Degree-penalized contact processes

Bartha, Zsolt and Komjathy, Julia and Valesin, Daniel (2026) Degree-penalized contact processes. FORUM OF MATHEMATICS SIGMA, 14. No. e6. ISSN 2050-5094

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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 d(u) infects its neighboring vertex v with degree d(v) with rate lambda / f(d(u), d(v)) for some positive function f. In the case f(du, 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 the degree distribution D of the GW tree. degrees When mu >= 1 , the process goes extinct for all distributions D for all sufficiently small lambda>0 ; degrees When mu in [1/2, 1) , and the tail of D weakly follows a power law with tail-exponent less than 1-mu , the process survives globally but not locally for all lambda small enough; degrees When mu in [1/2, 1) , and E[D1-mu]<infinity, the process goes extinct almost surely, for all lambda small enough; degrees When mu <1/2 , and D is heavier than stretched exponential with stretch-exponent 1-2 mu , the process survives (locally) with positive probability for all lambda>0 . We also study the product case, where f(d(u), d(v))=(d(u) d(v))<^>mu . In that case, the situation for mu < 1/2 is the same as the one described above, but mu >= 1/2 always leads to a subcritical contact process for small enough lambda > 0 on all graphs. Furthermore, for finite random graphs with prescribed degree sequences, we establish the corresponding phase transitions in terms of the length of survival.

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