Distributed Adaptive Control for Synchronization in Directed Complex Networks

The dynamics of a complex network is generally very complicated due to the self-dynamics of the node and their interactions. Many existing conditions for reaching certain desirable dynamics in a complex network require global information of the network, for example, the spectrum of its Laplacian matrix. A challenging problem is how the network structure affects the network dynamics in a distributed way especially for directed networks, which is still unclear today. This paper investigates the impact of the network structure on a general directed complex network and develops a scheme to change the weights in a local manner to achieve a desired behavior. In particular, network synchronization is investigated, for which some distributed adaptive laws are designed on the coupling weights for reaching synchronization in a directed complex network. It is found that the directed spanning tree structure is helpful while the aggregation and circular structures are harmful for achieving network synchronization. It is also found that the designed adaptive scheme on the weight of an edge in the network depends on the relative states between the terminal node and both of its inner and outer neighboring nodes, which reveals that the local information exchange in the terminal nodes is very important for reaching synchronization in directed complex networks.