We give the first polynomial-time, differentially node-private, and robust algorithm for estimating the edge density of Erd\H{o}s-R\'enyi random graphs and their generalization, inhomogeneous random graphs. We further prove information-theoretical lower bounds, showing that the error rate of our algorithm is optimal up to logarithmic factors. Previous algorithms incur either exponential running time or suboptimal error rates. Two key ingredients of our algorithm are (1) a new sum-of-squares algorithm for robust edge density estimation, and (2) the reduction from privacy to robustness based on sum-of-squares exponential mechanisms due to Hopkins et al. (STOC 2023).

翻译：我们提出了首个多项式时间、节点差分隐私且鲁棒的算法，用于估计Erd\H{o}s-R\'enyi随机图及其推广——非齐次随机图的边密度。我们进一步证明了信息论下界，表明该算法的误差率在对数因子内达到最优。先前算法要么需要指数运行时间，要么具有次优误差率。我们算法的两个关键组成部分是：（1）一种用于鲁棒边密度估计的新平方和算法，以及（2）基于Hopkins等人（STOC 2023）提出的平方和指数机制实现的从隐私到鲁棒性的规约。