Exact solution of bond percolation on small arbitrary graphs

TL;DR

This paper presents an exact iterative method for determining the component size distribution in small arbitrary graphs after random edge removal, with applications across various scientific fields.

Contribution

It introduces a novel set of equations that precisely solve for component distributions and can predict node partitions in undirected graphs, advancing percolation theory.

Findings

Exact component size distribution for small graphs

Predicts node partition distributions in undirected graphs

Applicable to large finite graphs and multiple disciplines

Abstract

We introduce a set of iterative equations that exactly solves the size distribution of components on small arbitrary graphs after the random removal of edges. We also demonstrate how these equations can be used to predict the distribution of the node partitions (i.e., the constrained distribution of the size of each component) in undirected graphs. Besides opening the way to the theoretical prediction of percolation on arbitrary graphs of large but finite size, we show how our results find application in graph theory, epidemiology, percolation and fragmentation theory.