On local equilibrium equations for clustering states

TL;DR

This paper demonstrates that local equilibrium equations, including TAP and belief propagation equations, have solutions in the colorable phase of graph coloring and similar zero-cost solution phases in other optimization problems, revealing clustering states.

Contribution

It extends the understanding of local equilibrium equations to the colorable phase and other zero-cost solution phases, linking solutions to clustering states in random graphs.

Findings

Solutions exist in the colorable phase of the coloring problem.

Solutions are associated with clusters of configurations in random graphs.

Almost everywhere in the uncolored phase, solutions or quasi-solutions are found.

Abstract

In this note we show that local equilibrium equations (the generalization of the TAP equations or of the belief propagation equations) do have solutions in the colorable phase of the coloring problem. The same results extend to other optimization problems where the solutions has cost zero (e.g. K-satisfiability). On a random graph the solutions of the local equilibrium equations are associated to clusters of configurations (clustering states). On a random graph the local equilibrium equations have solutions almost everywhere in the uncolored phase; in this case we have to introduce the concept quasi-solution of the local equilibrium equations.