From brain dynamics and neuronal firing, to power grids or financial markets, synchronization of networked  units is the collective behavior characterizing the normal functioning of most natural and man made systems. As an order parameter (typically the coupling strength in each link of the network) increases, a transition occurs between a fully disordered and gaseous-like phase (where the units evolve in a totally incoherent manner) to an ordered  or solid-like phase (in which, instead, all units follow the same trajectory in time). With the only help of eigenvalues and eigenvectors of the graph's Laplacian matrix, I show that the transition to synchronization of a generic networked dynamical system can be entirely predicted and completely characterized. In particular, the transition is made of a well defined sequence of events, each of which corresponds to either the nucleation of one (or several) cluster(s) of synchronized nodes or to the merging of multiple synchronized clusters into a single one. The network's nodes involved in each of such clusters can be exactly identified, and the value of the coupling strength at which such events are taking place (and therefore, the complete events' sequence) can be rigorously ascertained. We moreover clarify that the synchronized clusters are formed by those nodes which are  indistinguishable at the eyes of any other network's vertex, and as so they receive the same dynamical input from the rest of the network. Therefore, such clusters are more general subsets of nodes than those defined by the graph's symmetry orbits, and at the same time more specific than those described by network's equitable partitions. Finally, we present large scale simulations which show how accurate are our predictions in describing the synchronization transition of both synthetic and real-world large size networks, and we even report that the observed sequence of clusters is preserved in heterogeneous networks made of slightly non identical systems.

The COVID-19 pandemic forced an unprecedented response from health authorities worldwide. It also elicited a huge effort from the scientific community that rushed to improve and make more accurate already existent tools in network epidemiology. Research has produced many key results that combine theoretical models with data-driven simulations and network science tools. Here we show results that correspond to different stages of the COVID-19 pandemic in order to illustrate the trade-off between what models can tell and where more detailed data-driven modeling is useful, even if data is incomplete. Our models are tailored to work with networks of interactions at different levels, including mobility and population data from Europe and the U.S., and allow estimating the effectiveness of customary public interventions on the spread of COVID-19. We conclude by identifying the most pressing needs in terms of models and data for better preparedness for future similar pandemic scenarios.

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