  The study of network synchronization has attracted increasing attention
recently. 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 show 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 more links connecting the
nodes in the same layer (horizontal links) will lead to longer synchronization
time, while the increasing number of links connecting nodes in neighboring
layers (vertical links) will accelerate the synchronization. Our findings
provide insights into the roles of horizontal and vertical links in
synchronizing process, and suggest that the spectral analysis is helpful yet
insufficient for the understanding of network synchronization.
