### cs代写|复杂网络代写complex network代考|COMPLEX NETWORK SYSTEMS

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• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
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• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## cs代写|复杂网络代写complex network代考|COMPLEX NETWORK SYSTEMS

Far from being separate entities, many natural, social, and engineering systems can be considered as CNSs associated with tight interactions among neighboring entities within them $[3,10,18,29,39,50,92,120,130,141,175,194,201,211,219]$. Roughly speaking, a CNS refers to a networking system that consists of lots of interconnected agents, where each agent is an elementary element or a fundamental unit with detailed contents depending on the nature of the specific network under consideration [175]. For example, the Internet is a CNS of routers and computers connected by various physical or wireless links. The cell can be described by a CNS of chemicals connected by chemical interactions. The scientific citation network is a CNS of papers and books linked by citations among them. The WeChat social network is a CNS whose agents are users and whose edges represent the relationships among users, to name just a few.

With the aid of coordination with neighboring individuals, a CNS can exhibit fascinating cooperative behaviors far beyond the individuals’ inherent properties. Prototypical cooperative behaviors include synchronization $[38,95,101,177]$, consensus $[76,118,128]$, swarming $[48,115]$, flocking $[117,161]$. In this book, we focus on the CNSs which include complex networks and MASs as special cases. A lot of new

research challenges have been raised about understanding the emergence mechanisms responsible for various collective behaviors as well as global statistical properties of CNSs $[3,15,114,178]$. Network science, as a strong interdisciplinary research field, has been established at the first several years of the 21 st century [110]. It is increasingly recognized that a detailed study on cooperative dynamics of CNSs would not only help researchers understand the evolution mechanism for macroscopical cooperative behaviors, but also prompt the application of network science to solve various engineering problems, e.g., design of distributed sensor networks [135], formation control of multiple unmanned aerial vehicles [37], distributed localization [89], and load assignment of multiple energy storage units in modern power grid [191].

Among the various cooperative behaviors of CNSs, synchronization of complex networks and consensus of MASs are the most fundamental yet most important ones. Synchronization of complex networks exhibits the cooperative behavior that the states of all entities within these networks achieve an agreement on some quantities of interest. Compared with stability analysis of an isolated control plant, synchronization behavior analysis in CNSs are much more challenging as the synchronization process is determined by the evolution of network topology as well as the inherent dynamics of individual units within these network systems $[96,102,121,198,199]$. As a topic closely related to synchronization of complex networks, the consensus of MASs has recently gained much attention from various research fields, especially the system science, control theory, and electrical engineering communities $[22,65,88,116,128]$. In the remainder of this chapter, we will review some existing results on achieving synchronization of complex networks and consensus of MASs over dynamically changing communication topologies.

## cs代写|复杂网络代写complex network代考|DEFINITIONS OF SYNCHRONIZATION AND CONSENSUS

Before moving forward, the definition of consensus of MASs is given. Moreover, the synchronization of complex networks can be defined similarly.

Consider an MAS which consists of $N$ agents. Without loss of generality, we label the $N$ agents as agents $1, \ldots, N$, respectively. The dynamics of agent $i, i=1, \ldots, N$, are represented by
$$\dot{x}{i}(t)=f\left(t, x{i}(t), u_{i}(t)\right),$$
where $x_{i}(t) \in \mathbb{R}^{n}$ and $u_{i}(t) \in \mathbb{R}^{m}$ represent, respectively, the state and the control input, $f(\cdot, \cdot, \cdot):\left[t_{0},+\infty\right) \times \mathbb{R}^{n} \times \mathbb{R}^{m} \mapsto \mathbb{R}^{n}$ represents the nonlinear dynamics of agent i. A particular case is the general linear time-invariant MASs with the dynamics of agent $i$ are described by
$$\dot{x}{i}(t)=A x{i}(t)+B u_{i}(t), i=1, \ldots, N,$$
where $A \in \mathbb{R}^{n \times n}$ and $B \in \mathbb{R}^{n \times m}$ represent, respectively, the state matrix and control input matrix. For convenience, throughout this book, we call MAS (1.1) to represent the MAS whose dynamics are described by (1.1).

Definition $1.1$ Consensus of the $M A S(1.1)$ is said to be achieved if for arbitrary initial conditions $x_{i}\left(t_{0}\right), i=1, \ldots, N$,
$$\lim {t \rightarrow \infty}\left|x{i}(t)-x_{j}(t)\right|=0, i, j=1, \ldots, N .$$
The definition of consensus for MAS (1.1) given by Eq. (1.3) does not concern about the final consensus states. However, it is sometimes important to make the states of all agents in the considered MASs to finally converge to some predesigned trajectory, especially from the viewpoint of controlling various complex engineering systems. To ensure the states of all agents in MAS (1.1) converge to some desired states, a target system (may be virtual) is introduced to the network (1.1) as
$$\dot{s}(t)=f(t, s(t))$$
for some given initial value $s\left(t_{0}\right) \in \mathbb{R}^{n}$. Under this scenario, we call agent $i$ whose dynamics are described by (1.1) the follower $i, i=1, \ldots, N$, and call the agent whose dynamics are described by (1.4) the leader.

## cs代写|复杂网络代写complex network代考|SYNCHRONIZATION OF COMPLEX NETWORKS WITH SWITCHING TOPOLOGIES

In the field of complex networks’ synchronization with switching topologies, a wide range of research has been recently focused on dealing with issues related to the switchings and their effects on synchronization.

There has been increasing recognition that each topology candidate’s properties and the switching strategy for topologies play essential roles in achieving synchronization for complex networks with switching topologies. The analytical approaches for synchronization of continuous- and discrete-time complex networks with switching topologies are generally different. Mathematically, the continuous-time complex network with switching topologies is a special kind of those with time-varying topology. However, it is preliminarily assumed in some existing works on synchronization of continuous-time network systems with time-varying topology that the connection links evolve continuously over time with a known bound for the changing rate [103] or with a time-varying Laplacian matrix being simultaneously diagonalizable [11]. Thus, the techniques developed in these works to solve synchronization problem of complex networks with special time-varying topology are generally hard to apply to that with switching topologies, especially to the case with directed switching topologies.

Specifically, averaging-based approaches were developed to analyze synchronization of continuous-time complex networks with fast switching topologies $[7,140]$ while multiple Lyapunov functions (MLFs)-based approaches were developed to analyze synchronization of continuous-time complex networks with slowly switching topologies (especially for the case with directed switching topologies) [190]. Furthermore, MLFLs-based approaches were usually employed to analyze synchronization of continuous-time complex networks with switching topologies under delayed or sampled-data coupling [90,187]. Common Lyapunov function (CLF)- and functional (CLFL)-based approaches are applicable only to some special continuous-time complex networks with switching topologies such as each possible topology candidate is undirected [222].

For discrete-time CNSs with switching topologies, global synchronization for nonautonomous linear complex networks with randomly switching topologies was studied in [200] by developing a kind of approaches from ergodicity theory for nonhomogeneous Markovian chains. A method based on the Hajnal diameter of infinite coupling matrices was proposed in [97] to analyze the local synchronizability of a class of discrete-time complex networks with directed switching topologies. Synchronization of discrete-time complex networks with undirected switching topologies and impulsive controller was studied in [73] by constructing MLFs. Globally almost sure synchronization for discrete-time complex networks with switching topologies was investigated in [51] by using the super-martingale convergence theorem. For more recent related works, one can refer to the survey.

## cs代写|复杂网络代写complex network代考|DEFINITIONS OF SYNCHRONIZATION AND CONSENSUS

X˙一世(吨)=F(吨,X一世(吨),在一世(吨)),

X˙一世(吨)=一个X一世(吨)+乙在一世(吨),一世=1,…,ñ,

s˙(吨)=F(吨,s(吨))

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