Modeling memory effects in activity-driven networks

Activity-driven networks (ADNs) have recently emerged as a powerful paradigm to study the temporal evolution of stochastic networked systems. All the information on the time-varying nature of the system is encapsulated into a constant activity parameter, which represents the propensity to generate connections. This formulation has enabled the scientific community to perform effective analytical studies on temporal networks. However, the hypothesis that the whole dynamics of the system is summarized by constant parameters might be excessively restrictive. Empirical studies suggest that activity evolves in time, intertwined with the system evolution, causing burstiness and clustering phenomena. In this paper, we propose a novel model for temporal networks, in which a self-excitement mechanism governs the temporal evolution of the activity, linking it to the evolution of the networked system. We investigate the effect of self-excitement on the epidemic inception by comparing the epidemic threshold of a Susceptible-Infected-Susceptible model in the presence and in the absence of the self-excitement mechanism. Our results suggest that the temporal nature of the activity favors the epidemic inception. Hence, neglecting self-excitement mechanisms might lead to harmful underestimation of the risk of an epidemic outbreak. Extensive numerical simulations are presented to support and extend our analysis, exploring parameter heterogeneities and noise, transient dynamics, and immunization processes. Our results constitute a first, necessary step toward a theory of ADNs that accounts for memory effects in the network evolution.