Belief Propagation on replica symmetric random factor graph models

TL;DR

This paper proves that for certain random factor graph models satisfying static replica symmetry, the Belief Propagation algorithm accurately computes the free energy, confirming physics predictions.

Contribution

It rigorously establishes the validity of Belief Propagation for Poisson and regular random factor graphs under static replica symmetry.

Findings

Belief Propagation messages asymptotically satisfy the equations

Free energy density equals the Bethe free energy

Results confirm physics-based conjectures

Abstract

According to physics predictions, the free energy of random factor graph models that satisfy a certain "static replica symmetry" condition can be calculated via the Belief Propagation message passing scheme [Krzakala et al., PNAS 2007]. Here we prove this conjecture for two general classes of random factor graph models, namely Poisson random factor graphs and random regular factor graphs. Specifically, we show that the messages constructed just as in the case of acyclic factor graphs asymptotically satisfy the Belief Propagation equations and that the free energy density is given by the Bethe free energy formula.