Griffiths phases and localization in hierarchical modular networks.

Géza Ódor,Ronald Dickman,Gergely Ódor

doi:10.1038/srep14451

Abstract

We study variants of hierarchical modular network models suggested by Kaiser and Hilgetag [ Front. in Neuroinform., 4 (2010) 8] to model functional brain connectivity, using extensive simulations and quenched mean-field theory (QMF), focusing on structures with a connection probability that decays exponentially with the level index. Such networks can be embedded in two-dimensional Euclidean space. We explore the dynamic behavior of the contact process (CP) and threshold models on networks of this kind, including hierarchical trees. While in the small-world networks originally proposed to model brain connectivity, the topological heterogeneities are not strong enough to induce deviations from mean-field behavior, we show that a Griffiths phase can emerge under reduced connection probabilities, approaching the percolation threshold. In this case the topological dimension of the networks is finite, and extended regions of bursty, power-law dynamics are observed. Localization in the steady state is also shown via QMF. We investigate the effects of link asymmetry and coupling disorder, and show that localization can occur even in small-world networks with high connectivity in case of link disorder.

Highlights

We study variants of hierarchical modular network models suggested by Kaiser and Hilgetag [ Front. in Neuroinform., 4 (2010) 8] to model functional brain connectivity, using extensive simulations and quenched mean-field theory (QMF), focusing on structures with a connection probability that decays exponentially with the level index
We explore the dynamic behavior of the contact process (CP) and threshold models on networks of this kind, including hierarchical trees
We investigate the dynamical behavior of spreading models on hierarchical modular networks embedded in two-dimensional space, with long links whose probability decays as a power-law with distance

Summary

Introduction

We study variants of hierarchical modular network models suggested by Kaiser and Hilgetag [ Front. in Neuroinform., 4 (2010) 8] to model functional brain connectivity, using extensive simulations and quenched mean-field theory (QMF), focusing on structures with a connection probability that decays exponentially with the level index. While in the small-world networks originally proposed to model brain connectivity, the topological heterogeneities are not strong enough to induce deviations from mean-field behavior, we show that a Griffiths phase can emerge under reduced connection probabilities, approaching the percolation threshold. In this case the topological dimension of the networks is finite, and extended regions of bursty, power-law dynamics are observed. The possibility of self-tuning is well known in statistical physics; the paradigm of self-organized criticality (SOC) has been studied since the pioneering work of[10] Simple homogeneous models such as the stochastic sandpile exhibit criticality with power laws both in statics and dynamics. Many networks exhibit the small-world property and so have an infinite topological dimension, naturally occurring networks are always finite, exhibit cutoffs, and GPs can be expected as a consequence of inhomogeneous topology[17]

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