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Eigenvector localization, dynamical correlations and epidemic thresholds on random networks with degree correlations

Banff International Research Station for Mathematical Innovation and Discovery

I will present comparisons between large-scale stochastic simulations and mean-field theories for the epidemic thresholds and prevalence of the susceptible-infected-susceptible (SIS) model on networks with power-law degree distributions and degree correlations. We confirm the vanishing of the threshold regardless of the correlation pattern and degree exponent. The thresholds are compared with heterogeneous mean-field (HMF), quenched mean-field (QMF) and pair quenched mean-field (PQMF) theories where the degree correlation patterns are explicitly considered. The PQMF, which additionally reckons dynamical correlations, outperforms the other two theories and its level of quantitative success depends on the type of degree correlation (assortative, disassortative or uncorrelated). Furthermore, we observe a strong correlation between the success of PQMF theory and the properties of the principal eigenvector such as the inverse participation ration (IPR) and the spectral gap. If the IPR is large and tends to a finite value at the limit of large networks the PQMF predictions deviate from numerical simulations. Otherwise, if the IPR is small, PQMF theory shows an excellent match with the simulations. Finally, the epidemic prevalence near to the critical point and the corresponding critical exponents are compared with both QMF theory and exact results.

Mathematics; Probability theory and stochastic processes; Statistics

video/mp4

Author affiliation: Universidade Federal de Viçosa

2019-11-18

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/72295

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/