Heterogeneous excitable systems exhibit Griffiths phases below hybrid phase transitions
Géza Ódor, Beatriz de Simoni
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, and 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 a Griffiths phase in 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 the case of the 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.