Epidemic processes and spectral properties of complex networks

Romualdo Pastor-Satorras (Universitat Politecnica de Catalunya, Spain)

The discernment of the mechanisms that activate spreading processes on heterogeneous substrates is a pivotal issue, with practical applications ranging from the containment of epidemic outbreaks to the viral spreading of rumors and beliefs. The interest on the effects of heterogeneity has been brought about by the observation that social contact networks (the natural substrate for most human epidemic processes) are generally strongly heterogeneous, observation that has led to the introduction of complex network theory in the quantitative analysis of epidemic spreading. In this context, the nature of those mechanisms translates on simple epidemic models in setting the epidemic threshold λ_c for some rate of infection λ (the spreading rate), separating a phase in which the spreading affects a finite fraction of the population from a state in which only a vanishingly small fraction is hit. The research effort is thus focused on a twofold objective: The identification of the activation mechanisms as a function of the network topology, and the determination of the functional form of the epidemic threshold. In this talk we will show that, for the simple susceptible-infected-susceptible (SIS) model, the steady state is sustained by different mechanisms, depending on the degree of heterogeneity of the network, and we will discuss the relation of the these mechanisms and the associated epidemic thresholds with the spectral properties of the network. Reasoning in analogy with the observations for epidemics processes will allow us to obtain results regarding the spectral properties of complex networks.