Percolation on random graphs with a fixed degree sequence

TL;DR

This paper analyzes bond percolation on random graphs with fixed degree sequences, characterizing when a giant component persists after edge deletion, emphasizing the role of the degree distribution's tail.

Contribution

It provides a new characterization of the critical condition for the emergence of a giant component in sparse graphs without relying on the configuration model.

Findings

Giant component presence depends on the tail of the degree distribution.

The critical condition is characterized by the degree distribution's tail behavior.

Results apply to sparse degree sequences without traditional configuration model restrictions.

Abstract

We consider bond percolation on random graphs with given degrees and bounded average degree. In particular, we consider the order of the largest component after the random deletion of the edges of such a random graph. We give a rough characterisation of those degree distributions for which bond percolation with high probability leaves a component of linear order, known usually as a giant component. We show that essentially the critical condition has to do with the tail of the degree distribution. Our proof makes use of recent technique introduced by Joos et al. [FOCS 2016, pp. 695--703], which is based on the switching method and avoids the use of the classic configuration model as well as the hypothesis of having a limiting object. Thus our results hold for sparse degree sequences without the usual restrictions that accompany the configuration model.