The winner takes it all but one

TL;DR

This paper analyzes competing infections spreading on large random graphs with infinite-mean degrees, showing that typically one infection dominates almost entirely, regardless of passage time distributions.

Contribution

It demonstrates that in graphs with power-law degrees (exponent between 1 and 2), one infection almost surely outcompetes the other, extending previous models to infinite-mean degree scenarios.

Findings

One infection dominates almost all vertices with high probability.

Both infections have a positive chance of winning regardless of passage times.

Results hold for erased configuration models and degree conditioning.

Abstract

We study competing first passage percolation on graphs generated by the configuration model with infinite-mean degrees. Initially, two uniformly chosen vertices are infected with type 1 and type 2 infection, respectively, and the infection then spreads via nearest neighbors in the graph. The time it takes for the type 1 (resp. 2) infection to traverse an edge $e$ is given by a random variable $X_1(e)$ (resp. $X_2(e)$) and, if the vertex at the other end of the edge is still uninfected, it then becomes type 1 (resp. 2) infected and immune to the other type. Assuming that the degrees follow a power-law distribution with exponent $\tau \in (1,2)$, we show that, with high probability as the number of vertices tends to infinity, one of the infection types occupies all vertices except for the starting point of the other type. Moreover, both infections have a positive probability of winning regardless of the passage times distribution. The result is also shown to hold for the erased configuration model, where self-loops are erased and multiple edges are merged, and when the degrees are conditioned to be smaller than $n^\alpha$ for some $\alpha > 0$.