Relaxation of synchronization on complex networks

We study collective synchronization in a large number of coupled oscillators on various complex networks. In particular, we focus on the relaxation dynamics of the synchronization, which is important from the viewpoint of information transfer or the dynamics of system recovery from a perturbation. We measure the relaxation time τ that is required to establish global synchronization by varying the structural properties of the networks. It is found that the relaxation time in a strong-coupling regime (K> Kc) logarithmically increases with network size N, which is attributed to the initial random phase fluctuation given by O (N-1/2). After elimination of the initial-phase fluctuation, the relaxation time is found to be independent of the system size; this implies that the local interaction that depends on the structural connectivity is irrelevant in the relaxation dynamics of the synchronization in the strong-coupling regime. The relaxation dynamics is analytically derived in a form independent of the system size, and it exhibits good consistency with numerical simulations. As an application, we also explore the recovery dynamics of the oscillators when perturbations enter the system.