An exact formula for percolation on higher-order cycles

TL;DR

This paper derives exact formulas for the size of the giant connected component in graphs with higher-order cycles under bond percolation, providing analytical thresholds and insights applicable to various scientific fields.

Contribution

It introduces an exact analytical solution for percolation on graphs with higher-order cycles, including weak cycles and cliques, expanding understanding beyond simple pairwise interactions.

Findings

Exact formulas for GCC size in higher-order cycle graphs

Analytical percolation thresholds derived for these models

Insights into graphicality conditions for complex networks

Abstract

We present exact solutions for the size of the giant connected component (GCC) of graphs composed of higher-order homogeneous cycles, including weak cycles and cliques, following bond percolation. We use our theoretical result to find the location of the percolation threshold of the model, providing analytical solutions where possible. We expect the results derived here to be useful to a wide variety of applications including graph theory, epidemiology, percolation and lattice gas models as well as fragmentation theory. We also examine the Erd\H{o}s-Gallai theorem as a necessary condition on the graphicality of configuration model networks comprising higher-order clique sub-graphs.