Aligning Multiple Inhomogeneous Random Graphs: Fundamental Limits of Exact Recovery

TL;DR

This paper investigates the fundamental limits of exactly recovering node correspondences across multiple inhomogeneous random graphs, proposing an algorithm that leverages transitivity and analyzing its optimality and thresholds.

Contribution

It introduces a matching algorithm for multiple correlated graphs, providing sufficient conditions for exact recovery and establishing their necessity in homogeneous cases, revealing the power of multi-graph information.

Findings

The proposed algorithm achieves exact recovery under certain conditions.

Necessary and sufficient conditions are characterized for homogeneous graphs.

Transitivity enables exact matching even when pairwise matching fails.

Abstract

This work studies fundamental limits for recovering the underlying correspondence among multiple correlated graphs. In the setting of inhomogeneous random graphs, we present and analyze a matching algorithm: first partially match the graphs pairwise and then combine the partial matchings by transitivity. Our analysis yields a sufficient condition on the problem parameters to exactly match all nodes across all the graphs. In the setting of homogeneous (Erd\H{o}s-R\'enyi) graphs, we show that this condition is also necessary, i.e. the algorithm works down to the information theoretic threshold. This reveals a scenario where exact matching between two graphs alone is impossible, but leveraging more than two graphs allows exact matching among all the graphs. Converse results are also given in the inhomogeneous setting and transitivity again plays a role. Along the way, we derive independent results about the k-core of inhomogeneous random graphs.