The Study Of Synchronization And Pinning Control Of The Complex Network With Subnets

In recent years, with the rapid development of information technology and science technology, coupled with extensive cross-penetration of the interdiscipline, complex networks peremptorily have become a hot research field of present-day academia and get the attention and favor of the researchers from various fields. This paper mainly discusses the development of complex network and its general research situation. Here, we briefly introduce several important statistical parameters of complex networks, and give detailed description of the network synchronization dynamics, synchronizability judgement and several important complex network models. Finally, through the application of matrix theory and the nonlinear system and control theory, combined with computer software (Matlab and C language) simulation, we have deep studied the synchronizability and pinning control of the complex network with subnets. The main research work are as follows:1. With the BA scale-free network model, we investigate respectively the effects of the interior average links of the subnetworks, the external links of the subnetworks, and subnet nodes on the synchronizability of a whole network. Based on the study of the eigenvalues of the network coupling matrix, we find that the increase of the interior average links of the small subnet or the external links of the subnetworks both can improve the whole network’s synchronizability. However, the increase of the nodes of the small subnet can worsen the synchronizability of the whole network. Finally, with the Lorenz oscillator as the individual node, we build a network composing by two different size BA scale-free subnets. The simulation results on this network have verified the correctness of the conclusions on the synchronizability of the network.2. Based on the asymmetric weighted method, we investigate the effects of the topologies of the subnets on synchronizability of a whole network. Three topologies which are BA scale-free network, NW small world network and ER random graph are discussed, respectively. Based on the study of the eigenvalues of the network coupling matrix, we find that the synchronizability of a whole network is mainly determined by the topology of the small subnet. When the asymmetric couplings from larger to smaller subnet is dominated, the whole network can achieve the best synchronizability with the small subnet being a BA scale-free network. And this kind of network can get the worst synchronizability when the asymmetrical couplings from smaller to larger subnet is dominated. Meanwhile, we find that the synchronization is firstly achieved in the subnets and then between the subnets. Finally, with the Lorenz oscillator as the individual node, we build the networks composing by different topological subnets. The simulation results on these networks have verified the correctness of the conclusions on the synchronizability of the networks.3. Using the complex network with the symmetric weighted BA scale-free subnets, we investigate its pinning controllability. By controlling some important nodes which are selected by a certain percentage, we discuss the pinning controllability of the complex network when several different methods are applied to choose controlled nodes and different percentages are used to allocate controlled nodes in subnets. Through the detailed study, we find that for the complex network with subnets the selection methods of controlled nodes have important effect on the pinning controllability. Meanwhile, when we choose the controlled nodes, we should select respectively the nodes with big degree from different subnets, which can achieve effective control of each subnet, and then achieve better control of the whole network. And there exists an optimal allocation proportion to achieve optimal pinning controllability, which corresponds to the equal percents for the two subnets.To a certain extent, the complex networks with subnets in this paper reflect the characteristics of most of the real networks. The obtained conclusions have a certain theoretical value and instruction significance for us to build and control the actual complex networks and provide some theoretical basis for the further research of network with more complex structure.