Griffiths phases in infinite-dimensional, non-hierarchical modular networks

Griffiths phase (GP) generated by the heterogeneities on modular networks has
recently been suggested to provide a mechanism, rid of fine parameter tuning,
to explain the critical behavior of the brain. One conjectured requirement was
that the network of modules must be hierarchically organized and possess finite
topological dimension. We investigate the dynamical behavior of an activity
spreading model evolving in heterogeneous random networks with highly modular
structure, organized non-hierarchically. We observe that loosely coupled
modules act as effective rare-regions slowing down the extinction of activation.
As a consequence, we find extended control parameter regions with continuously
changing dynamical exponents for single network realizations in the thermodynamic
limit, as in a real GP. The avalanche size distributions of spreading events
exhibit robust power-law tails. Our findings relax the requirement of a
hierarchical organization of the modular structure, which can help to
rationalize the criticality of modular systems in the framework of GPs.