Study Of The Topologies And Synchronization Of The Oscillator Networks Based On The Spectrum Analysis

In recent years, domestic and foreign scientists have raised the study upsurge on the studying complex network. In nature and human society, the network is everywhere.the complex network can truely reflect some important topology structures and statistical properties in the real world complex systems. The study of complex network not only has the academic significance, but also has the important application value, it can provide a new approach to understand the nature and know the real world for humans. With the research scholars of various subject areas to the study of complex networks gradually, they began to explore the relationship between the structure properties of complex networks and network dynamics behavior. In the study of the dynamics of the network, at present, the main concerned is the study of synchronization behavior of complex networks.This paper studied the network synchronization with the Kuramoto phase oscillators being use as an individual node. In the studying, combined with Kuramoto oscillators model, the order parameter was taken as an index for scaling the network synchronizability. We studied and analyzed the synchronization performance of KM oscillators network and weighted oscillators network. By studying complex oscillator network synchronization, we can better analyze and understand the reality of the actual network synchronization problems. The main content of this paper is as follows:1. We studied the relationship between the eigenvalue spectrum of complex networks and network synchronization performance. Firstly, we introduced the concept of the complex network adjacency matrix and Laplacian matrix.Then,we introduced the concept of network synchronization and the relationship between synchronization and the eigenvalue spectrum of complex networks. Finally, using the knowledge of matrix theory, starting from the basic ER random network and BA scale-free network model we studied the difference of the eigenvalue spectral properties and the spectral density of the Uniform network and the non-uniform network. Complex networks’spectral analysis and spectral density analysis can provide a new method for study and analysis the topology structures of the actual network.2. We studied the relationship between the topology structures of the basic oscillator networks model and its synchronizability. Firstly,we made synchronization simulation using Kuramoto oscillators network for BA scale-free network,and studied the effect of the coupling strength on the synchronizability of BA scale-free network and obtained the critical coupling strength for global synchronization and the evolutive of each oscillator’s phase during the process of network synchronization. Then, we compared the synchronizability of three kinds of networks which are ER random networks, NW small world networks and BA scale-free networks combining with the Kuramoto oscillators model. Study shows that BA scale-free networks are prone to appear partial synchronization, but ER random networks are much easier to realize global synchronization and the NW small-world networks are relative difficult to realize synchronization among the three topologies. Different network’s topology structures affect network synchronizability.3. We studied weighted oscillator networks synchronization performance and the improvement of network synchronizability. Firstly, we introduced two typical weighted network model, which are BBV model and WANG model.And then,we studied the synchronization performance of the two weighted oscillator networks combining with the Kuramoto oscillators model. Study shows that the WANG networks’synchronizability is the best because WANG networks model has the topology structures which have uniformity weight distribution and elatively small power-law properties of the node strength,and the two weighted network synchronizability are better than not entitled to the BA scale-free networks. Finally, we used MZK way to make general BA scale-free network been weighted, the study result indicated, when the parameters are β=-1,the weighted BA scale-free oscillator network synchronizability is significantly improved.