Robust Estimation for Random Graphs

TL;DR

This paper introduces robust spectral algorithms for estimating the edge probability in Erdős-Rényi graphs with adversarial node corruption, achieving near-optimal accuracy within certain corruption levels.

Contribution

The paper presents a computationally-efficient spectral method for robustly estimating graph parameters under adversarial corruption, improving upon canonical estimators and approaching theoretical limits.

Findings

Spectral algorithm estimates $p$ with accuracy depending on $ ilde O(rac{ ext{sqrt}(p(1-p))}{n} + rac{ ext{corruption}}{ ext{sqrt}(n)})$ for $ ext{corruption} < 1/60$.

An inefficient algorithm achieves similar accuracy for all $ ext{corruption} < 1/2$, matching the information-theoretic limit.

Proven lower bounds show the algorithms' errors are nearly optimal up to logarithmic factors.

Abstract

We study the problem of robustly estimating the parameter $p$ of an Erd\H{o}s-R\'enyi random graph on $n$ nodes, where a $\gamma$ fraction of nodes may be adversarially corrupted. After showing the deficiencies of canonical estimators, we design a computationally-efficient spectral algorithm which estimates $p$ up to accuracy $\tilde O(\sqrt{p(1-p)}/n + \gamma\sqrt{p(1-p)} /\sqrt{n}+ \gamma/n)$ for $\gamma < 1/60$. Furthermore, we give an inefficient algorithm with similar accuracy for all $\gamma <1/2$, the information-theoretic limit. Finally, we prove a nearly-matching statistical lower bound, showing that the error of our algorithms is optimal up to logarithmic factors.