The Efficiency of Synchronization Dynamics and the Role of Network Syncreactivity

Synchronization of coupled oscillators is a fundamental process in both natural and artificial networks. While much work has investigated the asymptotic stability of the synchronous solution, the fundamental question of the efficiency of the synchronization dynamics has received far less attention. Here we address this question in terms of both coupling efficiency and energy efficiency. We use our understanding of the transient dynamics towards synchronization to design a coupling-efficient and energy-efficient synchronization strategy, which varies the coupling strength dynamically, instead of using the same coupling strength at all times. Our proposed synchronization strategy is able in both simulation and in experiments to synchronize networks by using an average coupling strength that is significantly lower (and, when there is an upper bound on the coupling strength, significantly higher) than what is needed for the case of constant coupling. In either case, the improvement can be of orders of magnitude. In order to characterize the effects of the network topology on the transient dynamics towards synchronization, we propose the concept of network syncreactivity. This is distinct from the previously introduced network synchronizability, which describes the ability of a network to synchronize asymptotically. We classify real-world examples of complex networks in terms of both their synchronizability and syncreactivity.