Information-Theoretic Thresholds for the Alignments of Partially Correlated Graphs

TL;DR

This paper establishes the fundamental information-theoretic limits for recovering vertex alignments in partially correlated Erdős-Rényi graphs, identifying thresholds for partial and exact recovery of the hidden vertex correspondence.

Contribution

It introduces a new model of partially correlated Erdős-Rényi graphs and derives optimal thresholds for both partial and exact graph alignment recovery.

Findings

Existence of an optimal rate for partial recovery of correlated nodes

Identification of a threshold for exact recovery

Development of correlated functional digraphs for analysis

Abstract

This paper studies the problem of recovering the hidden vertex correspondence between two correlated random graphs. We propose the partially correlated Erd\H{o}s-R\'enyi graphs model, wherein a pair of induced subgraphs with a certain number are correlated. We investigate the information-theoretic thresholds for recovering the latent correlated subgraphs and the hidden vertex correspondence. We prove that there exists an optimal rate for partial recovery for the number of correlated nodes, above which one can correctly match a fraction of vertices and below which correctly matching any positive fraction is impossible, and we also derive an optimal rate for exact recovery. In the proof of possibility results, we propose correlated functional digraphs, which partition the edges of the intersection graph into two types of components, and bound the error probability by lower-order cumulant generating functions. The proof of impossibility results build upon the generalized Fano's inequality and the recovery thresholds settled in correlated Erd\H{o}s-R\'enyi graphs model.