An improved Belief Propagation algorithm finds many Bethe states in the random field Ising model on random graphs

TL;DR

This paper introduces an enhanced Belief Propagation algorithm that efficiently finds many Bethe states in the random field Ising model on random graphs, revealing complex solution landscapes at phase transitions.

Contribution

The authors develop a new BP scheme using maximal solutions to identify multiple stable fixed points, improving ground state approximation and exploring the solution space complexity.

Findings

The new BP algorithm finds many stable fixed points in the critical region.

Number of fixed points grows with system size at phase transition.

The method efficiently approximates the ground state in complex energy landscapes.

Abstract

We first present an empirical study of the Belief Propagation (BP) algorithm, when run on the random field Ising model defined on random regular graphs in the zero temperature limit. We introduce the notion of maximal solutions for the BP equations and we use them to fix a fraction of spins in their ground state configuration. At the phase transition point the fraction of unconstrained spins percolates and their number diverges with the system size. This in turn makes the associated optimization problem highly non trivial in the critical region. Using the bounds on the BP messages provided by the maximal solutions we design a new and very easy to implement BP scheme which is able to output a large number of stable fixed points. On one side this new algorithm is able to provide the minimum energy configuration with high probability in a competitive time. On the other side we found that the number of fixed points of the BP algorithm grows with the system size in the critical region. This unexpected feature poses new relevant questions on the physics of this class of models.