Abstract

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 a control 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). The transition between such two phases can be discontinuous and irreversible, or smooth, continuous, and reversible. The first case is known as Explosive Synchronization, and refers to an abrupt onset of synchronization following an infinitesimally small change in the control parameter.  The second case is the most commonly observed one,  and corresponds to a second-order phase transition, resulting in intermediate states emerging in between the two phases. Namely, the path to synchrony is characterized by a sequence of events: structured states emerge made of different functional modules (or clusters), each one evolving in unison. In my talk, I will assume that, during the transition, the synchronous solution of each of such clusters does not differ substantially from that of the entire network and, under such an approximation, I will  introduce a (simple, effective, and limited in computational demand) method which is able to: i) predict the entire sequence of events  that are taking place during the transition, ii) identify exactly which graph's node belongs to each of the emergent clusters, and iii) provide a well approximated calculation of the critical coupling strength value at which each of such clusters is observed to synchronize. I will also demonstrate that, under the assumed approximation, the sequence of events is in fact universal, in that it is independent of the specific dynamical system operating in each network's node and depends, instead, only on the graph's structure.