Complex networks : from topology to dynamics

Abstract

Complex networks are ubiquitous in our real world, ranging from biological, social, to man-made networks. Regardless of the nature of a complex network, the topology of the network can be represented by nodes and edges in a certain connectivity structure. Therefore, topological structures of networks are the main issues for study in science and engineering. This thesis presents our research and developments on the concerned issues of modeling, synchronization and control of various complex networks. More precisely, in the thesis, we firstly propose a multi-local-world (MLW) model to describe the topology of complex networks with a prominent localization property. As an application, we applied it to model the Internet topology. Based on some careful analysis on the Internet topology, we found that the MLW model is better for describing the AS-level Internet topology than other models such as the BA, GBA, Fitness, and HOT models. Secondly, we investigated the synchronizability of several typical complex network models, i.e., regular network models, random network models, small-world network models, and scale-free network models, based on two commonly used synchronization criteria. By investigating the effects of some factors on the synchronizability of complex networks, we found that there are some critical factors that have significant influence on the synchronizability of scale-free networks. Also, our studies show that the maximum betweenness is not a good indicator for the synchronizability of a scale-free network, which invalidates some existing claims. On the other hand, considering the effect of delayed couplings, we also investigated the synchronization of dynamical networks with delays. Thirdly, we studied the relationship between the dynamical behaviors and topologies of complex networks, and found that synchronizing an evolving dynamical network can lead the network behave as a scale-free network. Finally, we discussed the issue of controlling a complex network onto its equilibrium point by using the technique of delayed feedback pinning control. We found that the stability conditions of complex networks can be described by a Linear Matrix Inequality (LMI). On the other hand, we compared two typical schemes of this control method, i.e., specifically pinning scheme and randomly pinning scheme. We found that specifically pinning scheme is more effective than the random one in stabilizing scale-free networks in general.