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.
Highlights
Phase transitions in genuine nonequilibrium systems have often been investigated among the reaction-diffusion (RD) type of models exhibiting absorbing states [1,2]
For simplicity we modeled the inhibitions by the introduction of links with negative weight contribution in the threshold comparison rule given by Eq (1), we think our results are transferred to the case of inhibitory nodes
In conclusion we provided numerical evidence that strong heterogeneity effects in networks, coming from the modular structure, can result in a Griffiths phase (GP) even if the topological dimension is high, where mean-field scaling would be expected
Summary
Phase transitions in genuine nonequilibrium systems have often been investigated among the reaction-diffusion (RD) type of models exhibiting absorbing states [1,2]. The brain can operate at different regimes close to the critical point, which can provide the desired advantages for biological systems Another possible resolution for the above controversy is the operation at a transition of hybrid type. Like the integrate-and-fire models of the brain [41], are suggested to describe other phenomena, like power grids [42,43,44], crack and fracture formation [45], contagion [46], etc In these models HPT can emerge naturally, and the present results can be relevant. Heterogeneity effects are very common in nature and result in dynamical criticality in extended GPs, in the case of quasistatic quenched disorder approximation [47] This leads to avalanche size and time distributions, with nonuniversal PL tails. By extending our model we will show that the proposed mechanism is very general, providing an explanation for the observed wide range of scale-free behavior below the transition point
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