Abstract In this paper 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 u with degree dᵤ infects its neighboring vertex v with degree dᵥ with rate / f (dᵤ, dᵥ) for some positive function f. In the case f (dᵤ, dᵥ) = (dᵤ, dᵥ) ^ for some 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 (at 1/2) and the degree distribution D of the GW tree. • When 1, the process goes extinct for all distributions D for all sufficiently small >0 ; • When 1/2, 1), and the tail of D weakly follows a power law with tail-exponent less than 1-, the process survives globally but not locally for all small enough; • When [1/2, 1), and E[D^1- <, the process goes extinct almost surely, for all small enough; • When <1/2, and D is heavier than stretched exponential with stretch-exponent
Bartha et al. (Thu,) studied this question.