Percolation on sparse networks

TL;DR

This paper introduces a message passing framework to analyze percolation on sparse networks, providing exact and highly accurate predictions for cluster sizes and thresholds, applicable to network resilience and disease spread modeling.

Contribution

It reformulates percolation as a message passing process, deriving equations for cluster sizes and thresholds, especially on sparse networks with few short loops.

Findings

Exact calculations for percolation properties on sparse networks.

High accuracy of the message passing approach even with many short loops.

Percolation threshold related to the inverse of the leading eigenvalue of the non-backtracking matrix.

Abstract

We study percolation on networks, which is used as a model of the resilience of networked systems such as the Internet to attack or failure and as a simple model of the spread of disease over human contact networks. We reformulate percolation as a message passing process and demonstrate how the resulting equations can be used to calculate, among other things, the size of the percolating cluster and the average cluster size. The calculations are exact for sparse networks when the number of short loops in the network is small, but even on networks with many short loops we find them to be highly accurate when compared with direct numerical simulations. By considering the fixed points of the message passing process, we also show that the percolation threshold on a network with few loops is given by the inverse of the leading eigenvalue of the so-called non-backtracking matrix.