Bootstrap percolation on geometric inhomogeneous random graphs

TL;DR

This paper investigates how bootstrap percolation spreads on geometric inhomogeneous random graphs, revealing a phase transition influenced by geometry and providing insights into controlling infection spread with minimal edge removal.

Contribution

It is the first mathematical analysis of the role of geometry in bootstrap percolation on geometric scale-free networks, including phase transition and infection containment strategies.

Findings

Identified a phase transition in infection spread based on initial infection rate.

Determined the speed of infection spread in the supercritical case.

Showed how to contain infection by removing few edges.

Abstract

Geometric inhomogeneous random graphs (GIRGs) are a model for scale-free networks with underlying geometry. We study bootstrap percolation on these graphs, which is a process modelling the spread of an infection of vertices starting within a (small) local region. We show that the process exhibits a phase transition in terms of the initial infection rate in this region. We determine the speed of the process in the supercritical case, up to lower order terms, and show that its evolution is fundamentally influenced by the underlying geometry. For vertices with given position and expected degree, we determine the infection time up to lower order terms. Finally, we show how this knowledge can be used to contain the infection locally by removing relatively few edges from the graph. This is the first time that the role of geometry on bootstrap percolation is analysed mathematically for geometric scale-free networks.