Paths towards synchronization: analytical treatment of completely connected networks

Abstract

Abstract We describe the transient regime of synchronizing flows defined over a network, introducing a codification of the paths towards synchronization. In all cases, the collection of paths toward synchronization defines a combinatorial structure: the transition diagram. In this paper, besides introducing this general framework, we study in full detail the case of completely connected networks where a monotonic convergence towards synchronization takes place. For these systems we are able to supply a detailed analytical description of the transition diagram describing both, the Laplacian flow and a special case of the Kuramoto model on the same network. This constitutes a first instance of a full symbolic description of the path towards synchronization and illustrates the hidden complexity of the transient regime.