Optimal recovery of correlated Erd\H{o}s-R\'enyi graphs

TL;DR

This paper investigates the limits of recovering vertex correspondences between two correlated Erdős-Rényi graphs, establishing bounds that depend on graph parameters and revealing conditions for optimal partial recovery.

Contribution

It introduces a novel connection between graph recoverability and load balancing, providing asymptotic bounds for partial recovery in correlated Erdős-Rényi graphs.

Findings

Derived upper and lower bounds for recoverable vertex pairs

Characterized asymptotic optimal recovery fraction for most parameters

Connected graph recoverability to load distribution in intersection graph

Abstract

For two unlabeled graphs $G_1,G_2$ independently sub-sampled from an Erd\H{o}s-R\'enyi graph $\mathbf G(n,p)$ by keeping each edge with probability $s$, we aim to recover \emph{as many as possible} of the corresponding vertex pairs. We establish a connection between the recoverability of vertex pairs and the balanced load allocation in the true intersection graph of $ G_1 $ and $ G_2 $. Using this connection, we analyze the partial recovery regime where $ p = n^{-\alpha + o(1)} $ for some $ \alpha \in (0, 1] $ and $ nps^2 = \lambda = O(1) $. We derive upper and lower bounds for the recoverable fraction in terms of $ \alpha $ and the limiting load distribution $ \mu_\lambda $ (as introduced in \cite{AS16}). These bounds coincide asymptotically whenever $ \alpha^{-1} $ is not an atom of $ \mu_\lambda $. Therefore, for each fixed $ \lambda $, our result characterizes the asymptotic optimal recovery fraction for all but countably many $ \alpha \in (0, 1] $.