Heterogeneous excitable systems exhibit Griffiths phases below hybrid phase transitions

TL;DR

This paper demonstrates that Griffiths phases, characterized by slow activity decay, occur in high-dimensional modular networks with threshold dynamics, due to activity fragmentation by network modules, relevant for brain and epidemic models.

Contribution

It provides numerical evidence of Griffiths phases in high-dimensional hierarchical modular networks with threshold models, highlighting the role of network fragmentation.

Findings

Griffiths phases are present below hybrid phase transitions in modular networks.

Network modularity causes activity fragmentation, leading to Griffiths phases.

The phenomenon persists with inhibitory links and refractory states.

Abstract

In $d > 2$ dimensional, homogeneous threshold models discontinuous transition occur, but the mean-field solution provides $1/t$ power-law activity decay and other power-laws, thus it is called mixed-order or hybrid type. It has recently been shown that the introduction of quenched disorder rounds the discontinuity and second order phase transition and Griffiths phases appear. Here we provide numerical evidence, that even in case of high graph dimensional hierarchical modular networks the Griffiths phase of the $K=2$ threshold model is present below the hybrid phase transition. This is due to the fragmentation of the activity propagation by modules, which are connected via single links. This provides a widespread mechanism in case of threshold type of heterogeneous systems, modeling the brain or epidemics for the occurrence of dynamical criticality in extended Griffiths phase parameter spaces. We investigate this in synthetic modular networks with and without inhibitory links as well as in the presence of refractory states.