Belief propagation for networks with loops

TL;DR

This paper introduces an improved belief propagation method capable of accurately computing probabilities in networks with short loops, overcoming a major limitation of traditional message passing techniques.

Contribution

The authors derive a novel belief propagation algorithm that efficiently handles networks with short loops, providing accurate probability, entropy, and partition function calculations.

Findings

Significantly improves accuracy over standard methods on real and synthetic networks

Provides reliable calculations of entropy and partition functions in loopy networks

Demonstrates effectiveness using the Ising model as a case study

Abstract

Belief propagation is a widely used message passing method for the solution of probabilistic models on networks such as epidemic models, spin models, and Bayesian graphical models, but it suffers from the serious shortcoming that it works poorly in the common case of networks that contain short loops. Here we provide a solution to this long-standing problem, deriving a belief propagation method that allows for fast calculation of probability distributions in systems with short loops, potentially with high density, as well as giving expressions for the entropy and partition function, which are notoriously difficult quantities to compute. Using the Ising model as an example, we show that our approach gives excellent results on both real and synthetic networks, improving significantly on standard message passing methods. We also discuss potential applications of our method to a variety of other problems.