Rare regions of the Susceptible Infected Susceptible model on Barab\'asi-Albert networks

TL;DR

This paper investigates rare-region effects in SIS models on weighted Barabási-Albert networks, demonstrating how network inhomogeneities influence epidemic dynamics and the effectiveness of spectral analysis and QMF approximation.

Contribution

It extends previous work by analyzing SIS models on weighted scale-free networks, showing how quenched disorder and topology induce slow dynamics and rare-region effects.

Findings

QMF reliably detects activity clustering in networks.

Epidemic threshold vanishes in the thermodynamic limit.

Disassortative weights lead to localization effects.

Abstract

I extend a previous work to Susceptible-Infected-Susceptible (SIS) models on weighted Barab\'asi-Albert scale-free networks. Numerical evidence is provided that phases with slow, power-law dynamics emerge as the consequence of quenched disorder and tree topologies studied previously with the Contact Process. I compare simulation results with spectral analysis of the networks and show that the quenched mean-field (QMF) approximation provides a reliable, relatively fast method to explore activity clustering. This suggests that QMF can be used for describing rare-region effects due to network inhomogeneities. Finite size study of the QMF shows the expected disappearance of the epidemic threshold $\lambda_c$ in the thermodynamic limit and an inverse participation ratio $\sim 0.25$, meaning localization in case of disassortative weight scheme. Contrary, for the multiplicative weights and the unweighted trees this value vanishes in the thermodynamic limit, suggesting only weak rare region effects in agreement with the dynamical simulations. Strong corrections to the mean-field behavior in case of disassortative weights explains the concave shape of the order parameter $\rho(\lambda)$ at the transition point. Application of this method to other models may reveal interesting rare-region effects, Griffiths Phases as the consequence of quenched topological heterogeneities.