Localization transition, Lifschitz tails and rare-region effects in network models

Abstract

Effects of heterogeneity in the suspected-infected-susceptible model on networks are investigated using quenched mean-field theory. The emergence of localization is described by the distributions of the inverse participation ratio and compared with the rare-region effects appearing in simulations and in the Lifschitz tails. The latter, in the linear approximation, is related to the spectral density of the Laplacian matrix and to the time dependent order parameter. I show that these approximations indicate correctly Griffiths Phases both on regular one-dimensional lattices and on small world networks exhibiting purely topological disorder. I discuss the localization transition that occurs on scale-free networks at $\gamma =3$ degree exponent.

arXiv comment9 pages, 9 figures, accepted version in PRE

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• Does the brain behave like a (complex) network? I. Dynamics
• Griffiths effects of the susceptible-infected-susceptible epidemic model on random power-law networks
• Efficient sampling of spreading processes on complex networks using a composition and rejection algorithm
• An overview of epidemic models with phase transitions to absorbing states running on top of complex networks
• Phase transition of the susceptible-infected-susceptible dynamics on time-varying configuration model networks
• Sudden spreading of infections in an epidemic model with a finite seed fraction
• Localization in random bipartite graphs: numerical and empirical study
• The Griffiths Phase on Hierarchical Modular Networks with Small-world Edges