Partial Recovery of Erd\H{o}s-R\'enyi Graph Alignment via $k$-Core Alignment

TL;DR

This paper establishes conditions under which partial recovery of graph alignment is possible in sparse Erdős-Rényi graphs, introducing a $k$-core alignment estimator and providing matching bounds.

Contribution

It introduces the $k$-core alignment estimator for partial graph alignment recovery and proves its effectiveness with matching converse bounds.

Findings

Partial recovery is achievable when the intersection's average degree tends to infinity.

Exact alignment is possible when the average degree grows faster than log of vertices.

The $k$-core estimator effectively finds alignments with minimum degree $k$.

Abstract

We determine information theoretic conditions under which it is possible to partially recover the alignment used to generate a pair of sparse, correlated Erd\H{o}s-R\'enyi graphs. To prove our achievability result, we introduce the $k$-core alignment estimator. This estimator searches for an alignment in which the intersection of the correlated graphs using this alignment has a minimum degree of $k$. We prove a matching converse bound. As the number of vertices grows, recovery of the alignment for a fraction of the vertices tending to one is possible when the average degree of the intersection of the graph pair tends to infinity. It was previously known that exact alignment is possible when this average degree grows faster than the logarithm of the number of vertices.