On the Performance of a Canonical Labeling for Matching Correlated Erdős-Rényi Graphs
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Graph matching in two correlated random graphs refers to the task of identifying the correspondence between vertex sets of the graphs. Recent results have characterized the exact information-theoretic threshold for graph matching in correlated Erdős-Rényi graphs. However, very little is known about the existence of efficient algorithms to achieve graph matching without seeds. In this work we identify a region in which a straightforward O(n^2 n)-time canonical labeling algorithm, initially introduced in the context of graph isomorphism, succeeds in matching correlated Erdős-Rényi graphs. The algorithm has two steps. In the first step, all vertices are labeled by their degrees and a trivial minimum distance matching (i.e., simply sorting vertices according to their degrees) matches a fixed number of highest degree vertices in the two graphs. Having identified this subset of vertices, the remaining vertices are matched using a matching algorithm for bipartite graphs.
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