  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.
