We present a comparison between stochastic simulations and mean-field theories for the epidemic threshold of the susceptible-infected-susceptible (SIS) model on correlated networks (both assortative and disassortative) with power-law degree distribution P (k) ∼ k −γ . We confirm the vanishing of the threshold regardless of the correlation pattern and the degree exponent γ. Thresholds determined numerically are compared with quenched mean-field (QMF) and pair quenched mean-field (PQMF) theories. Correlations do not change the overall picture: QMF and PQMF provide estimates that are asymptotically correct for large size for γ < 5/2, while they only capture the vanishing of the threshold for γ > 5/2, failing to reproduce quantitatively how this occurs. For a given size, PQMF 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 γ < 5/2, while QMF provides an estimate of the epidemic threshold that is asymptotically exact, it fails to reproduce the singularity of the prevalence around the transition.

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Localization phenomena permeate many branches of physics playing a fundamental role on dynamical processes evolving on heterogeneous networks. These localization analyses are frequently grounded, for example, on eigenvectors of adjacency or non-backtracking matrices which emerge in theories of dynamic processes near to an active to inactive phase transition. We advance in this problem gauging nodal activity to quantify the localization in dynamical processes on networks whether they are near to a transition or not. The method is generic and applicable to theory, stochastic simulations, and real data. We investigate spreading processes on a wide spectrum of networks, both analytically and numerically, showing that nodal activity can present complex patterns depending on the network structure. Using annealed networks we show that a localized state at the transition and an endemic phase just above it are not incompatible features of a spreading process. We also report that epidemic prevalence near to the transition is determined by the delocalized component of the network even when the analysis of the inverse participation ratio indicates a localized activity. Also, dynamical processes with distinct critical exponents can be described by the same localization pattern. Turning to quenched networks, a more complex picture, depending on the type of activation and on the range of degree exponent, is observed and discussed. Our work paves an important path for investigation of localized activity in spreading and other processes on networks.

We investigate a fermionic susceptible-infected-susceptible model with mobility of infected individuals on uncorrelated scale-free networks with power-law degree distributions P (k) ∼ k −γ of exponents 2 < γ < 3. Two diffusive processes with diffusion rate D of an infected vertex are considered. In the standard diffusion, one of the nearest-neighbors is chosen with equal chance while in the biased diffusion this choice happens with probability proportional to the neighbor's degree. A non-monotonic dependence of the epidemic threshold on D with an optimum diffusion rate D * , for which the epidemic spreading is more efficient, is found for standard diffusion while monotonic decays are observed in the biased case. The epidemic thresholds go to zero as the network size is increased and the form that this happens depends on the diffusion rule and degree exponent. We analytically investigated the dynamics using quenched and heterogeneous mean-field theories. The former presents, in general, a better performance for standard and the latter for biased diffusion models, indicating different activation mechanisms of the epidemic phases that are rationalized in terms of hubs or max k-core subgraphs.Nowadays, we live in an interwoven world where information, goods, and people move through a complex structure with widely diversified types of interactions such as on-line friendship and airport connections. These and many other systems of completely distinct nature can be equally suited in a theoretical representation called complex networks, in which the elements are represented by vertices and the interactions among them by edges connecting these vertices.The study of epidemic processes on complex networks represents one of the cornerstones in modern network science and can aid the prevention (or even stimulation) of disease or misinformation spreading. The relevance of the interplay between diffusion and epidemic spreading in real systems is self-evident since hosts of infectious agents, such as people and mobile devices, are constantly moving, being the carriers that promote the quick transition from a localized outbreak to a large scale epidemic scenario. In this work, we perform a theoretical analysis and report nontrivial roles played by mobility of infected agents on the efficiency of epidemic spreading running on the top of complex networks. We expect that our results will render impacts for forthcoming research related to the area.