Aligning random graphs with a sub-tree similarity message-passing algorithm

TL;DR

This paper introduces a message-passing algorithm for aligning sparse Erdős-Rényi random graphs and its extension to graphs with prescribed degree distributions, demonstrating its effectiveness through extensive simulations.

Contribution

The paper presents a novel polynomial-time message-passing algorithm for graph alignment in the sparse regime and extends it to more general correlated random graphs.

Findings

Algorithm achieves partial recovery in certain parameter ranges

Effective in sparse Erdős-Rényi graphs with constant average degree

Extended to correlated graphs with prescribed degree distributions

Abstract

The problem of aligning Erd\"os-R\'enyi random graphs is a noisy, average-case version of the graph isomorphism problem, in which a pair of correlated random graphs is observed through a random permutation of their vertices. We study a polynomial time message-passing algorithm devised to solve the inference problem of partially recovering the hidden permutation, in the sparse regime with constant average degrees. We perform extensive numerical simulations to determine the range of parameters in which this algorithm achieves partial recovery. We also introduce a generalized ensemble of correlated random graphs with prescribed degree distributions, and extend the algorithm to this case.