Private Edge Density Estimation for Random Graphs: Optimal, Efficient and Robust

TL;DR

This paper introduces the first efficient, differentially private algorithm for estimating edge density in Erdős-Rényi and inhomogeneous random graphs, achieving near-optimal accuracy with polynomial runtime.

Contribution

It presents a novel polynomial-time, differentially private algorithm for edge density estimation with robustness, improving over previous methods with exponential time or suboptimal error.

Findings

Algorithm is differentially private and robust

Error rate is optimal up to logarithmic factors

Runs in polynomial time

Abstract

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).