Abstract
Griffiths phases (GPs), generated by the heterogeneities on modular networks, have recently been suggested to provide a mechanism, rid of fine parameter tuning, to explain the critical behavior of complex systems. One conjectured requirement for systems with modular structures was that the network of modules must be hierarchically organized and possess finite dimension. We investigate the dynamical behavior of an activity spreading model, evolving on heterogeneous random networks with highly modular structure and 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, preserved after finite size analyses, as in a real GP. The avalanche size distributions of spreading events exhibit robust power-law tails. Our findings relax the requirement of hierarchical organization of the modular structure, which can help to rationalize the criticality of modular systems in the framework of GPs.
Highlights
A recurrent feature of complex systems is the presence of critical states, in which spatial and temporal correlations diverge[1,2]
Griffiths phases (GPs) is the consequence of rare regions (RRs), consisting of local supercritical domains, which occur with small probability but sustaining activity for long times
We have analyzed the activity spreading of the SIS model in loosely connected, non-hierarchical modular networks and observed extended regions of non-universal scaling behavior for intrinsic [Figs 4(a), 5(a) and 6] and topological [Fig. 4(b)] disorders, using density decay and spreading analysis
Summary
A recurrent feature of complex systems is the presence of critical states, in which spatial and temporal correlations diverge[1,2]. Some evidences suggest that the brain network is heterogeneous and present hierarchical modular organization, in which modules are themselves composed by modular substructures at distinct levels[10,24,25] These inspired Moretti and Muñoz[26] to investigate activity spreading models on hierarchical modular networks of finite dimension, for which GPs and extended critical regions were observed (see also27,28) and to conjecture that the brain criticality could be effected by quenched disorder without fine parameter tuning. Optimal fluctuation theory[31] and simulations provided extended critical regions on heterogeneous networks of finite size constrained to averages over independent network samples[30] This inspired us to investigate the dynamical behavior of the continuous time Markovian susceptible-infected-susceptible (SIS) model, which has been used to describe activity or information spreading in socio-technological and biological systems[26,38,39,40], on loosely coupled network of modules. Our results point out that we can relax the requirement of hierarchical organization and large-world[22,26] for the existence of GPs on modular networks, these factors certainly enhance RR effects
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
