Rare regions of the susceptible-infected-susceptible model on Barabási-Albert networks

Abstract

I extend a previous work to susceptible-infected-susceptible (SIS) models on weighted Barabási-Albert scale-free networks. Numerical evidence is provided that phases with slow, power-law dynamics emerge as the consequence of quenched disorder and tree topologies studied previously with the contact process. I compare simulation results with spectral analysis of the networks and show that the quenched mean-field (QMF) approximation provides a reliable, relatively fast method to explore activity clustering. This suggests that QMF can be used for describing rare-region effects due to network inhomogeneities. Finite-size study of the QMF shows the expected disappearance of the epidemic threshold λ c in the thermodynamic limit and an inverse participation ratio ∼0.25, meaning localization in case of disassortative weight scheme. Contrarily, for the multiplicative weights and the unweighted trees, this value vanishes in the thermodynamic limit, suggesting only weak rare-region effects in agreement with the dynamical simulations. Strong corrections to the mean-field behavior in case of disassortative weights explains the concave shape of the order parameter ρ(λ) at the transition point. Application of this method to other models may reveal interesting rare-region effects, Griffiths phases as the consequence of quenched topological heterogeneities.

Critical dynamics on a large human Open Connectome network

“…This introduces disassortativity, enhancing RR effects [19], besides the modularity [23]. However, inhibitory links increase the heterogeneity so drastically, that a full equalization of the internal sensitivity may not be an obligatory condition for finding Griffiths effects.…”

“…It has been conjectured that network heterogeneity * odor@mfa.kfki.hu can cause GPs if the topological (graph) dimension D, defined by N r ∼ r D , where N r is the number of (j ) nodes within topological distance r = d(i,j ) from an arbitrary origin (i), is finite [16]. This hypothesis was pronounced for the contact process (CP) [17], but subsequent studies found numerical evidence for its validity in the case of more general spreading models [18][19][20]. Recently, a GP has been reported in synthetic brain networks [21][22][23] with finite D. At first sight this seems to exclude relevant disorder effects in the so-called small-world network models.…”

Slow epidemic extinction in populations with heterogeneous infection rates

“…This is probably because weights are randomly distributed over the network, thus high-degree nodes always spread the disease (it is very unlikey that all links attached to hubs have very low weights). Instead, assigning weights according to the topology of the network may induce rare-region effects, as it was shown in [16][17][18] using Barabasi-Albert trees with disassortative weighting. It would be worthwhile to perform a deeper analysis to study how relaxations are affected by other properties of real contact networks, such as topological and temporal correlations.…”

“…Other studies have introduced heterogeneity at the individual level, by assigning power law intertime events [13,14], node-dependent infection rates [15], or topology-dependent weight patterns [16][17][18]. In our model heterogeneity is at the interaction level, by means of link-dependent infection rates which are not correlated with the topology of the network.…”

Spectral analysis and slow spreading dynamics on complex networks

“…This is based on optimal fluctuation theory and simulations of the contact process (CP) [21,22] on Erdős-Rényi (ER) [20] and on generalized small world networks [23][24][25]. In the case of networks with an infinite topological dimension, like the Barabási-Albert (BA) [26] graph, slow dynamics has been found only in tree networks and weighting schemes, that suppress the information propagation among hubs [27,28].…”