Percolation and Epidemic Processes in One-Dimensional Small-World Networks

TL;DR

This paper rigorously establishes phase transition thresholds for bond percolation and epidemic processes in one-dimensional small-world networks, revealing key epidemiological phenomena despite the model's simplicity.

Contribution

It provides the first fully rigorous thresholds for percolation and SIR epidemic processes in small-world graphs, linking mathematical results to epidemiological insights.

Findings

Identifies precise percolation thresholds in small-world networks.

Demonstrates epidemic spread through local outbreaks and super-spreader events.

Highlights the impact of random connections on epidemic dynamics.

Abstract

We obtain tight thresholds for bond percolation on one-dimensional small-world graphs, and apply such results to obtain tight thresholds for the \emph{Independent Cascade} process and the \emph{Reed-Frost} process in such graphs. These are the first fully rigorous results establishing a phase transition for bond percolation and SIR epidemic processes in small-world graphs. Although one-dimensional small-world graphs are an idealized and unrealistic network model, a number of realistic qualitative epidemiological phenomena emerge from our analysis, including the epidemic spread through a sequence of local outbreaks, the danger posed by random connections, and the effect of super-spreader events.