Message passing on networks with loops

Abstract

In this paper we offer a solution to a long-standing problem in the study of networks. Message passing is a fundamental technique for calculations on networks and graphs. The first versions of the method appeared in the 1930s and over the decades it has been applied to a wide range of foundational problems in mathematics, physics, computer science, statistics, and machine learning, including Bayesian inference, spin models, coloring, satisfiability, graph partitioning, network epidemiology, and the calculation of matrix eigenvalues. Despite its wide use, however, it has long been recognized that the method has a fundamental flaw: it only works on networks that are free of short loops. Loops introduce correlations that cause the method to give inaccurate answers at best, and to fail completely in the worst cases. Unfortunately, almost all real-world networks contain many short loops, which limits the usefulness of the message passing approach. In this paper we demonstrate how to rectify this shortcoming and create message passing methods that work on any network. We give two example applications, one to the percolation properties of networks and the other to the calculation of the spectra of sparse matrices.

Belief propagation for networks with loops

“…In (20), we previously described message passing schemes for percolation models and spectral calculations on loopy networks. Here, we extend this approach to the solution of general probabilistic models.…”

Percolation on feature-enriched interconnected systems

“…In this context, the fraction of removed nodes are usually thought of as failures or attacks, and the largest connected component after the perturbation has a functional interpretation assumed to be the part of the network that is still operative. Therefore, percolation in this type of topologies has brought a deeper understanding of the robustness and resilience of real-world networked systems 10 – 13 , as well as, from a fundamental perspective, it has provided new analytical techniques 14 , 15 and interesting phenomenology from the standpoint of statistical physics 16 – 19 .…”

Random graphs with arbitrary clustering and their applications

“…These generating function models are based on joint degree distributions that partition the edges of a node into either treelike or triangle motifs. There have also been recent advances in this endeavour using the related method of message passing [2,18].…”