On the Characterization of Families of Synchronizing Graphs for Kuramoto Coupled Oscillators

Kuramoto model of coupled oscillators represents situations where several individual agents interact and reach a collective behavior. The interaction is naturally described by a interconnection graph. Frequently, the desired performance is the synchronization of all the agents. Almost global synchronization means that the desire objective is reached for every initial conditions, with the possible exception of a zero Lebesgue measure set. This is a useful concept, specially when global synchronization can not be stated, due, for example, to the existence of multiple equilibria. In this survey article, we give an analysis of the influence of the interconnection graph on this dynamical property. We present in a ordered way several known and new results that help on the characterization of what we have called synchronizing topologies.