A model for infection on graphs

TL;DR

This paper models how network topology influences virus spread and information dissemination among mobile agents performing random walks, revealing the impact of graph structure on infection probability and phase transitions in heterogeneous networks.

Contribution

It introduces a simple infection model on graphs to analyze the effect of topology on infection spread, including homogeneous and power-law graphs, with new phase transition insights.

Findings

Coincidence time inversely proportional to number of nodes in homogeneous graphs.

Existence of phase transition in coincidence time on power-law graphs.

Network topology significantly affects infection probability.

Abstract

We address the question of understanding the effect of the underlying network topology on the spread of a virus and the dissemination of information when users are mobile performing independent random walks on a graph. To this end we propose a simple model of infection that enables to study the coincidence time of two random walkers on an arbitrary graph. By studying the coincidence time of a susceptible and an infected individual both moving in the graph we obtain estimates of the infection probability. The main result of this paper is to pinpoint the impact of the network topology on the infection probability. More precisely, we prove that for homogeneous graph including regular graphs and the classical Erdos-Renyi model, the coincidence time is inversely proportional to the number of nodes in the graph. We then study the model on power-law graphs, that exhibit heterogeneous connectivity patterns, and show the existence of a phase transition for the coincidence time depending on the parameter of the power-law of the degree distribution.