Matching Graphs with Community Structure: A Concentration of Measure Approach

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.