Percolation on random graphs

TL;DR

This paper investigates the phase transition in percolation on various random graph models, analyzing how network damage affects connectivity and the differences between static and dynamic graph behaviors.

Contribution

It reviews recent advances in understanding percolation near criticality on rank-1 inhomogeneous and dynamic random graphs, highlighting the impact of inhomogeneity.

Findings

Different behaviors in static and dynamic random graphs near criticality

Inhomogeneity significantly influences percolation phase transition

Scaling laws for component sizes near criticality

Abstract

Percolation is a model for random damage to a network. It is one of the simplest models that displays a phase transition: when the network is severely damaged, it falls apart in many small connected components, while if the damage is light, connectivity is hardly affected. We study the location and nature of the phase transition on random graphs. In particular, we focus on the connectivity structure close to, or below, criticality, where components display intricate scaling behaviour such that a typical connected component has bounded size, while the average and maximal connected component sizes grow like powers of the network size. We review the recent progress that has been made in two important settings: random graphs whose expected adjacency matrix is close to being rank-1, the most prominent examples being the configuration model and rank-1 inhomogeneous random graphs, and dynamic random graphs, i.e., random graphs that grow with time, such as uniform and preferential attachment models. Remarkably, these two settings behave rather differently. In all cases, the inhomogeneity of the underlying random graph on which we perform percolation is of crucial importance.