  In this paper, matching pairs of random graphs under the community structure
model is considered. The problem emerges naturally in various applications such
as privacy, image processing and DNA sequencing. A pair of randomly generated
labeled graphs with pairwise correlated edges are considered. It is assumed
that the graph edges are generated based on the community structure model.
Given the labeling of the edges of the first graph, the objective is to recover
the labels in the second graph. The problem is considered under two scenarios:
i) with side-information where the community membership of the nodes in both
graphs are known, and ii) without side-information where the community
memberships are not known. A matching scheme is proposed which operates based
on typicality of the adjacency matrices of the graphs. Achievability results
are derived which provide theoretical guarantees for successful matching under
specific assumptions on graph parameters. It is observed that for the proposed
matching scheme, the conditions for successful matching do not change in the
presence of side-information. Furthermore, a converse result is derived which
characterizes a set of graph parameters for which matching is not possible.
