Rare region effects in the contact process on networks

TL;DR

This paper investigates how quenched disorder and topological heterogeneity in networks lead to Griffiths phases and slow relaxation in the contact process, with implications for understanding propagation in complex systems.

Contribution

It demonstrates the emergence of Griffiths phases due to both intrinsic disorder and topological heterogeneity in networks, extending understanding of slow dynamics in complex systems.

Findings

Griffiths phases occur in Erdős-Rényi networks with node-dependent infection rates.

Topological heterogeneity alone can induce Griffiths phases even with constant rates.

Slow relaxation phenomena are observed in various network topologies, including generalized small-world networks.

Abstract

Networks and dynamical processes occurring on them have become a paradigmatic representation of complex systems. Studying the role of quenched disorder, both intrinsic to nodes and topological, is a key challenge. With this in mind, here we analyse the contact process, i.e. the simplest model for propagation phenomena, with node-dependent infection rates (i.e. intrinsic quenched disorder) on complex networks. We find Griffiths phases and other rare region effects, leading rather generically to anomalously slow (algebraic, logarithmic, etc.) relaxation, on Erd\H{o}s-R\'enyi networks. We predict similar effects to exist for other topologies as long as a non-vanishing percolation threshold exists. More strikingly, we find that Griffiths phases can also emerge --even with constant epidemic rates-- as a consequence of mere topological heterogeneity. In particular, we find Griffiths phases in finite dimensional networks as, for instance, a family of generalized small-world networks. These results have a broad spectrum of implications for propagation phenomena and other dynamical processes on networks, and are relevant for the analysis of both models and empirical data.