## 数学代写|复杂网络代写complex networks代考|COS496

statistics-lab™ 为您的留学生涯保驾护航 在代写复杂网络complex networks方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写复杂网络complex networks方面经验极为丰富，各种代写复杂网络complex networks相关的作业也就用不着说。

## 数学代写|复杂网络代写complex networks代考|Martingales

A sequence of integrable random variables $\left{M(t): t \in \mathbb{Z}{+}\right}$is called adapted to an increasing family of $\sigma$-fields $\left{\mathcal{F}_t: t \in \mathbb{Z}{+}\right}$if $M(t)$ is $\mathcal{F}t$-measurable for each $t$. The sequence is called a martingale if $\mathrm{E}\left[M(t+1) \mid \mathcal{F}_t\right]=M(t)$ for all $t \in \mathbb{Z}{+}$, and a supermartingale if $\mathrm{E}\left[M(t+1) \mid \mathcal{F}t\right] \leq M(t)$ for $t \in \mathbb{Z}{+}$.

A martingale difference sequence $\left{Z(t): t \in \mathbb{Z}{+}\right}$is an adapted sequence of random variables such that the sequence $M(t)=\sum{i=0}^t Z(i), t \geq 0$, is a martingale.
The following result is basic:
Theorem 1.3.4. (Martingale Convergence Theorem) Let $M$ be a supermartingale, and suppose that
$$\sup t \mathrm{E}[|M(t)|]<\infty .$$ Then ${M(t)}$ converges to a finite limit with probability one. If ${M(t)}$ is a positive, real valued supermartingale then by the smoothing property of conditional expectations (1.10), $$\mathrm{E}[|M(t)|]=\mathrm{E}[M(t)] \leq \mathrm{E}[M(0)]<\infty, \quad t \in \mathbb{Z}{+}$$
Hence we have as a direct corollary to the Martingale Convergence Theorem
Theorem 1.3.5. A positive supermartingale converges to a finite limit with probability one.

Since a positive supermartingale is convergent, it follows that its sample paths are bounded with probability one. The following result gives an upper bound on the magnitude of variation of the sample paths of both positive supermartingales, and general martingales.

## 数学代写|复杂网络代写complex networks代考|Markov models

The Markov chains that we consider evolve on a countable state space, denoted X. The chain itself is denoted $\boldsymbol{X}=\left{X(t): t \in \mathbb{Z}_{+}\right}$, with transition law defined by the transition matrix $P$ :
$$\mathrm{P}{X(t+1)=y \mid X(0), \ldots, X(t)}=P(x, y), \quad x, y \in \mathrm{X} .$$

Examples of Markov chains include both the reflected and unreflected random walks defined in Section 1.3.3. The independence of the $\mathcal{E}$ guarantees the Markovian property (1.16).

The transition matrix is viewed as a (possibly infinite-dimensional) matrix. Likewise, a function $c: \mathrm{X} \rightarrow \mathbb{R}$ can be viewed as a column vector, and we can express conditional expectations as a matrix-vector product,
$$\mathrm{E}[c(X(t+1)) \mid X(t)=x]=P c(x):=\sum_{y \in \mathbf{X}} P(x, y) c(y), \quad x \in \mathrm{X} .$$
More generally, the matrix product is defined inductively by $P^0(x, y)=\mathbf{1}{{x=y}}$ and for $n \geq 1$, $$P^n(x, y)=\sum P(x, z) P^{n-1}(z, y), \quad x, y \in \mathbf{X} .$$ Based on this we obtain the representation, $$\mathrm{E}[c(X(t+n)) \mid X(t)=x]=P^n c(x), \quad x \in \mathrm{X}, t \geq 0, n \geq 1 .$$ Central to the theory of Markov chains is the following generalization, known as the strong Markov property. Recall that a random time $\tau$ is called a stopping time if there exists a sequence of functions $f_n: \mathrm{X}^{n+1} \rightarrow{0,1}, n \geq 0$, such that the event ${\tau=n}$ can be expressed as a function of the first $n$ samples of $\boldsymbol{X}$, $${\tau=n}=f_n(X(0), \ldots, X(n)), \quad n \geq 0 .$$ We write this as ${\tau=n} \in \mathcal{F}_n$, where $\left{\mathcal{F}_k: k \geq 0\right}$ is the filtration generated by $\boldsymbol{X}$. We let $\mathcal{F}\tau$ denote the $\sigma$-field generated by the events “before $\tau$ “: that is,
$$\mathcal{F}_\tau:=\left{A: A \cap{\tau \leq n} \in \mathcal{F}_n, n \geq 0\right}$$

## 数学代写|复杂网络代写complex networks代考|Martingales

$$\sup t \mathrm{E}[|M(t)|]<\infty .$$然后${M(t)}$收敛到一个概率为1的有限极限。如果${M(t)}$是一个正的实值上鞅，则根据条件期望(1.10)的平滑性质，$$\mathrm{E}[|M(t)|]=\mathrm{E}[M(t)] \leq \mathrm{E}[M(0)]<\infty, \quad t \in \mathbb{Z}{+}$$

## 数学代写|复杂网络代写complex networks代考|Markov models

$$\mathrm{P}{X(t+1)=y \mid X(0), \ldots, X(t)}=P(x, y), \quad x, y \in \mathrm{X} .$$

$$\mathrm{E}[c(X(t+1)) \mid X(t)=x]=P c(x):=\sum_{y \in \mathbf{X}} P(x, y) c(y), \quad x \in \mathrm{X} .$$

## 广义线性模型代考

statistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

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。

## 数学代写|复杂网络代写complex networks代考|SSIE641

statistics-lab™ 为您的留学生涯保驾护航 在代写复杂网络complex networks方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写复杂网络complex networks方面经验极为丰富，各种代写复杂网络complex networks相关的作业也就用不着说。

## 数学代写|复杂网络代写complex networks代考|Random walks

Random walks are used in this book to model the cumulative arrival process to a network, as well as cumulative service at a buffer. The reflected random walk is a model for storage and queueing systems.

Both are defined by taking successive sums of independent and identically distributed (i.i.d.) random variables.
Definition 1.3.1. Random Walks
Suppose that $\boldsymbol{X}=\left{X(k) ; k \in \mathbb{Z}{+}\right}$is a sequence of random variables defined by, $$X(k+1)=X(k)+\mathcal{E}(k+1), \quad k \in \mathbb{Z}{+}$$
where $X(0) \in \mathbb{R}$ is independent of $\mathcal{E}$, and the sequence $\mathcal{E}$ is i.i.d., taking values in $\mathbb{R}$. Then $\boldsymbol{X}$ is called a random walk on $\mathbb{R}$.
Suppose that the stochastic process $Q$ is defined by the recursion,
$$Q(k+1)=[Q(k)+\mathcal{E}(k+1)]{+}:=\max (0, Q(k)+\mathcal{E}(k+1)), \quad k \in \mathbb{Z}{+},$$
where again $Q(0) \in \mathbb{R}$, and $\mathcal{E}$ is an i.i.d. sequence of random variables taking values in $\mathbb{R}$. Then $Q$ is called the reflected random walk. .

Consider the following two models for comparison: For a fixed constant $a>0$, let $L^u$ denote the uniform distribution on the interval $[0, a]$, and $L^d$ the discrete distribution supported on the two points ${a / 3, a}$ with
$$L^d{a / 3}=1-L^d{a}=3 / 4 .$$

## 数学代写|复杂网络代写complex networks代考|Renewal processes

Renewal processes are used to model service-processes as well as arrivals to a network in standard books on queueing theory $[114,23]$. The general renewal process is defined as follows.
Definition 1.3.2. Renewal process
Let ${\mathcal{E}(1), \mathcal{E}(2), \ldots}$ be a sequence of independent and identical random variables with distribution function $\Gamma$ on $\mathbb{R}{+}$, and let $T$ denote the associated random walk defined by $T(n)=\mathcal{E}(1)+\cdots+\mathcal{E}(n), n \geq 1$, with $T(0)=0$. Then the (undelayed) renewal process is the continuous-time stochastic process, taking values in $\mathbb{Z}{+}$, defined by,
$$R(t)=\max {n: T(n) \leq t} .$$
The sample paths of a renewal process are piecewise constant, with jumps at the renewal times ${T(n): n \geq 1}$.

A renewal process $R$ takes on integer values and is non-decreasing, so that the quantity $R\left(t_1\right)-R\left(t_0\right), t_0, t_1 \in \mathbb{R}_{+}$, can be used to model the number of arrivals during the time-interval $\left(t_0, t_1\right]$, or the number of service completions for a server that is busy during this time-interval.

The most important example of a renewal process is the standard Poisson process, in which the process $\mathcal{E}$ has an exponential marginal distribution. The Poisson process is also another example of a stochastic process with independent increments, whose distribution is expressed as follows: For each $k \geq 0$ and $0 \leq t_0 \leq t_1<\infty$,
$$\mathrm{P}\left{R\left(t_1\right)-R\left(t_0\right)=k\right}=\frac{\left(\mu\left(t_1-t_0\right)\right)^k}{k !} e^{\mu\left(t_1-t_0\right)} .$$
Proposition 1.3.3 summarizes some basic results. More structure is described in Asmussen [23].

## 数学代写|复杂网络代写complex networks代考|Random walks

1.3.1.定义随机漫步

$$Q(k+1)=[Q(k)+\mathcal{E}(k+1)]{+}:=\max (0, Q(k)+\mathcal{E}(k+1)), \quad k \in \mathbb{Z}{+},$$

$$L^d{a / 3}=1-L^d{a}=3 / 4 .$$

## 数学代写|复杂网络代写complex networks代考|Renewal processes

1.3.2.定义更新流程

$$R(t)=\max {n: T(n) \leq t} .$$

$$\mathrm{P}\left{R\left(t_1\right)-R\left(t_0\right)=k\right}=\frac{\left(\mu\left(t_1-t_0\right)\right)^k}{k !} e^{\mu\left(t_1-t_0\right)} .$$

## 广义线性模型代考

statistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

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。

## 数学代写|复杂网络代写complex networks代考|CS-E5740

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## 数学代写|复杂网络代写complex networks代考|Linear programs

In the theory of linear programming the standard primal problem is defined as the optimization problem,
$$\begin{array}{lrll} \max & c^{\mathrm{T}} x & & \ \text { s.t. } & \sum_j a_{i j} x_j & \leq b_i, & \text { for } i=1, \ldots, m ; \ & x_j & \geq 0, & \text { for } j=1, \ldots, n . \end{array}$$
Its dual is the linear program,
$$\begin{array}{lrl} \text { min } b^{\mathrm{T}} w & \ \text { s.t. } & \sum_j a_{j i} w_j \geq c_i, & \text { for } i=1, \ldots, n ; \ & w_j \geq 0, & \text { for } j=1, \ldots, m . \end{array}$$
The primal is usually written in matrix notation, $\max c^{\mathrm{T}} x$ subject to $A x \leq b, x \geq 0$; and the dual as $\min b^{\mathrm{T}} w$ subject to $A^{\mathrm{T}} w \geq c, w \geq 0$.

Any linear programming problem can be placed in the standard form (1.8). For example, a minimization problem can be reformulated as a maximization problem by changing the sign of the objective function. An equality constraint $y=b$ can be represented as two inequality constraints, $y \leq b$ and $-y \leq-b$. In the resulting dual one finds that the two corresponding variables can be replaced by one variable that is unrestricted in sign.

## 数学代写|复杂网络代写complex networks代考|Some Probability Theory

Until Part III this book requires little knowledge of advanced topics in probability. It is useful to outline some of this advanced material here since, for example, the Law of Large Numbers and the Central Limit Theorem for martingales and renewal processes serves as motivation for the idealized network models developed in Parts I and II.
The starting point of probability theory is the probability space, defined as the triple $(\Omega, \mathcal{F}, \mathrm{P})$ with $\Omega$ an abstract set of points, $\mathcal{F}$ a $\sigma$-field of subsets of $\Omega$, and $\mathrm{P}$ a probability measure on $\mathcal{F}$. A mapping $X: \Omega \rightarrow \mathrm{X}$ is called a random variable if
$$X^{-1}{B}:={\omega: X(\omega) \in B} \in \mathcal{F}$$
for all sets $B \in \mathcal{B}(\mathrm{X})$ : that is, if $X$ is a measurable mapping from $\Omega$ to $\mathrm{X}$.
Given a random variable $X$ on the probability space $(\Omega, \mathcal{F}, \mathrm{P})$, we define the $\sigma$ field generated by $X$, denoted $\sigma{X} \subseteq \mathcal{F}$, to be the smallest $\sigma$-field on which $X$ is measurable.

If $X$ is a random variable from a probability space $(\Omega, \mathcal{F}, \mathrm{P})$ to a general measurable space $(\mathrm{X}, \mathcal{B}(\mathrm{X}))$, and $h$ is a real valued measurable mapping from $(\mathrm{X}, \mathcal{B}(\mathrm{X}))$ to the real line $(\mathbb{R}, \mathcal{B}(\mathbb{R}))$ then the composite function $h(X)$ is a real-valued random variable on $(\Omega, \mathcal{F}, \mathrm{P})$ : note that some authors reserve the term “random variable” for such real-valued mappings. For such functions, we define the expectation as
$$\mathrm{E}[h(X)]=\int_{\Omega} h(X(\omega)) \mathrm{P}(d w)$$
The set of real-valued random variables $Y$ for which the expectation is well-defined and finite is denoted $L^1(\Omega, \mathcal{F}, \mathrm{P})$. Similarly, we use $L^{\infty}(\Omega, \mathcal{F}, \mathrm{P})$ to denote the collection of essentially bounded real-valued random variables $Y$; That is, those for which there is a bound $M$ and a set $A_M \subset \mathcal{F}$ with $\mathrm{P}\left(A_M\right)=0$ such that ${\omega:|Y(\omega)|>M} \subseteq A_M$.
Suppose that $Y \in L^1(\Omega, \mathcal{F}, \mathrm{P})$ and $\mathcal{G} \subset \mathcal{F}$ is a sub- $\sigma$-field of $\mathcal{F}$. If $\hat{Y} \in$ $L^1(\Omega, \mathcal{G}, \mathrm{P})$ and satisfies
$$\mathrm{E}[Y Z]=\mathrm{E}[\hat{Y} Z] \quad \text { for all } Z \in L_{\infty}(\Omega, \mathcal{G}, \mathrm{P})$$
then $\hat{Y}$ is called the conditional expectation of $Y$ given $\mathcal{G}$, and denoted $\mathrm{E}[Y \mid \mathcal{G}]$. The conditional expectation defined in this way exists and is unique (modulo P-null sets) for any $Y \in L^1(\Omega, \mathcal{F}, \mathrm{P})$ and any sub $\sigma$-field $\mathcal{G}$.
Suppose now that we have another $\sigma$-field $\mathcal{H} \subset \mathcal{G} \subset \mathcal{F}$. Then
$$\mathrm{E}[Y \mid \mathcal{H}]=\mathrm{E}[\mathrm{E}[Y \mid \mathcal{G}] \mid \mathcal{H}] .$$

## 数学代写|复杂网络代写complex networks代考|Linear programs

$$\begin{array}{lrll} \max & c^{\mathrm{T}} x & & \ \text { s.t. } & \sum_j a_{i j} x_j & \leq b_i, & \text { for } i=1, \ldots, m ; \ & x_j & \geq 0, & \text { for } j=1, \ldots, n . \end{array}$$

$$\begin{array}{lrl} \text { min } b^{\mathrm{T}} w & \ \text { s.t. } & \sum_j a_{j i} w_j \geq c_i, & \text { for } i=1, \ldots, n ; \ & w_j \geq 0, & \text { for } j=1, \ldots, m . \end{array}$$

## 数学代写|复杂网络代写complex networks代考|Some Probability Theory

$$X^{-1}{B}:={\omega: X(\omega) \in B} \in \mathcal{F}$$

$$\mathrm{E}[h(X)]=\int_{\Omega} h(X(\omega)) \mathrm{P}(d w)$$

## 广义线性模型代考

statistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

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。

## 计算机代写|复杂网络代写complex network代考|Cohesive Subgroups or Communities as Block Models

statistics-lab™ 为您的留学生涯保驾护航 在代写复杂网络complex network方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写复杂网络complex network代写方面经验极为丰富，各种代写复杂网络complex network相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 计算机代写|复杂网络代写complex network代考|Cohesive Subgroups or Communities as Block Models

The abundance of diagonal block models or modular structures makes modularity a concept so important that it is often studied outside the general framework of block modeling. One explanation may be that in social networks it may even be the dominant blocking structure. The reason may be that homophily [16], i.e., the tendency to form links with agents similar to oneself, is a dominant mechanism in the genesis of social networks. Recall, however, that the concept of functional roles in networks is much wider than mere cohesiveness as it specifically focuses on the inter-dependencies between groups of nodes. Modularity or community structure, emphasizing the absence of dependencies between groups of nodes is only one special case. It may also be that the concept of modularity appeals particularly to physicists because it is reminiscent of the reductionist approach of taking systems apart into smaller subsystems that has been so successful in the natural sciences.

Nevertheless, in the literature, there is no generally accepted definition of what a community or module actually is. A variety of definitions exist that all imply that members of a community are more densely connected among themselves than to the rest of the network. Two approaches exist to tackle the problem. Either, one starts with a definition of what a community is in the first place and then searches for sets of nodes that match this definition. Or one can use a heuristic approach by designing an algorithm and define a community as whatever this algorithm outputs. Both of these approaches differ in one fundamental way: When starting from a definition of community, it often occurs that some nodes in the network will not be placed into any community. The algorithmic approaches on the other hand will generally partition the set of vertices such that all nodes are found in some community. Whether all nodes need to be assigned into a community needs to be decided by the researcher and may determine which definitions and methods are useful in the analysis of actual data. With these considerations in mind we shall briefly review the approaches taken in the literature.

## 计算机代写|复杂网络代写complex network代考|Sociological Definitions

The study of community structure has a long tradition in the field of sociology and it comes as no surprise that the example that sparked the interest of physicists in the field was a sociological one $[17,18]$. Alternatively to community, the term cohesive subgroup is often used to subsume a number of definitions that emphasize different aspects of the problem. These can be grouped into definitions based on reachability, nodal degree or the comparison of within to outside links [11].

Cliques are complete subgraphs, such that every member is connected to every other member in the clique. An n-clique is a maximal subgraph, such that the geodesic distance $d(i, j)$ between any two members $i, j$ is smaller or equal to $n$. Naturally, cliques are 1-cliques. Note that the shortest path may also run through nodes not part of the n-clique, such that the diameter of an n-clique may be larger than $n$. An n-clan denotes an n-clique with diameter less or equal to $n$. Naturally, all n-clans are also n-cliques. Alternatively, an $n$-club is a maximal subgraph of diameter $n$.

These definitions are problematic in several ways. Cliques can never get larger than the smallest degree among the member nodes which limits these communities to be generally very small in large networks with limited degrees. The other definitions relying on distances are problematic if the network possesses the small world property. The overlap of such communities will generally be as large as a typical group.

Another group of definitions is based on the degree of the members of a community. A $k$-plex is a maximal subgraph of $n$ nodes, such that each member has at least $n-k$ connections to other nodes in the k-plex. This definition is less strict than that of a clique as it allows some links to be missing. At the same time, a k-plex only contains nodes with minimum degree $d \geq(n-k)$. A $k$-core denotes a maximal subgraph, such that each node has at least $k$ connections to other members of the k-core.

Here again, the size of k-plexes is limited by the degrees of the nodes. K-cores are problematic also because they disregard all nodes with degree smaller than $k$ even if they have all their connections to nodes within this core.

# 复杂网络代写

## 有限元方法代写

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## MATLAB代写

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。