Description
We study degree-penalized contact processes on Galton-Watson (GW) trees and the configuration model. The model we consider is a modification of the usual contact process on a graph. In particular, each vertex can be either infected or healthy. When infected, each vertex heals at rate one. Also, when infected, a vertex $v$ with degree $d_v$ infects its neighboring vertex $u$ with degree $d_u$ with rate $\lambda / f(d_u, d_v)$ for some positive function $f$. In the case $f(d_u, d_v)=\max(d_u, d_v)^\mu$ for some $\mu>0$, the infection is slowed down to and from high degree vertices. This is in line with arguments used in social network science: people with many contacts do not have the time to infect their neighbors at the same rate as people with fewer contacts. We show that new phase transitions occur in terms of the parameter $\mu$ (at 1/2 and 1) and the degree distribution $D$ of the GW tree (which matters in the regime $1/2<\mu<1$). Different phases are characterized by the long-term behavior of the infection process, which can exhibit extinction, global survival or local survival. We also study the choice $f(x,y)=(xy)^\mu$, in which case we obtain a simpler phase diagram, as $\mu>1/2$ always leads to a subcritical process. Furthermore, for finite random graphs with prescribed degree sequences, we establish the corresponding phase transitions in terms of the length of survival.
Joint work with Júlia Komjáthy and Daniel Valesin.