The contact process on random hyperbolic graphs: metastability and critical exponents

TL;DR

This paper studies the contact process on hyperbolic random graphs with heavy-tailed degree distributions, revealing universal critical exponents and metastability phenomena as the graph size grows.

Contribution

It demonstrates the universality of the critical exponent for the contact process on hyperbolic graphs and establishes metastability results for finite graph models.

Findings

Non-extinction probability decays as a power law with exponent depending on hi

Critical exponent matches that of the configuration model, indicating universality

Finite hyperbolic graphs exhibit metastability as size increases

Abstract

We consider the contact process on the model of hyperbolic random graph, in the regime when the degree distribution obeys a power law with exponent $\chi \in(1,2)$ (so that the degree distribution has finite mean and infinite second moment). We show that the probability of non-extinction as the rate of infection goes to zero decays as a power law with an exponent that only depends on $\chi$ and which is the same as in the configuration model, suggesting some universality of this critical exponent. We also consider finite versions of the hyperbolic graph and prove metastability results, as the size of the graph goes to infinity.