Spectral Graph Matching and Regularized Quadratic Relaxations II

Zhou Fan, Cheng Mao, Yihong Wu, Jiaming Xu

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 a 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 Erdős-Rényi graphs with edge correlation coefficient \(1-\sigma ^2\) and average degree at least \({\text {polylog}}(n)\), we show that GRAMPA exactly recovers the latent vertex correspondence with high probability when \(\sigma \lesssim 1/{\text {polylog}}(n)\). 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.