The nodes and links connecting them are the two main geometric elements of a complex network. If the nodes and links change over time, all the nodes and all the links are regarded the dynamic systems, respectively, and thus, a complex dynamical network (CDN) can be regarded to be composed of the two interconnected subsystems: the nodes subsystem (NS) and the links subsystem (LS). The outer synchronization happens in two CDNs, which means that the state error between the two NSs converges to zero. However, the dynamics of LS is ignored in the existing literature about the outer synchronization. In addition, the existing literature rarely consider the unknown interaction between two CDNs into the outer synchronization. In view of this, this paper investigates the outer synchronization of two CDNs with the unknown interaction. By using the dynamics of LSs and synthesizing the adaptive control scheme, the outer synchronization is asymptotically achieved. The results in this paper show that the outer synchronization happens when the two LSs can track, respectively, the synthesized auxiliary dynamic tracking targets. Finally, the numerical simulation is given to show the effectiveness of the theoretical results in this paper.

The nodes and their connection relationships (CRs) are the two main parts for a complex dynamical network (CDN). In existing theoretical studies about the outer synchronization, the nodes are considered as the main part of synchronization phenomena mainly associated by coupling effect of CRs between nodes. However, if the CRs between nodes are time-varying, they can also be regarded as one dynamic system coupled with the nodes, and thus their state may evolve with time and maybe assist the nodes to achieve the outer synchronization. From the angle of large-scale systems, a CDN can be regarded as two interconnected subsystems, one of which is the node subsystem (NS) and the other is the link subsystem (LS). Hence, how the whole dynamic of LS contributes to the outer synchronization of NS is one of worthy research problem. In this paper, the two CDNs are considered with the unknown interaction. In each CDN, the dynamics of NS is modelled as the vector differential equation, the LS is modelled as the Riccati matrix differential equation, and the two kinds of differential equations are coupled with each other. By employing the above dynamic models of CDNs and the synthesized coupling terms in the two LSs, the adaptive controller of NS is synthesized for the response CDN. The results show that the outer synchronization happens when the two LSs tracking the synthesized auxiliary dynamic tracking targets. Finally, the numerical simulation is given to show the effectiveness of the theoretical results in this paper.