The influence of global cues and local coupling on the rate of synchronization in the presence of time delays

The influence of global cues and local coupling on the rate of synchronization is analyzed in the presence of delayed interaction. First we give a delay-dependent synchronization condition. Then we prove that a stronger global cue always leads to a faster synchronization rate, even when it is only connected to a small number of nodes and the time delays in different channels are nonidentical. The local coupling is proven to have no influence on the synchronization rate when the global cue affects the whole network uniformly and delays in different channels are identical and small. This points out a way to elucidate complex synchronization properties in, e.g., biological networks. At last, using a published software DDE-BIFTOOL, synchronization rates are numerically calculated in the case study to confirm the theoretical predictions.