐私程度 -- -- 而不是以往。在此新范围中,我们发现我们提议的参数估算标准估算值的计算值显示一个有趣的阶段过渡性过渡,即其趋近的趋近速度,其排序结果将显示其最接近的精确性,其最高性值将显示其最高性能的精确性度,其最高性值的计算结果将显示为美元值的准确性。