On Seeded Subgraph-to-Subgraph Matching: The ssSGM Algorithm and Matchability Information Theory

The subgraph-subgraph matching problem is, given a pair of graphs and a positive integer <i>K</i>, to find <i>K</i> vertices in the first graph, <i>K</i> vertices in the second graph, and a bijection between them, so as to minimize the number of adjacency disagreements across the bijection; it is “seeded” if some of this bijection is fixed. The problem is intractable, and we present the ssSGM algorithm, which uses Frank-Wolfe methodology to efficiently find an approximate solution. Then, in the context of a generalized correlated random Bernoulli graph model, in which the pair of graphs naturally have a core of <i>K</i> matched pairs of vertices, we provide and prove mild conditions for the subgraph-subgraph matching problem solution to almost always be the correct <i>K</i> matched pairs of vertices. Supplementary materials for this article are available online.