Color-avoiding percolation in edge-colored Erd\H{o}s-R\'enyi graphs

TL;DR

This paper analyzes a variant of color-avoiding percolation on edge-colored Erdős-Rényi graphs, deriving explicit formulas for the size of the giant component and other connected components under certain conditions.

Contribution

It extends the color-avoiding percolation model to edge-colored Erdős-Rényi graphs and provides explicit formulas for component sizes using branching process probabilities.

Findings

Convergence of component size fractions under mild assumptions

Explicit formulas for the giant component size

Results specialized for the two-colored case

Abstract

We study a variant of the color-avoiding percolation model introduced by Krause et al., namely we investigate the color-avoiding bond percolation setup on (not necessarily properly) edge-colored Erd\H{o}s-R\'{e}nyi random graphs. We say that two vertices are color-avoiding connected in an edge-colored graph if after the removal of the edges of any color, they are in the same component in the remaining graph. The color-avoiding connected components of an edge-colored graph are maximal sets of vertices such that any two of them are color-avoiding connected. We consider the fraction of vertices contained in color-avoiding connected components of a given size as well as the fraction of vertices contained in the giant color-avoiding connected component. Under some mild assumptions on the color-densities, we prove that these quantities converge and the limits can be expressed in terms of probabilities associated to edge-colored branching process trees. We provide explicit formulas for the limit of the normalized size of the giant color-avoiding component, and in the two-colored case we also provide explicit formulas for the limit of the fraction of vertices contained in color-avoiding connected components of a given size.