Spectral analysis and slow spreading dynamics on complex networks

TL;DR

This paper uses spectral analysis of the adjacency matrix within quenched mean-field theory to predict slow epidemic spreading and Griffiths phases in complex networks, validated by simulations on various graph types.

Contribution

It demonstrates how spectral decomposition can accurately predict epidemic thresholds and rare-region effects, including Griffiths phases, in different complex network models.

Findings

QMF predicts epidemic thresholds accurately for ER graphs.

Rare-region effects and Griffiths phases occur in fragmented and weighted networks.

Spectral analysis reveals slow dynamics in Barabási-Albert networks with aging connections.

Abstract

The Susceptible-Infected-Susceptible (SIS) model is one of the simplest memoryless system for describing information/epidemic spreading phenomena with competing creation and spontaneous annihilation reactions. The effect of quenched disorder on the dynamical behavior has recently been compared to quenched mean-field (QMF) approximations in scale-free networks. QMF can take into account topological heterogeneity and clustering effects of the activity in the steady state by spectral decomposition analysis of the adjacency matrix. Therefore, it can provide predictions on possible rare-region effects, thus on the occurrence of slow dynamics. I compare QMF results of SIS with simulations on various large dimensional graphs. In particular, I show that for Erd\H os-R\'enyi graphs this method predicts correctly the epidemic threshold and the rare-region effects. Griffiths Phases emerge if the graph is fragmented or if we apply strong, exponentially suppressing weighting scheme on the edges. The latter model describes the connection time distributions in the face-to-face experiments. In case of generalized Barab\'asi-Albert type of networks with aging connections strong rare-region effects and numerical evidence for Griffiths Phase dynamics are shown.