Synchronization Path In The Scale-Free Network

With the development of technology, more and more people use the network concept study of nature and society. With further research, it is proposed various types of networks, such as regular networks, random networks, deterministic networks, small-world networks and scale-free networks.People are interested in the dynamical behavior of characteristics of complex networks, where synchronization is a very important dynamic phenomena. Gardenes et al studied the scale-free networks where the positive correlation between local heterogeneity and dynamics enables the network to explosive synchronization. Thomas Kaue Dal’Maso Peron, who worked on the model Gardenes proposed, found that synchronization was improved by introducing a time delay coupling in the model. They found the existence of a time delay can contribute to the occurrence of an explosive in synchronization. However, their study did not consider nonlinear correlation of local heterogeneity and dynamics characteristics. Since previous studies of positive underlying assumptions can not find the reason that some local heterogeneity (oscillators’degrees) plays an important role in the explosive synchronization, nonlinear underlying assumptions have been made in this topic and can solve this problems.Details are as follows:a nonlinear distribution between the degree of oscillators and the initial frequency oscillator, introducing a new variable parameter β to describe the relationship between network structure and dynamics. We found that when β is small, synchronous path is continuous, with the increase of β, the synchronization path becomes a first-order phase transition and is not continuous, also known as the explosion of synchronization. We also found that the critical coupling strength of synchronization and desynchronization are related to β. Both the critical coupling strength and the regions with the hysteresis increases with the increase of β. Another addition to the conditions of the oscillator frequency distribution follows the degree, we also calculated when the initial frequency distribution follows Lorentz distribution. The synchronization transition may be either a continuous one or a discontinuous one depending on the correlation and that strong correlation always postpones both the transitions from the incoherent state to a synchronous one and the transition from a synchronous state to the incoherent one.Section of this paper is as follows:The first chapter describes the background knowledge of complex networks, some basic knowledge of synchronization and oscillator synchronization behavior on the evolution of complex networks. The second chapter describes the generation mechanism of the Barabasi-Albert (BA) network, presents the background of BA and analyzes characteristics of network. In the third chapter, we have a detailed description of the Kuramoto model and existing researches on the Kuramoto model. Chapter4, we introduce the nonlinear correlation model presented in this paper, and the simulation results are explained and analyzed. The full text of the summary is in the Chapter5, to purpose a brief introduction, significance of this research and the development of proposed future.