Griffiths phases on complex networks

TL;DR

This paper investigates how quenched disorder and topological heterogeneity in complex networks lead to Griffiths phases, causing slow relaxation in dynamical processes like the Contact Process.

Contribution

It demonstrates the emergence of Griffiths phases due to quenched disorder and topological heterogeneity in complex networks, a novel insight into network dynamics.

Findings

Griffiths phases occur in Erdős-Rényi networks with quenched disorder.

Topological heterogeneity can induce Griffiths phases without disorder.

Anomalously slow relaxation behaviors are observed in studied networks.

Abstract

Quenched disorder is known to play a relevant role in dynamical processes and phase transitions. Its effects on the dynamics of complex networks have hardly been studied. Aimed at filling this gap, we analyze the Contact Process, i.e. the simplest propagation model, with quenched disorder on complex networks. We find Griffiths phases and other rare region effects, leading rather generically to anomalously slow (algebraic, logarithmic, ...) relaxation, on Erd\H os-R\'enyi networks. Similar effects are predicted to exist for other topologies with a finite percolation threshold. More surprisingly, we find that Griffiths phases can also emerge in the absence of quenched disorder, as a consequence of topological heterogeneity in networks with finite topological dimension. These results have a broad spectrum of implications for propagation phenomena and other dynamical processes on networks.