Spectral Graph Matching and Regularized Quadratic Relaxations II: Erdős-Rényi Graphs and Universality
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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 Erdős-Rényi graphs with edge correlation coefficient 1-σ^2 and average degree at least polylog(n), we show that GRAMPA exactly recovers the latent vertex correspondence with high probability when σ≲ 1/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.
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