Competing first passage percolation on random graphs with finite variance degrees

TL;DR

This paper analyzes how two competing infections spread on random graphs with finite variance degrees, revealing that equal infection rates lead to a random coexistence fraction, while unequal rates favor the stronger infection.

Contribution

It provides a rigorous analysis of competing first passage percolation on configuration model graphs with finite second moment degrees, characterizing the asymptotic infection distribution.

Findings

Equal infection rates result in a random coexistence fraction between 0 and 1.

The infection with higher rate dominates when rates differ.

Results hold for both multigraphs and simple graphs with the given degree sequence.

Abstract

We study the growth of two competing infection types on graphs generated by the configuration model with a given degree sequence. Starting from two vertices chosen uniformly at random, the infection types spread via the edges in the graph in that an uninfected vertex becomes type 1 (2) infected at rate $\lambda_1$ ($\lambda_2$) times the number of nearest neighbors of type 1 (2). Assuming (essentially) that the degree of a randomly chosen vertex has finite second moment, we show that if $\lambda_1=\lambda_2$, then the fraction of vertices that are ultimately infected by type 1 converges to a continuous random variable $V\in(0,1)$, as the number of vertices tends to infinity. Both infection types hence occupy a positive (random) fraction of the vertices. If $\lambda_1\neq \lambda_2$, on the other hand, then the type with the larger intensity occupies all but a vanishing fraction of the vertices. Our results apply also to a uniformly chosen simple graph with the given degree sequence.