A Spectral Approach to Analyzing Belief Propagation for 3-Coloring

TL;DR

This paper provides a spectral analysis of belief propagation (BP) for 3-coloring, establishing its convergence properties on complex graphs with planted solutions, and explaining how BP distinguishes between color permutations.

Contribution

It offers the first rigorous spectral analysis of BP for graph coloring on complex structures, extending understanding beyond tree-like graphs.

Findings

BP convergence relates to spectral properties of the graph

BP breaks symmetry among color permutations

First rigorous analysis of BP on complex graph structures

Abstract

Contributing to the rigorous understanding of BP, in this paper we relate the convergence of BP to spectral properties of the graph. This encompasses a result for random graphs with a ``planted'' solution; thus, we obtain the first rigorous result on BP for graph coloring in the case of a complex graphical structure (as opposed to trees). In particular, the analysis shows how Belief Propagation breaks the symmetry between the $3!$ possible permutations of the color classes.