On Graph Matching Using Generalized Seed Side-Information

TL;DR

This paper proposes a graph matching method leveraging generalized seed side-information, such as ambiguity sets, to improve label recovery in correlated Erdős-Rényi graphs, with theoretical guarantees and necessary conditions.

Contribution

It introduces a new matching strategy using joint typicality that generalizes seeded graph matching with ambiguity sets and provides both sufficient and necessary conditions for success.

Findings

Proposed a joint typicality-based matching strategy.

Derived sufficient conditions for successful label recovery.

Established necessary conditions using Fano-type bounds.

Abstract

In this paper, matching pairs of stocahstically generated graphs in the presence of generalized seed side-information is considered. The graph matching problem emerges naturally in various applications such as social network de-anonymization, image processing, DNA sequencing, and natural language processing. A pair of randomly generated labeled Erdos-Renyi graphs with pairwise correlated edges are considered. It is assumed that the matching strategy has access to the labeling of the vertices in the first graph, as well as a collection of shortlists -- called ambiguity sets -- of possible labels for the vertices of the second graph. The objective is to leverage the correlation among the edges of the graphs along with the side-information provided in the form of ambiguity sets to recover the labels of the vertices in the second graph. This scenario can be viewed as a generalization of the seeded graph matching problem, where the ambiguity sets take a specific form such that the exact labels for a subset of vertices in the second graph are known prior to matching. A matching strategy is proposed which operates by evaluating the joint typicality of the adjacency matrices of the graphs. Sufficient conditions on the edge statistics as well as ambiguity set statistics are derived under which the proposed matching strategy successfully recovers the labels of the vertices in the second graph. Additionally, Fano-type arguments are used to derive general necessary conditions for successful matching.