Local adaptive heterogeneous synchronization for interdependent networks with delayed coupling

Abstract

With the development of the networks, the coupling between networks has become increasingly significant. Here, the networks can be described as interdependent networks. An interdependent network can have two different kinds of links, a connectivity link and a dependency link, which are fundamental properties of interdependent networks. During the past several years, interdependent complex network science has attracted a great deal of attention. This is mainly because the rapid increase in computing power has led to an information and communication revolution. Investigating and improving our understanding of interdependent networks will enable us to make the networks (such as infrastructures) we use in daily life more efficient and robust. As a significant collective behavior, synchronization phenomena and processes are common in nature and play a vital role in the interaction between dynamic units. At the same time, the time delay problem is an important issue to be investigated, especially in biological and physical networks. As a matter of fact, time delays exist commonly in the real networks. A signal or influence traveling through a network is often associated with time delay. In this paper, the local adaptive heterogeneous synchronization is investigated for interdependent networks with delayed coupling consisting of two sub-networks, which are one-by-one inter-coupled. The delays exist both in the intra-coupling and in the inter-coupling between two sub-networks, the intra-coupling and inter-coupling relations of the networks satisfy the requirements for nonlinearity and smoothness, and the nodes between two sub-networks have different dynamical systems, namely heterogeneous systems. Based on the Lyapunov stability theory, linear matrix inequality, and adaptive control technique, with proper controllers and adaptive laws for the networks, the sufficient conditions are proposed to synchronize the sub-networks of the interdependent networks into heterogeneous isolated systems, respectively. In order to illustrate the main results of the theoretical analysis clearly, some numerical simulations for an interdependent network with NW small world sub-network and BA sub-network are presented, in which each sub-network has 100 nodes and the heterogeneous systems are Lorenz and Rössler systems. The numerical simulations show that using the controllers and adaptive laws proposed, the network obtains the local heterogeneous synchronization quickly, that is, the nodes of two sub-networks are synchronized into Lorenz and Rössler systems separately. Thus, they verify the feasibility and correctness of the proposed techniques. It is worth noting that the presented results are delay-independent. In the future, our research will be directed to the further investigation of the delay-dependent synchronization of interdependent networks by using the current results as a basis.