title: Griffiths phases and localization in hierarchical modular networks.
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.
authors.affiliation: MTA-MFA-EK Research Institute for Technical Physics and Materials Science, H-1121 Budapest, P.O. Box 49, Hungary.
authors.forename: Géza
authors.initials: G
authors.lastname: Ódor
authors.affiliation: Departamento de Fisica and National Institute of Science and Technology of Complex Systems, ICEx, Universidade Federal de Minas Gerais, Caixa Postal 702, 30161-970, Belo Horizonte - Minas Gerais, Brazil.
authors.forename: Ronald
authors.initials: R
authors.lastname: Dickman
authors.affiliation: Massachusetts Institute of Technology, 77 Massachusetts Avenue Cambridge, MA 02139-4307, USA.
authors.forename: Gergely
authors.initials: G
authors.lastname: Ódor