Griffiths phases and localization in hierarchical modular networks

TL;DR

This paper investigates how hierarchical modular networks, inspired by brain connectivity, exhibit Griffiths phases and localization phenomena, especially near the percolation threshold, using simulations and mean-field theory.

Contribution

It demonstrates the emergence of Griffiths phases and localization in hierarchical networks with decaying connection probabilities, extending understanding of dynamics in brain-like structures.

Findings

Griffiths phases appear near the percolation threshold in these networks.

Localization occurs even in highly connected small-world networks with link disorder.

Extended power-law dynamics are observed in networks with finite topological dimension.

Abstract

We study variants of hierarchical modular network models suggested by Kaiser and Hilgetag [Frontiers in Neuroinformatics, 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.