Spectral Graph Matching and Regularized Quadratic Relaxations II: Erd\H{o}s-R\'enyi Graphs and Universality

TL;DR

This paper proves the universality of spectral graph matching guarantees for Erdős-Rényi graphs using the GRAMPA algorithm, extending previous results from Gaussian weights to a broader class of models.

Contribution

It establishes the universality of exact recovery guarantees for the GRAMPA algorithm on Erdős-Rényi graphs, generalizing prior Gaussian-specific results.

Findings

GRAMPA achieves exact recovery with high probability for Erdős-Rényi graphs when edge correlation is sufficiently high.

A variant of GRAMPA based on quadratic programming also guarantees recovery under similar conditions.

The analysis employs resolvent representations and local laws for sparse Wigner matrices.

Abstract

We analyze a new spectral graph matching algorithm, GRAph Matching by Pairwise eigen-Alignments (GRAMPA), for recovering the latent vertex correspondence between two unlabeled, edge-correlated weighted graphs. Extending the exact recovery guarantees established in the companion paper for Gaussian weights, in this work, we prove the universality of these guarantees for a general correlated Wigner model. In particular, for two Erd\H{o}s-R\'enyi graphs with edge correlation coefficient $1-\sigma^2$ and average degree at least $\operatorname{polylog}(n)$, we show that GRAMPA exactly recovers the latent vertex correspondence with high probability when $\sigma \lesssim 1/\operatorname{polylog}(n)$. Moreover, we establish a similar guarantee for a variant of GRAMPA, corresponding to a tighter quadratic programming relaxation of the quadratic assignment problem. Our analysis exploits a resolvent representation of the GRAMPA similarity matrix and local laws for the resolvents of sparse Wigner matrices.