Synchronization on effective networks

The study of network synchronization has attracted increasing attentionrecently. In this paper, we strictly define a class of networks, namely effective networks, which are synchronizable and orientable networks. We can prove that all the effective networks with the same size have the same spectra, and are of the best synchronizability according to the master stability analysis. However, it is found that the synchronization time for different effective networks can be quite different. Further analysis shows that the key ingredient affecting the synchronization time is the maximal depth of an effective network: the larger depth results in a longer synchronization time. The secondary factor is the number of links. The increasing number of links connecting nodes in the same layer (horizontal links) will lead to longer synchronization time, whereas the increasing number of links connecting nodes in neighboring layers (vertical links) will accelerate the synchronization. Our analysis of the relationship between the structure and synchronization properties of the original and effective networks shows that the purely directed effective network can provide an approximation of the original weighted network with normalized input strength. Our findings provide insights into the roles of depth, horizontal and vertical links in the synchronizing process, and suggest that the spectral analysis is helpful yet insufficient for the comprehensive understanding of network synchronization.