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

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.