Percolation in Random Graphs of Unbounded Rank

TL;DR

This paper studies bootstrap percolation in complex random graphs with clustering, providing a fixed point characterization of infection spread and an algorithm for neural network computation.

Contribution

It introduces a new class of unbounded rank random graphs with clustering and analyzes contagion dynamics using a fixed point approach.

Findings

Final infected fraction is given by a fixed point of a non-linear operator.

An algorithm leveraging neural networks efficiently computes the fixed point.

Criteria based on Fréchet derivative determine global versus local spread.

Abstract

Bootstrap percolation in (random) graphs is a contagion dynamics among a set of vertices with certain threshold levels. The process is started by a set of initially infected vertices, and an initially uninfected vertex with threshold $k$ gets infected as soon as the number of its infected neighbors exceeds $k$. This process has been studied extensively in so called \textit{rank one} models. These models can generate random graphs with heavy-tailed degree sequences but they are not capable of clustering. In this paper, we treat a class of random graphs of unbounded rank which allow for extensive clustering. Our main result determines the final fraction of infected vertices as the fixed point of a non-linear operator defined on a suitable function space. We propose an algorithm that facilitates neural networks to calculate this fixed point efficiently. We further derive criteria based on the Fr\'echet derivative of the operator that allows one to determine whether small infections spread through the entire graph or rather stay local.