The winner takes it all

TL;DR

This paper analyzes competing infection spread on scale-free networks, showing that one infection eventually dominates with high probability, and the winner is random, regardless of infection rates.

Contribution

It demonstrates that in scale-free networks with degree exponent between 2 and 3, one infection type almost surely dominates, extending understanding of competing processes on complex networks.

Findings

One infection dominates almost surely in large networks.

The winning infection is random, with positive probability for either type.

Results hold even with multiple initial infection points.

Abstract

We study competing first passage percolation on graphs generated by the configuration model. At time 0, vertex 1 and vertex 2 are infected with the type 1 and the type 2 infection, respectively, and an uninfected vertex then becomes type 1 (2) infected at rate $\lambda_1$ ($\lambda_2$) times the number of edges connecting it to a type 1 (2) infected neighbor. Our main result is that, if the degree distribution is a power-law with exponent $\tau\in(2,3)$, then, as the number of vertices tends to infinity and with high probability, one of the infection types will occupy all but a finite number of vertices. Furthermore, which one of the infections wins is random and both infections have a positive probability of winning regardless of the values of $\lambda_1$ and $\lambda_2$. The picture is similar with multiple starting points for the infections.