The contact process on Scale-Free Percolation

TL;DR

This work analyzes the contact process on Scale-Free Percolation graphs restricted to finite boxes, establishing metastable extinction times that grow exponentially in the box size in two regimes determined by the tail parameter: ultra-small-world $\gamma\in(1,2)$ yields $\tau_{\mathcal{G}_n}\ge e^{c n}$, while small-world $\gamma>2$ yields $\tau_{\mathcal{G}_n}\ge e^{c n (\log n)^{-A}}$ with a logarithmic correction. The proofs hinge on a constellation-based construction that embeds many high-degree stars with short connecting paths, coupled with a generalized Mountford metastability framework and a multi-scale geometric analysis. The results illuminate how degree tails and spatial embedding jointly govern epidemic persistence on spatial scale-free networks, and they situate the SFP model within the broader landscape of metastability phenomena alongside HRG and related spatial models. The paper also discusses non-extinction probabilities on the infinite graph, adapting prior methods to the soft, spatial setting, and clarifies how the asymptotics of persistence depend on the regime parameter $\gamma$.

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 $β$. We study the extinction time $τ_{G_n}$ of the contact process on the graph restricted to a d-dimensional box of volume n, starting from full occupancy. In the regime $β\in (2, 3)$, 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 $τ_{G_n}$ is exponential in n. Our main contribution, though, deals with the case $β\geq 3$, where the degrees have finite variance and the graph is small-world. We prove that also in this case $τ_{G_n}$ 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.

Figures (4)

• Figure 1: Value of the critical percolation parameter (left) and graph distances (right) for SFP in $\mathbb{R}^d$, $d \geq 1$, for different values of $\alpha$ and $\tau$.
• Figure 2: Realizations of SFP with $n=1000$ vertices in a square box $[0,1]^2$ for $\alpha=2.5$, $\tau=2.2$ so that $\gamma=1.5$ (left) and $\alpha=3$, $\tau=4$ so that $\gamma=4.5$ (right). The percolation parameter $\rho$ was chosen such that both graphs had a similar number of edges (average degree approximately equal to 10).
• Figure 3: Representation of a (10,3,2)-constellation with $|J|=16$ within a simulation of SFP in dimension 2, obtained with a modified version of the simulator described in blasius2022efficiently. The set of distinguished vertices $J$ is drawn in red. For each of these vertices, a set of 10 neighbours is drawn in orange, and the paths linking the distinguished vertices are drawn in green.
• Figure 4: Partitioning scheme of $[0,n^{1/d})^d\times (1,\infty)$ used in the proof of Proposition \ref{['prop:graphe etoile']}, in dimension 1. As the level $k$ increases, the weight layers $({\rm e}^{kL\alpha/\gamma},{\rm e}^{(k+1)L\alpha/\gamma})$ become larger to compensate for the increasing scarcity of high-weight vertices. For $k\ge 1$, boxes $B_{k+1,\mathtt{v}}$ have $2^d$ children boxes denoted by $B_{k,2\mathtt{v}+e}$ for $e\in\{0,1\}^d$.

Theorems & Definitions (25)

• Theorem 1.1
• Theorem 1.2
• Remark 1.3
• definition 1
• Proposition 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• Proposition 3.1