Seeded graph matching for large stochastic block model graphs

Graph matching is an increasingly important problem in inferential graph statistics. There are no known efficient exact graph matching algorithms, though current approximate algorithms achieve excellent performance on numerous benchmarks, with complexity O(n^3) (n the number of vertices to be matched). Herein, we present a novel approximate seeded graph matching algorithm specifically designed to match very large graphs. Our algorithm, the LSGM algorithm, combines spectral graph embedding with existing state-of-the-art seeded graph matching procedures. We prove that modestly correlated, large stochastic block model random graphs are correctly matched through the joint procedure of spectral embedding and graph matching utilizing very few seeds. We show that under very mild conditions, our algorithm has complexity O(dn^2), with potential for significantly faster speed in the sparse regime, ($d$ the embedding dimension), and demonstrate the effectiveness of LSGM in recovering the unknown alignment in simulated and real data examples.