### Rumour Spreading

Most of real-world temporal networks are generated by non-Markovian processes, i.e. by processes that include memory effects. We analyzed a real world dataset of mobile phone calls and provided a simple empirical characterization of the effects of memory in its microscopic dynamical evolution. Considering the empirical evidences, we defined a novel generative model for time-varying networks with memory. The model mirrors many of the structural properties observed in the real network, like degree and weight heterogeneities, and shows the spontaneous emergence of non-trivial connectivity patterns characterized by strong and weak ties. We characterize the effects of non- Markovian and heterogeneous connectivity patterns on rumor spreading processes. Interestingly, we find that strong ties are responsible for constraining the rumor diffusion within localized groups of individuals. This evidence points out that strong ties may have an active role in weakening the spreading of information by constraining the dynamical process in clumps of strongly connected social groups.

The non-Markovian properties of real-world temporal networks have also been investigated. In particular, we have discussed which properties of a temporal network can lead to the slow-down or speed-up of propagation processes, and related these effects to the spectral properties of the network. We have indeed used a novel causality-preserving time-aggregated representation to analyze temporal networks from the perspective of spectral graph theory and provide one of the first analytical explanations for the frequently observed slow-down of information diffusion in empirical non-Markovian temporal networks. Using our approach we derived an analytical prediction for the magnitude of this slow-down and we validated our prediction against empirical data sets. Counter-intuitively, we further showed that non-Markovian properties could result in a speed-up of information diffusion that can be related to the spectral properties of the underlying temporal network.

Overall, the presented results underline the subtleties inherent to the analysis of dynamical processes in exogenous time-varying networks. No one-fits-all picture exists, and a classification of dynamical process behaviour calls for a thorough analysis of each particular processes and networks considered.

Time Varying Networks and the weakness of strong ties

Causality-driven slow-down and speed-up of diffusion in non-Markovian temporal networks