Abstract
We present a comparison between stochastic simulations and mean-field theories for the epidemic threshold of the susceptible-infected-susceptible model on correlated networks (both assortative and disassortative) with a power-law degree distribution . We confirm the vanishing of the threshold regardless of the correlation pattern and the degree exponent . Thresholds determined numerically are compared with quenched mean-field (QMF) and pair quenched mean-field (PQMF) theories. Correlations do not change the overall picture: The QMF and PQMF theories provide estimates that are asymptotically correct for large sizes for , while they only capture the vanishing of the threshold for , failing to reproduce quantitatively how this occurs. For a given size, PQMF theory is more accurate. We relate the variations in the accuracy of QMF and PQMF predictions with changes in the spectral properties (spectral gap and localization) of standard and modified adjacency matrices, which rule the epidemic prevalence near the transition point, depending on the theoretical framework. We also show that, for , while QMF theory provides an estimate of the epidemic threshold that is asymptotically exact, it fails to reproduce the singularity of the prevalence around the transition.
Highlights
Metabolic chains of protein interactions [1], collaborations among scientists, co-starring in a movie [2], and person-toperson contacts [3] are all examples of interacting systems that can be modeled using complex networks [2]
For γ < 5/2, while quenched mean-field (QMF) theory provides an estimate of the epidemic threshold that is asymptotically exact, it fails to reproduce the singularity of the prevalence around the transition
In this work we investigate the ability of the aforementioned approaches (HMF, QMF, and pair quenched mean-field (PQMF)) to quantitatively predict the value of the epidemic threshold for both uncorrelated and correlated networks generated using the Weber-Porto model [35] and for real-world topologies
Summary
Metabolic chains of protein interactions [1], collaborations among scientists, co-starring in a movie [2], and person-toperson contacts [3] are all examples of interacting systems that can be modeled using complex networks [2]. A basic approach to investigate dynamical processes on networks is the heterogeneous mean-field (HMF) theory, in which degree heterogeneity and correlations are taken into account through the distributions P(k) and P(k|k ), respectively [8,9,17,18]. In spite of being qualitatively correct, QMF theory is not able to accurately predict the effective finite-size epidemic threshold in this case [25]. In this work we investigate the ability of the aforementioned approaches (HMF, QMF, and PQMF) to quantitatively predict the value of the epidemic threshold for both uncorrelated and correlated networks generated using the Weber-Porto model [35] and for real-world topologies.
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