Title: On the Performance of a Canonical Labeling for Matching Correlated Erdős-Rényi Graphs,
Abstract: 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 $O(n^2\log
n)$-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.