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

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  • Statistical Inference 统计推断
  • Statistical Computing 统计计算
  • Advanced Probability Theory 高等概率论
  • Advanced Mathematical Statistics 高等数理统计学
  • (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

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.


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.


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|COMPLEX NETWORK SYSTEMS


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

许多自然、社会和工程系统远不是独立的实体,而是可以被视为与其中相邻实体之间紧密相互作用相关的 CNS[3,10,18,29,39,50,92,120,130,141,175,194,201,211,219]. 粗略地说,CNS 是指由许多相互连接的代理组成的网络系统,其中每个代理是一个基本元素或基本单元,其详细内容取决于所考虑的特定网络的性质[175]。例如,互联网是通过各种物理或无线链路连接的路由器和计算机的 CNS。细胞可以通过化学相互作用连接的化学中枢神经系统来描述。科学引文网络是论文和书籍之间通过引文链接的 CNS。微信社交网络是一个CNS,其代理是用户,其边代表用户之间的关系,仅举几例。

借助与邻近个体的协调,CNS 可以表现出远远超出个体固有属性的迷人合作行为。典型的合作行为包括同步[38,95,101,177], 共识[76,118,128], 蜂拥而至[48,115], 植绒[117,161]. 在本书中,我们重点关注包含复杂网络和 MAS 的 CNS 作为特例。很多新的

关于理解负责各种集体行为的出现机制以及 CNS 的全局统计特性的研究挑战已经提出[3,15,114,178]. 网络科学作为一个强大的跨学科研究领域,在 21 世纪头几年已经确立[110]。人们越来越认识到,对 CNS 合作动力学的详细研究不仅有助于研究人员了解宏观合作行为的演化机制,而且还可以促进网络科学应用于解决各种工程问题,例如分布式传感器网络的设计[135]。 ],多架无人机编队控制[37],分布式定位[89],现代电网中多个储能单元的负荷分配[191]。

在 CNS 的各种协作行为中,复杂网络的同步和 MAS 的共识是最基本也是最重要的。复杂网络的同步展示了这些网络中所有实体的状态就某些感兴趣的数量达成一致的合作行为。与孤立控制设备的稳定性分析相比,CNS 中的同步行为分析更具挑战性,因为同步过程取决于网络拓扑的演变以及这些网络系统中各个单元的固有动态[96,102,121,198,199]. 作为与复杂网络同步密切相关的课题,MASs 的共识最近引起了各个研究领域的广泛关注,尤其是系统科学、控制理论和电气工程界。[22,65,88,116,128]. 在本章的剩余部分,我们将回顾一些关于实现复杂网络同步和 MAS 在动态变化的通信拓扑上的共识的现有结果。


在继续之前,给出了 MAS 共识的定义。此外,可以类似地定义复杂网络的同步。

考虑一个 MAS,它由ñ代理。不失一般性,我们标记ñ代理人作为代理人1,…,ñ, 分别。代理的动态一世,一世=1,…,ñ, 表示为

在哪里X一世(吨)∈Rn和在一世(吨)∈R米分别表示状态和控制输入,F(⋅,⋅,⋅):[吨0,+∞)×Rn×R米↦Rn表示代理 i 的非线性动力学。一个特例是具有代理动力学的一般线性时不变 MAS一世被描述为

在哪里一个∈Rn×n和乙∈Rn×米分别表示状态矩阵和控制输入矩阵。为方便起见,在整本书中,我们称 MAS (1.1) 来表示其动力学由 (1.1) 描述的 MAS。


由方程式给出的 MAS (1.1) 共识的定义。(1.3) 不关心最终共识状态。然而,有时重要的是使所考虑的 MAS 中的所有代理的状态最终收敛到某个预先设计的轨迹,特别是从控制各种复杂工程系统的角度来看。为了确保 MAS (1.1) 中所有代理的状态收敛到一些期望的状态,将目标系统(可能是虚拟的)引入网络 (1.1) 为

对于一些给定的初始值s(吨0)∈Rn. 在这种情况下,我们称代理一世其动力学由(1.1)跟随者描述一世,一世=1,…,ñ, 并调用由 (1.4) 描述动态的智能体为领导者。




具体来说,开发了基于平均的方法来分析具有快速切换拓扑的连续时间复杂网络的同步[7,140]同时开发了多个基于 Lyapunov 函数 (MLF) 的方法来分析具有缓慢切换拓扑的连续时间复杂网络的同步(特别是对于有向切换拓扑的情况)[190]。此外,基于 MLFL 的方法通常用于分析在延迟或采样数据耦合下具有切换拓扑的连续时间复杂网络的同步 [90,187]。基于通用 Lyapunov 函数 (CLF) 和泛函 (CLFL) 的方法仅适用于一些具有切换拓扑的特殊连续时间复杂网络,例如每个可能的拓扑候选者都是无向的 [222]。

对于具有切换拓扑的离散时间CNS,在[200]中研究了具有随机切换拓扑的非自治线性复杂网络的全局同步,通过开发一种非齐次马尔可夫链的遍历性理论方法。在[97]中提出了一种基于无限耦合矩阵的Hajnal直径的方法来分析一类具有定向切换拓扑的离散时间复杂网络的局部同步性。[73] 通过构建 MLF 研究了具有无向开关拓扑和脉冲控制器的离散时间复杂网络的同步。在[51]中使用超鞅收敛定理研究了具有交换拓扑的离散时间复杂网络的全局几乎确定的同步。更多近期相关作品,

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统计代写请认准statistics-lab™. statistics-lab™为您的留学生涯保驾护航。







术语 广义线性模型(GLM)通常是指给定连续和/或分类预测因素的连续响应变量的常规线性回归模型。它包括多元线性回归,以及方差分析和方差分析(仅含固定效应)。



有限元是一种通用的数值方法,用于解决两个或三个空间变量的偏微分方程(即一些边界值问题)。为了解决一个问题,有限元将一个大系统细分为更小、更简单的部分,称为有限元。这是通过在空间维度上的特定空间离散化来实现的,它是通过构建对象的网格来实现的:用于求解的数值域,它有有限数量的点。边界值问题的有限元方法表述最终导致一个代数方程组。该方法在域上对未知函数进行逼近。[1] 然后将模拟这些有限元的简单方程组合成一个更大的方程系统,以模拟整个问题。然后,有限元通过变化微积分使相关的误差函数最小化来逼近一个解决方案。





随机过程,是依赖于参数的一组随机变量的全体,参数通常是时间。 随机变量是随机现象的数量表现,其时间序列是一组按照时间发生先后顺序进行排列的数据点序列。通常一组时间序列的时间间隔为一恒定值(如1秒,5分钟,12小时,7天,1年),因此时间序列可以作为离散时间数据进行分析处理。研究时间序列数据的意义在于现实中,往往需要研究某个事物其随时间发展变化的规律。这就需要通过研究该事物过去发展的历史记录,以得到其自身发展的规律。


多元回归分析渐进(Multiple Regression Analysis Asymptotics)属于计量经济学领域,主要是一种数学上的统计分析方法,可以分析复杂情况下各影响因素的数学关系,在自然科学、社会和经济学等多个领域内应用广泛。


MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中,其中问题和解决方案以熟悉的数学符号表示。典型用途包括:数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发,包括图形用户界面构建MATLAB 是一个交互式系统,其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题,尤其是那些具有矩阵和向量公式的问题,而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问,这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展,得到了许多用户的投入。在大学环境中,它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域,MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要,工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数(M 文件)的综合集合,可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。



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