The contact process on Scale-Free Percolation

TL;DR

This paper analyzes the extinction time of the contact process on scale-free percolation graphs, showing exponential growth in the volume for different degree distribution regimes, using multi-scale and percolation techniques.

Contribution

It extends existing methods to analyze contact process extinction times on scale-free graphs with different degree exponents, including the case with finite variance.

Findings

Extinction time is exponential in graph volume for 2<β<3.

Extinction time also grows exponentially for β≥3, with a logarithmic correction.

The analysis combines multi-scale, chemical distance, and percolation methods.

Abstract

We consider the contact process on scale-free percolation, a spatial random graph model where the degree distribution of the vertices follows a power law with exponent $\beta$. We study the extinction time $\tau_{G_n}$ of the contact process on the graph restricted to a d-dimensional box of volume n, starting from full occupancy. In the regime $\beta \in (2, 3)$, where the degrees have finite mean but infinite variance and the graph exhibits the ultra-small world behaviour, we adapt the techniques of [Linker et al., 2021] to show that $\tau_{G_n}$ is exponential in n. Our main contribution, though, deals with the case $\beta \geq 3$, where the degrees have finite variance and the graph is small-world. We prove that also in this case $\tau_{G_n}$ grows exponentially, at least up to a logarithmic correction reflecting the sparser graph structure. The proof requires the generalization of a result from [Mountford et al., 2016] and combines a multi-scale analysis of the graph, the study of the chemical distance between vertices and percolation arguments.