  The study of complex networks with multi-weights has been a hot topic
recently. For a network with a single weight, previous studies have shown that
they can promote synchronization. But for complex networks with multi-weights,
there are no rigorous analysis to show that synchronization can be reached
faster. In this paper, the complex network is allowed to be directed, which
will make the synchronization analysis difficult for multiple couplings. In
virtue of the normalized left eigenvectors (NLEVec) corresponding to the zero
eigenvalue of coupling matrices, we prove that if the Chebyshev distance
between NLEVec is less than some value, which is defined as the allowable
deviation bound, then the synchronization and control will be realized with
sufficiently large coupling strengths, i.e., all coupling matrices do
accelerate synchronization. Moreover, adaptive rules are also designed for the
coupling strength.
