Synchronization Analysis And Computation In Dynamical Complex Networks

Generally speaking, a complex network is a large set of interconnected nodes, in which a node is a fundamental unit with specific contents. The recent decade has witnessed the birth of a new movement of interest and research on the study of complex networks throughout the world. The researchers in physics, biology, mathematics and computer science dedicate to the study of complex networks. In this dissertation, we apply the theory of dynamical systems and method of numerical computation to studying the synchronization of complex networks, explaining the relations between topological structures and synchronization. The main work is to study synchronization in delayed complex networks, the synchronization and bifurcation of weighted network and synchronization between two coupled networks. In details, the main contents of the dissertation are organized as follows:1. Synchronization analysis in delayed complex network is studied in Chapters 2 and 3, where some sufficient conditions on synchronization are derived. Firstly we expand the coupling delay by introducing the "delay vector" and "delay matrix"; secondly synchronization of complex networks with time delay and nonlinear inner-coupling function is also discussed, we use the linear matrix equality to obtain the theory of synchronization; the numerical examples taken here show the efficiency of the derived theory.2. Chapter 4 deals with the transition of synchronized states in weighted networks. In many works, the authors often confine the coupling matrix satisfying the dissipation condition (the sum of every row equals zero), and the synchronization state is determined only by the node function. In this Chapter, we needn't confine the sum of each row to be zero, but to equal a nonzero constant. Such a network can synchronize to a new state, not to that of limit set determined by the original node equation. It is interesting to find that the synchronized state appears bifurcation if we regard the constant as a bifurcation parameter.3. Synchronization between two coupled networks is discussed in Chapter 5. If network nodes are of similar properties, we can regard it as one network; otherwise, as more networks. For example, how the infectious diseases (Mad Cows, SARS, AIDS) spread between the human beings and animals, here regarding the human and animals as two networks. In this Chapter, we propose "network outer synchronization". If synchronization happens in a network, we may regard it as "inner synchronization"; while we may call it as "outer synchronization" if synchronization exists between two or more networks.4. Application part of network research is shown in Chapter 6. Based on the data in statistics and annual reports on China torch program in 2004-2005, we apply the network knowledge to construct the China Hi-Tec Park Networks (CHTPN). From the viewpoints of complex networks, firstly we investigate and initially analyze its topological properties, including the average path length, clustering coefficient and degree distribution. Secondly using the software CFinder, we analyze the evolvement of network community structures. The constructed networks basically show the trend of real CHTPNs through the numerical results. Chapter 7 summarizes the conclusions and gives further studies in the future.