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Chapter 1 Introduction 1.1 Background With the rapid development of modern technology�� the world has entered the age of networks. Typical examples of networks include the World Wide Web�� routes of airlines�� biological networks�� human relationships�� and so on [1]. As a special kind of network�� complex networked systems consisting of large groups of cooperating agents have made a significant impact on a broad range of applications including cooperative control of autonomous underwater vehicles (AUVs) [2]��scheduling of automated highway systems [3]��and congestion control in communication networks [4]. The study of complex networks can be traced back to Euler��s celebrated solution of the Konigsberg bridge problem in 1735��which is often regarded as the first true proof in the theory of networks. In the later 1950s�� a random-graph model was proposed by Paul Erdos and Alfred Renyi [5]�� which laid a solid foundation for modern network theory. Watts and Strogatz proposed a model of small-world networks in 1998 [6]; after that�� Barabasi and Albert proposed a model of scale-free networks in 1999 based on the preferential attachment [7]. These two works reveal small-world effect and scale-free property of complex networks and the reasons for the above phenomena. Over the past two decades�� complex dynamical networks have been widely exploited by researchers in various fields of physics [8]��mathematics [9]�� engineering [10��11]��biology [12]��and sociology [13]. What makes complex networked systems distinct from other kinds of systems is that they make it possible to deploy a large number of subsystems as a team to cooperatively carry out a prescribed task. Furthermore�� the most striking feature that can be observed in complex networked systems is their ability to show collective behavior that cannot be well explained in terms of the individual dynamics of each single node. Two significant kinds of cooperative behaviors are synchronization and consensus [9��14-18]��both of which mean that all agents reach an agreement on certain quantities of interest. The formal study of consensus dates back to 1974 [19]�� where a mathematical model was presented to describe how the group reaches an agreement. Another interesting discovery is the collective behavior of a group of birds exhibited in foraging or flight�� which is found by biologists in the observation of birds�� flocking [20]. If attention is paid�� one can find that consensus is a universal phenomenon in nature�� such as the shoaling behavior of fish [21]�� the synchronous flashing of fireflies [22]�� the swarming behavior of insects [20��23�� 24]�� and the herd behavior of land animals [25]. The key feature of consensus is how local communications and cooperations among agents��i.e.�� consensus protocols (or consensus algorithms)�� can lead to certain desirable global behavior [26-29]. Various models have been proposed to study the mechanism of multi-agent consensus problem [30-37]. In [38]�� the consensus problem was considered of a switched multi-agent system composed of continuous-time and discrete-time subsystems. The authors in [39] investigated consensus problems of a class of second-order continuous-time multi-agent systems with time-delay and jointly-connected topologies. Literature [40] focused on the mean square practical leader-following consensus of second-order nonlinear multiagent systems with noises and unmodeled dynamics. Synchronization�� as typical collective behavior and basic motion in nature�� means that the difference among the states of any two different subsystems goes to zero as time goes to infinity or time goes to a certain fixed value. Synchronization phenomena exist widely and can be found in different forms in nature and man-made systems�� such as fireflies�� synchronous flashing�� attitude alignment�� and the synchronized applause of audiences. To reveal the mechanism of synchronization of complex dynamical networks�� a vast volume of work has been done over the past few years. Before the appearance of small-world [6] and scale-free [7] network models�� Wu and Chua in [41��42] investigated synchronization of an array of linearly coupled systems and gave some effective synchronization criteria. In 1998�� Pecora and Carroll [43] proposed the concept of master stability function as synchronization criterion�� which revealed that synchronization highly depends on the coupling strategy or the topology of the network. In [14�� 44-46]�� synchronization in small-world and scale-free networks were studied in detail. Over the past few years�� different kinds of synchronization have been found and studied�� such as complete synchronization [14��41��42��47��48]�� cluster synchronization [49-52]�� phase synchronization [53]�� lag synchronization [54��55] and generalized synchronization [56]. In the literature�� most works on the consensus/synchronization of complex networks mainly focus on the analysis of network models with perfect communication�� in which it