Modeling And Dynamic Properties Of Complex Networks

Recently, complex networks attract more and more attentions from various fields of science and engineering. In this dissertation, we apply statistical method, nonlinear system theory, control theory and matrix theory to the research of complex networks, and we study the dynamic properties of complex networks, as well as the modeling of weighted complex networks. These studies are very important both in theory and in practical applications. By studying the dynamic properties of complex networks, on the one hand, we can understand and explain the dynamic properties presented in real-world networks, such as stability, synchronization and oscillation phenimena; and on the other hand, we can apply these theoretical results to some practical applications, for example, we can apply these results to the design of real networks to achieve good performance or to the control of real networks to achieve some desirable network behaviors that benefit the networks. In addition, many real-world networks are weighted networks with different weights in different connections. The unweighted network models studied in many existing literatures are simplified modeling of real networks, while weighted network models can provide more realistic and comprehensive descriptions of real networks. So, the importance of the modelling of weighted networks is clearly self-evident.The main contents and originalities in this paper can be summarized as follows: 1. Local stability and Hopf bifurcation in a delayed small-world network modelDue to limited transmitting speed, competition and congestions, usually there are time delays in complex networks. We study the local stability and Hopf bifurcation of a delayed small-world network model. We determine the stability of the bifurcating periodic solutions and directions of Hopf bifurcation by applying the center manifold theorem and the normal form theory. The studies of Hopf bifurcation in small-world networks are quite important. On the one hand, the bifurcations, which involve emergence of oscillatory behaviors, may provide an explanation for the parameter sensitivity observed in practice in many realistic small-world networks such as theInternet, the electrical power grids, and the biological neural networks; and on the other hand, if we understand more about the bifurcation behaviors of small-world networks, we can apply the existing effective bifurcation control method to achieve some desirable system behaviors that benefit the networks.2. Synchronization of complex networksThe synchronization has attracted increasing attentions due to its importance both in theory and in practical applications. In this dissertation we study the synchronization problems of delayed small-world networks of phase oscillators and coupled maps. And we study the synchronization of a general delayed complex network model by using Lyapunov-krasovskii functions and linear matrix inequalities (LMIs). We derive some easy-verified synchronization criteria. Further, we study the chaotic phase synchronization in small-world networks.3. On-off intermittency in small-world network of chaotic mapsWe study how the small-world topology would affect on-off intermittency of small-world networks of chaotic maps. When the globally coupled chaotic maps are synchronous, we fix the coupling coefficient. We find that by decreasing the connection probability gradually, when the probability slightly less than a critical value, the synchronous chaos is no longer stable and on-off intermittency appears. By further decreasing the probability, the intermittent dynamics will eventually be replaced by fully developed asynchronous chaos.4. Study on a neural network model with weighted small-world connections There are many biological neural networks that present small-world connections.But in most existing literatures, the authors studied neural network models with regular topologies. Note also that in most of the existing small-world network models no special weights in the connections were taken into account. However, this kind of simplified network models cannot well characterize biological neural networks, in which there are different weights in different synaptic connections. Here, we present a neural network model with small-world topology and with random weight values in different connections, and further investigate the stability of this model by using the Lyapunov function method and statistical method. We derive explicit relationship between the stability of small-world neural networks and the values of the network connection probability p and the number of neurons N.