The contact process on dynamical random trees with degree dependence

TL;DR

This paper studies how the contact process, modeling infection spread, behaves on dynamically evolving random trees with degree-dependent connection probabilities, revealing conditions for phase transition and survival.

Contribution

It introduces a degree-dependent dynamical percolation model for evolving graphs and analyzes the impact on the contact process, including phase transition criteria and survival probabilities.

Findings

Positive critical value for survival under certain conditions

Strong survival with positive probability on Galton-Watson trees with stretched exponential tails

Complete phase transition characterization for power-law offspring distributions

Abstract

The contact process is a simple model for the spread of an infection in a structured population. We investigate the case when the underlying structure evolves dynamically as a degree-dependent dynamical percolation model. Starting with a connected locally finite base graph we initially declare edges independently open with a connection probability that is allowed to depend on the degree of the adjacent vertices and closed otherwise. Edges are independently updated with a rate depending on the degrees and then are again declared open and closed with the same probabilities. We are interested in the contact process, where infections are only allowed to spread via open edges. Our aim is to analyse the impact of the update speed and the connection probability on the existence of a phase transition. For a general connected locally finite graph, our first result gives sufficient conditions for the critical value for survival to be strictly positive. Furthermore, in the setting of Bienaym\'e-Galton-Watson trees, we show that the process survives strongly with positive probability for any infection rate if the offspring distribution has a stretched exponential tail with an exponent depending on the connection probability and the update speed. In particular, if the offspring distribution follows a power law and the connection probability is given by a product kernel and the update speed exhibits polynomial behaviour, we provide a complete characterisation of the phase transition.