Synchronization and Control for Multiweighted and Directed Complex Networks

The study of complex networks with multiweights (CNMWs) has been a hot topic recently. For a network with a single weight, previous studies have shown that they can promote synchronization, but for CNMWs, there are no rigorous analyses about the role of coupling matrices. In this brief, the complex network is allowed to be directed, which is the main difference with previous studies and may make the synchronization analysis difficult for multiple couplings. At first, we prove that if the inner coupling matrices are all diagonal, then synchronization can be realized only if the weighted sum (or union) of multiple coupling matrices is strongly connected, which bridges the gap between single-weighted and multiweighted networks. Moreover, we also consider the case that inner coupling matrices are positive definite but not diagonal. We design two techniques for this hard problem. One technique is to decompose inner coupling matrices into diagonal matrices and residual matrices. The other one is to measure the similarity between outer coupling matrices. In virtue of the normalized left eigenvectors (NLEVecs) corresponding to the zero eigenvalue of coupling matrices, we prove that if the Chebyshev distance between NLEVec is less than some value, defined as the allowable deviation bound, then the synchronization and control will be realized with sufficiently large coupling strengths. Furthermore, adaptive rules are also designed for coupling strength.