Spectral clustering for divide-and-conquer graph matching

We present a parallelized bijective graph matching algorithm that leverages seeds and is designed to match very large graphs. Our algorithm combines spectral graph embedding with existing state-of-the-art seeded graph matching procedures. We justify our approach by proving that modestly correlated, large stochastic block model random graphs are correctly matched utilizing very few seeds through our divide-and-conquer procedure. We also demonstrate the effectiveness of our approach in matching very large graphs in simulated and real data examples. V. Lyzinski, D.L. Sussman, D.E. Fishkind, H. Pao, L. Chen, J.T. Vogelstein, Y. Park, C.E. Priebe, “Spectral Clustering for Divide-and-Conquer Graph Matching,” Parallel Computing, accepted for publication, 2015. V. Lyzinski Background Given two graphs, G1 = (V1, E1) and G2 = (V2, E2), the Graph Matching Problem (GMP) seeks an alignment between the vertex sets V1 and V2 that best preserves structure across the graphs. In bijective graph matching, we further assume |V1| = |V2| = n, and the alignment sought by GMP is a bijection between V1 and V2. Graph Matching Problem Find a bijection ψ : V1 → V2 minimizing the quantity ∣∣{ (i, j) ∈ V1 ×V1 s.t. [i ∼G1 j, ψ(i) G2 ψ(j)] or [i G1 j, ψ(i) ∼G2 ψ(j)]}∣∣, (1) i.e. the problem seeks to minimize the number of edge disagreements between G2 and “ψ(G1)”. Equivalently stated, if A and B are the respective adjacency matrices of G1 and G2, then this problem seeks to minimize ‖A − PBP‖F, over all permutation matrices P∈Π(n) := {n× n permutation matrices}, with ‖ · ‖F the matrix Frobenius norm. 3 / 10 Background In the seeded graph matching problem (SGMP), we further assume the presence of a latent alignment φ between the vertex sets of G1 and G2. Our task is to then efficiently leverage the information in a partial observation of the latent alignment, i.e. a seeding, to estimate the remaining latent alignment. Seeded Graph Matching Problem Given subsets of the vertices S1 ⊂ V1 and S2 ⊂ V2 called seeds with |S1| = |S2| = s and a bijective seeding function φS : S1 → S2, the task is to use φS to estimate φ by finding the bijection extending φS which minimizes (1).