Spectral Graph Matching and Regularized Quadratic Relaxations: Algorithm and Theory

Graph matching, also known as network alignment, aims at recovering the latent vertex correspondence between two unlabeled, edge-correlated weighted graphs. To tackle this task, we propose a spectral method, GRAph Matching by Pairwise eigen-Alignments (GRAMPA), which first constructs a similarity matrix as a weighted sum of outer products between all pairs of eigenvectors of the two graphs, and then outputs a matching by a simple rounding procedure. For a universality class of correlated Wigner models, GRAMPA achieves exact recovery of the latent matching between two graphs with edge correlation $1 - 1/\mathrm{polylog}(n)$ and average degree at least $\mathrm{polylog}(n)$. This matches the state-of-the-art guarantees for polynomial-time algorithms established for correlated Erd\H{o}s-R\'{e}nyi graphs, and significantly improves over existing spectral methods. The superiority of GRAMPA is also demonstrated on a variety of synthetic and real datasets, in terms of both statistical accuracy and computational efficiency.