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.

9 pages, 9 figures, accepted version in PRE

References in corpus (14)

• Thresholds for epidemic spreading in networks
• Small But Slow World: How Network Topology and Burstiness Slow Down Spreading
• Rare region effects at classical, quantum, and non-equilibrium phase transitions
• Epidemic thresholds of the Susceptible-Infected-Susceptible model on networks: A comparison of numerical and theoretical results
• Griffiths phases on complex networks
• Laplacian spectra of complex networks and random walks on them: Are scale-free architectures really important?
• Infinite-randomness critical point in the two-dimensional disordered contact process
• Langevin approach for the dynamics of the contact process on annealed scale-free networks
• Optimal network topologies: Expanders, Cages, Ramanujan graphs, Entangled networks and all that
• Slow dynamics and rare-region effects in the contact process on weighted tree networks
• Rare regions of the Susceptible Infected Susceptible model on Barabási-Albert networks
• Random elastic networks : strong disorder renormalization approach
• Superdiffusion in a class of networks with marginal long-range connections
• Rare regions and Griffiths singularities at a clean critical point: The five-dimensional disordered contact process

Cited by in corpus (4)

• Multiple phase transitions of the susceptible-infected-susceptible epidemic model on complex networks
• Does the brain behave like a (complex) network? I. Dynamics
• Localization in random bipartite graphs: numerical and empirical study
• The Griffiths Phase on Hierarchical Modular Networks with Small-world Edges