Synchronizability And Pinning Control Of Partially Interdependent Networks

In the past twenty years,there have been a lot of research results on complex dynamical networks.For the complex network synchronization,the results are mainly based on two aspects.One is starting from the network structure,judging and improving the synchronizability of the networks.The other is starting from the dynamic behavior,investigating the synchronization of the networks and proposing the synchronization sufficient condition.Interdependent networks,also known as ‘network of networks’,consist of a number of complex networks with different properties and different scales coupled by interdependent relationships.Concerning the above two aspects,in this paper,the synchronizability and synchronization of partially interdependent networks are investigated.The main contents and innovation points are listed as the following.(1)Based on the master stability function method,some theoretical results are derived for the synchronizability of partially interdependent networks.For bounded and unbounded synchronous regions and three different kinds of inter-coupling,we illustrate how factors of the topology can affect the synchronizability of networks with a Barabási-Albert(BA)scale-free and a Newman-Watts(NW)small-world subnet(simplified here as BA-NW network)and networks with a BA scale-free and a nearest-neighbor(NN)subnet(BA-NN network),respectively.When only the scale of the networks is increased,the synchronizability systematically reduces.If only increasing the inter-coupling strength or the number of interdependent nodes,the synchronizability keeps improving.Generally,we find that BA-NW networks have better synchronizability than BA-NN networks and positively correlated coupling is the best interconnection.Random inter-coupling,by contrast,is the best choice for BA-NN networks.(2)Based on the Lyapunov stability theory,pinning control technology,adaptivecontrol technology and linear matrix inequalities,in this paper,the heterogeneous adaptive pinning synchronization problems for partially interdependent networks are discussed,in which the coupling functions are linear and nonlinear functions,separately.With controllers on partial nodes in the networks,we propose some sufficient conditions so that each sub-network in the networks synchronizes to the isolated node system of them.The final numerical simulations prove the feasibility and effectiveness of the proposed theoretical results.