Exponential synchronization for a class of complex networks of networks with directed topology and time delay

The pinning synchronization problem for a class of complex networks of networks is investigated in this paper, in which the network topology is assumed to be directed. The concerned complex networks of networks are composed of both leaders' network and followers' networks (also called “subnetworks”), where the leaders' network and followers' networks are regarded as the nodes of the networks of networks, followers' networks can receive the information from leaders' network, but not the reverse. Then, by means of the Lyapunov stability theory, graph theory, Barbalat's Lemma and the Kronecker product techniques, some criteria are obtained to guarantee the globally exponential synchronization for the general complex networks of networks. In addition, theoretical results are verified through an illustrative example.