Analysis And Improvement Of The Dynamical System Synchronizability On Complex Network

Since D. J. Watts and S.H. Strogatz presented small-world network model in1998and A. L. Barabasi and R. Albert proposed Scale-free network model in1999, the complex network has been widely researched in the field of mathematics, physics, computer science, sociology and so on, and it becomes one of the most energetic cross subjects. Simply, the network is composed of a number of nodes and edges, which nodes are used to represent the different individuals in the real networks and the edges indicate a relationship between individuals. In fact, the complex network can be seen everywhere in our life, for example, we can abstract the network from the electricity network, railway network, road network, relationships and so on.The ultimate goal of studying complex networks is to understand the effect of topological properties on the dynamical behavior (such as transportation, transmission, game and synchronization, etc.). The study of the dynamic system synchronization property on complex network is one of the important issues. It has not only application value, but also value of scientific research. For example, it is widely applied in the transportation and information dissemination.In this paper, we used the eigenvalues of coupling matrix to measure synchronizability on theory analysis and numerical simulation method. We investigate the effect of average degrees on synchronizability and the structure propety of the network. We design a way to improve synchronizability of the network and we also present a method to obtained networks with optimal synchronizability. The main contents in this paper are as follows:(1) The network average degree is a very important network characteristic, so we explore the relationship between the average degree and synchronizability. It can be found that as the average degrees increased, the network synchronizability is improved. Compared with regular the nearest coupling network, the synchronizability of small-world network and BA scale-free network is much better. Surprisingly, we find that the synchronizability of BA Scale-free networks is similar to random network with the same average degree when the input degree is normalized for each degree.(2) We made an overview on the complex network, and analysed the various methods for improving the complex network synchronizability which proposed by scholars in detail. It can be found that some methods can make the network achieve optimal synchronizability, however, these methods have significant limitations, such as huge amount of computation and slow computing speed, or being not able to obtain optimal synchronizability. Thus, one method about local structure information of the network is introduced in our work, which uses the nodes’ degree to improve the network synchronizability. When the degrees of next nearest neighbors are taken into account, it is found that the network synchronizability is further improved, and achieves optimal state. Our method use only local strueture information but are useful for all kind of scale-free networks.(3) We proposed a new method for obtained effective networks with local structure information, compared with the methods which scholars have introduced, our method is simple in operation and the effective networks will hold the structure property of the original networks to some extent. It is a useful method for obtained effective networks, and to adjust the synchronizability of the scale-free networks.