Resumo: Quenched disorder is known to play a relevant role in dynamical processes and phase transitions. By studying the Contact Process (CP) we showed that Griffiths Phases (GP) and other rare region effects, leading rather generically to anomalously slow (algebraic, logarithmic,…) relaxation on Erdos-Renyi networks with explicit quenched disorder. More surprisingly, we found that GPs can also emerge solely as the consequence of topological heterogeneity on generalized small world networks exhibiting finite topological dimensions [1-3]. Similar power-law dynamics can also be observed on scale-free trees in case of disassortative weighting schemes, in the neighborhood of smeared phase transitions [4]. Recently I have pointed out that localization, described by quenched mean-field approximations is related to the existence of rare region effects and GPs in case of Susceptible Infected Susceptible (SIS) models on various complex networks [5-7], in particular on Barabasi-Albert type of networks with aging connections.
Bursty dynamics of agents is shown to appear at criticality or in extended GPs even in case of Poisson processes. I provide numerical evidence for power-law type of intercommunication time distributions by simulating the CP and SIS. This observation suggests that in case of non-stationary bursty systems the observed non-poissonian behavior can emerge as the consequence of an underlying hidden poissonian network process, which is either critical or exhibits strong rare-region effects. On contrary, in time varying networks rare-region effects do not cause deviation from the mean-field behavior and heterogeneity induced burstyness is absent [8].
[1] M. A. Munoz, R. Juhasz, C. Castellano, and G, Odor, Griffiths Phases on Complex Networks, Phys. Rev. Lett. 105, 128701 (2010)
[2] G. Odor, R. Juhasz, C. Castellano, M. A. Munoz, Griffiths phases in the contact process on complex networks, AIP Conf. Proc. 1332, Melville, New York (2011) p. 172-178. Non-equilibrium Statistical Physics Today, Proc. of the 11th Granada Seminar on Computational and Statistical Physics, La Herradura, Spain 13-17 Sept. 2010, Editors: P. L. Garrido, J. Marro, F. de los Santos.
[3] R. Juhasz, G. Odor, C. Castellano, M. A. Munoz, Rare region effects in the contact process on networks, Phys. Rev. E 85, 066125 (2012)
[4] G. Odor, R. Pastor-Satorras, Slow dynamics and rare-region effects in the contact process on weighted tree networks, Phys. Rev. E 86, 026117 (2012)
[5] Geza Odor, Rare regions of the susceptible-infected-susceptible model on Barabasi-Albert networks, Phys. Rev. E 87, 042132 (2013)
[6] Geza Odor, Slow dynamics of the contact process on complex networks, EPJ Web of Conferences 44, 04005 (2013)
[7] Geza Odor, Spectral analysis and slow spreading dynamics on complex networks, Phys. Rev. E 88 032109 (2013)
[8] G. Odor, Slow, bursty dynamics as the consequence of quenched network topologies, arXiv:1401.5303
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