Seeded Graph Matching for Correlated Erdos-Renyi Graphs
Vince Lyzinski, Donniell E. Fishkind, Carey E. Priebe; 15(108):3693−3720, 2014.
Graph matching is an important problem in machine learning and pattern recognition. Herein, we present theoretical and practical results on the consistency of graph matching for estimating a latent alignment function between the vertex sets of two graphs, as well as subsequent algorithmic implications when the latent alignment is partially observed. In the correlated ErdÅs-RÃ©nyi graph setting, we prove that graph matching provides a strongly consistent estimate of the latent alignment in the presence of even modest correlation. We then investigate a tractable, restricted-focus version of graph matching, which is only concerned with adjacency involving vertices in a partial observation of the latent alignment; we prove that a logarithmic number of vertices whose alignment is known is sufficient for this restricted-focus version of graph matching to yield a strongly consistent estimate of the latent alignment of the remaining vertices. We show how Frank-Wolfe methodology for approximate graph matching, when there is a partially observed latent alignment, inherently incorporates this restricted-focus graph matching. Lastly, we illustrate the relationship between seeded graph matching and restricted-focus graph matching by means of an illuminating example from human connectomics.
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