Generalized projective lag synchronization between delay-coupled networks under circumstance noise

It is well known that time delay and random noise are universal in complex networks. However, the research on the synchroniza- tion of coupled networks that are subjected to delay-coupling and noise perturbation is very rare. In this paper, for two delay-coupled complex networks with different node dynamics, different topological structures and different numbers of nodes, under circumstance noise, the generalized projective lag synchronization between two networks is proposed for the first time. First, a more realistic theoretical framework is constructed for the drive-response network synchronization. Second, according to the LaSalle-type theorem for stochastic differential delay equations, we rigorously prove that the generalized projective lag synchronization between the drive-response networks can be achieved almost surely, by introducing an appropriate controller. Furthermore, numerical simulation is employed to verify the theoretical analysis. The results indicate that the drive-response networks can indeed achieve generalized projective lag synchronization, and that the synchronization is independent of time delay and scaling factor. Moreover, the remarkable influences of the update gain and the coupling delay on synchronization speed are revealed through the numerical results.