Synchronization and consensus analysis of complex networks

Abstract

The last decade has witnessed the birth of a new movement of interest and research in the study of complex networks, i.e. networks whose structure is irregular, complex and dynamically evolving in time. The main focus has moved from the analysis of small networks to that of systems with thousands or millions of nodes, with a renewed attention to the properties of networks of dynamical units. This flurry of activity, triggered by two seminal papers, that by Watts and Strogatz on small-world networks, appeared in Nature in 1998, and that by Barabási and Albert on scale-free networks appeared one year later in Science, has seen the physics' community among the principal actors, and has been certainly induced by the increased computing powers and by the possibility to study the properties of a plenty of large databases of real networks. The research on complex networks begun with the effort of defining new concepts and measures to characterize the topology of real networks. The main result has been the identification of a series of unifying principles and statistical properties common to most of the real networks considered. Over the past decade, much of the interesting dynamical behavior of complex dynamical networks, such as synchronization, consensus and spatio-temporal chaos, has recently attracted increasing attention from researchers in different areas. Among these, synchronization and consensus techniques have been applied in many fields, such as secure communication, harmonic oscillation generation and parallel image processing. Hence, synchronization and consensus analysis for complex networks has important real-life application. The major work of this thesis is to analyze synchronization and consensus of complex networks. In Chapter 2, due to the fact that we usually have to consider some algebraic constraints of complex networks in modeling the real world problems, we aim to build singular hybrid coupled systems to describe complex networks with a special class of constraints. Based on a reference state and the Lyapunov stability, a sufficient condition is obtained ensuring the globally asymptotical synchronization of a class of singular hybrid coupled networks with finite-varying nonlinear perturbation. In Chapter 3, pinning synchronization of a directed network with Markovian jump and nonlinear perturbations is considered. By analyzing the structure of the network, a detailed pinning scheme is given to ensure the synchronization of all nodes in a directed network. This pinning scheme can overcome the difficulties of deciding which nodes needs to be pinned. This scheme can also identify the exact least number of pinned nodes for a directed network model. In addition, the time-varying polytopic directed network with Markovian jump is discussed. In Chapter 4, the consensus problem of multiagent nonlinear directed networks is discussed in the case that a multiagent nonlinear directed network does not have a spanning tree to reach the consensus of all nodes. By using the Lie algebra theory, a linear node-and-node pinning method is proposed to achieve a consensus for all nonlinear functions satisfying a given set of conditions. Based on some optimal algorithms, large-size networks are aggregated to small-size ones. Subsequently, by applying the principle minor theory to the small-size networks, a sufficient condition is given to reduce the number of controlled nodes. In Chapter 5, the consensus problem of a multi-agent directed network with nonlinear perturbations is investigated. Based on a reduced-order transformation, it is shown that the discussed multi-agent model cannot reach a consensus under a Hypoth esis even though the discussed network has a spanning tree. An impulsive approach is then introduced and a simple criterion is presented to guarantee the consensus of all agents in the multi-agent model.