First and second moments of the size distribution of bond percolation clusters on regular graphs

TL;DR

This paper derives universal bounds for the size distribution moments of bond percolation clusters on regular graphs, with applications to network resilience and cyber security, using branching process approximations.

Contribution

It introduces new bounds for the first and second moments of cluster sizes on regular graphs, applicable in different probabilistic regimes.

Findings

Universal upper bounds for cluster size moments derived

Bounds are accurate in regimes of small open or closed edge probabilities

Application demonstrated on Platonic solids

Abstract

Motivated by network resilience and insurance premiums in the context of cyber security, we derive universal upper bounds for the first and second moments of the size of bond percolation clusters on finite regular graphs. Thinking of the clusters as dynamical objects coupled with branching processes gives a first set of bounds that are accurate when the probability of an edge being open is small. Estimating the number of isolated vertices, we also obtain a second set of bounds that are accurate when the probability of an edge being closed is small. As an illustration, we apply our results to the first three Platonic solids.