分类：复杂网络代写complex networks代考

数学代写|复杂网络代写complex networks代考|COS496

Posted on 2023年8月29日2023年8月29日 by statistics-lab

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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,
$$
\mathcal{F}_\tau:=\left{A: A \cap{\tau \leq n} \in \mathcal{F}_n, n \geq 0\right}
$$

复杂网络代写

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

一个可积随机变量序列$\left{M(t): t \in \mathbb{Z}{+}\right}$被称为适应于一个不断增加的家族$\sigma$ -字段$\left{\mathcal{F}_t: t \in \mathbb{Z}{+}\right}$，如果$M(t)$是$\mathcal{F}t$，则每个$t$可测量。对于所有$t \in \mathbb{Z}{+}$序列称为鞅if $\mathrm{E}\left[M(t+1) \mid \mathcal{F}_t\right]=M(t)$，对于$t \in \mathbb{Z}{+}$序列称为上鞅if $\mathrm{E}\left[M(t+1) \mid \mathcal{F}t\right] \leq M(t)$。

一个鞅差分序列$\left{Z(t): t \in \mathbb{Z}{+}\right}$是一个随机变量的适应序列，使得序列$M(t)=\sum{i=0}^t Z(i), t \geq 0$是一个鞅。
下面的结果是基本的:
定理1.3.4。(鞅收敛定理)设$M$为上鞅，并设
$$
\sup t \mathrm{E}[|M(t)|]<\infty . $$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，$$ \mathrm{E}[|M(t)|]=\mathrm{E}[M(t)] \leq \mathrm{E}[M(0)]<\infty, \quad t \in \mathbb{Z}{+}
$$
因此我们得到了鞅收敛定理的一个直接推论
定理1.3.5。正上鞅收敛到有限极限的概率为1。

由于正上鞅是收敛的，因此它的样本路径有界的概率为1。下面的结果给出了正上鞅和一般鞅的样本路径变化幅度的上界。

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

我们考虑的马尔可夫链在可数状态空间上演化，记为x，链本身记为$\boldsymbol{X}=\left{X(t): t \in \mathbb{Z}_{+}\right}$，其转移律由转移矩阵$P$定义:
$$
\mathrm{P}{X(t+1)=y \mid X(0), \ldots, X(t)}=P(x, y), \quad x, y \in \mathrm{X} .
$$

马尔可夫链的例子包括1.3.3节中定义的反射和非反射随机漫步。$\mathcal{E}$的独立性保证了马尔可夫性(1.16)。

转换矩阵被看作是一个(可能是无限维的)矩阵。同样，一个函数$c: \mathrm{X} \rightarrow \mathbb{R}$可以看作是一个列向量，我们可以将条件期望表示为一个矩阵-向量积，
$$
\mathrm{E}[c(X(t+1)) \mid X(t)=x]=P c(x):=\sum_{y \in \mathbf{X}} P(x, y) c(y), \quad x \in \mathrm{X} .
$$
更一般地说，矩阵积由$P^0(x, y)=\mathbf{1}{{x=y}}$归纳定义，对于$n \geq 1$, $$ P^n(x, y)=\sum P(x, z) P^{n-1}(z, y), \quad x, y \in \mathbf{X} . $$，基于此，我们得到了表示，$$ \mathrm{E}[c(X(t+n)) \mid X(t)=x]=P^n c(x), \quad x \in \mathrm{X}, t \geq 0, n \geq 1 . $$马尔可夫链理论的核心是以下推广，称为强马尔可夫性质。回想一下，如果存在一个函数序列$f_n: \mathrm{X}^{n+1} \rightarrow{0,1}, n \geq 0$，则随机时间$\tau$称为停止时间，使得事件${\tau=n}$可以表示为$\boldsymbol{X}$, $$ {\tau=n}=f_n(X(0), \ldots, X(n)), \quad n \geq 0 . $$的第一个$n$样本的函数。我们将其写为${\tau=n} \in \mathcal{F}n$，其中$\left{\mathcal{F}_k: k \geq 0\right}$是$\boldsymbol{X}$产生的过滤。我们让$\mathcal{F}\tau$表示“$\tau$之前”的事件生成的$\sigma$字段: $$ \mathcal{F}\tau:=\left{A: A \cap{\tau \leq n} \in \mathcal{F}_n, n \geq 0\right}
$$

统计代写|复杂网络代写complex networks代考请认准statistics-lab™

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

随机过程代考

在概率论概念中，随机过程是随机变量的集合。若一随机系统的样本点是随机函数，则称此函数为样本函数，这一随机系统全部样本函数的集合是一个随机过程。实际应用中，样本函数的一般定义在时间域或者空间域。 随机过程的实例如股票和汇率的波动、语音信号、视频信号、体温的变化，随机运动如布朗运动、随机徘徊等等。

贝叶斯方法代考

贝叶斯统计概念及数据分析表示使用概率陈述回答有关未知参数的研究问题以及统计范式。后验分布包括关于参数的先验分布，和基于观测数据提供关于参数的信息似然模型。根据选择的先验分布和似然模型，后验分布可以解析或近似，例如，马尔科夫链蒙特卡罗 (MCMC) 方法之一。贝叶斯统计概念及数据分析使用后验分布来形成模型参数的各种摘要，包括点估计，如后验平均值、中位数、百分位数和称为可信区间的区间估计。此外，所有关于模型参数的统计检验都可以表示为基于估计后验分布的概率报表。

广义线性模型代考

广义线性模型（GLM）归属统计学领域，是一种应用灵活的线性回归模型。该模型允许因变量的偏差分布有除了正态分布之外的其它分布。

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。

机器学习代写

随着AI的大潮到来，Machine Learning逐渐成为一个新的学习热点。同时与传统CS相比，Machine Learning在其他领域也有着广泛的应用，因此这门学科成为不仅折磨CS专业同学的“小恶魔”，也是折磨生物、化学、统计等其他学科留学生的“大魔王”。学习Machine learning的一大绊脚石在于使用语言众多，跨学科范围广，所以学习起来尤其困难。但是不管你在学习Machine Learning时遇到任何难题，StudyGate专业导师团队都能为你轻松解决。

多元统计分析代考

基础数据: $N$ 个样本， $P$ 个变量数的单样本，组成的横列的数据表
变量定性: 分类和顺序；变量定量：数值
数学公式的角度分为: 因变量与自变量

时间序列分析代写

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。

回归分析代写

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，工具箱允许您学习和应用专业技术。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。

R语言代写	问卷设计与分析代写
PYTHON代写	回归分析与线性模型代写
MATLAB代写	方差分析与试验设计代写
STATA代写	机器学习/统计学习代写
SPSS代写	计量经济学代写
EVIEWS代写	时间序列分析代写
EXCEL代写	深度学习代写
SQL代写	各种数据建模与可视化代写

数学代写|复杂网络代写complex networks代考|SSIE641

Posted on 2023年8月29日2023年8月29日 by statistics-lab

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].

复杂网络代写

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

在这本书中使用随机漫步来模拟网络的累积到达过程，以及缓冲区的累积服务。反射随机漫步是存储和排队系统的一个模型。

两者都是通过取独立和同分布(i.i.d)随机变量的连续和来定义的。
1.3.1.定义随机漫步
假设$\boldsymbol{X}=\left{X(k) ; k \in \mathbb{Z}{+}\right}$是一个随机变量序列，定义为$$ X(k+1)=X(k)+\mathcal{E}(k+1), \quad k \in \mathbb{Z}{+}
$$
其中$X(0) \in \mathbb{R}$独立于$\mathcal{E}$，序列$\mathcal{E}$是id，取$\mathbb{R}$中的值。然后$\boldsymbol{X}$被称为$\mathbb{R}$上的随机漫步。
假设随机过程$Q$由递归定义，
$$
Q(k+1)=[Q(k)+\mathcal{E}(k+1)]{+}:=\max (0, Q(k)+\mathcal{E}(k+1)), \quad k \in \mathbb{Z}{+},
$$
这里还是$Q(0) \in \mathbb{R}$, $\mathcal{E}$是在$\mathbb{R}$中取值的随机变量的i.i.d序列。然后$Q$被称为反射随机漫步。

考虑以下两个模型进行比较:对于固定常数$a>0$，设$L^u$表示区间$[0, a]$上的均匀分布，$L^d$表示两个点${a / 3, a}$上支持的离散分布
$$
L^d{a / 3}=1-L^d{a}=3 / 4 .
$$

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

在排队理论$[114,23]$的标准书籍中，更新过程用于对服务过程以及网络到达进行建模。一般更新过程定义如下。
1.3.2.定义更新流程
设${\mathcal{E}(1), \mathcal{E}(2), \ldots}$为一个独立且相同的随机变量序列，在$\mathbb{R}{+}$上具有分布函数$\Gamma$，设$T$表示由$T(n)=\mathcal{E}(1)+\cdots+\mathcal{E}(n), n \geq 1$定义的相关随机游走，并使用$T(0)=0$。则(无延迟)更新过程为连续时间随机过程，取值为$\mathbb{Z}{+}$，定义为:
$$
R(t)=\max {n: T(n) \leq t} .
$$
更新过程的示例路径是分段常量，在更新时间${T(n): n \geq 1}$处有跳跃。

续订过程$R$采用整数值并且不递减，因此数量$R\left(t_1\right)-R\left(t_0\right), t_0, t_1 \in \mathbb{R}_{+}$可用于建模时间间隔$\left(t_0, t_1\right]$期间的到达数量，或者在此时间间隔内繁忙的服务器的服务完成数量。

更新过程最重要的例子是标准泊松过程，其中过程$\mathcal{E}$具有指数边际分布。泊松过程也是具有独立增量的随机过程的另一个例子，其分布表示为:对于每个$k \geq 0$和$0 \leq t_0 \leq t_1<\infty$，
$$
\mathrm{P}\left{R\left(t_1\right)-R\left(t_0\right)=k\right}=\frac{\left(\mu\left(t_1-t_0\right)\right)^k}{k !} e^{\mu\left(t_1-t_0\right)} .
$$
命题1.3.3总结了一些基本结果。Asmussen[23]描述了更多的结构。

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

随机过程代考

贝叶斯方法代考

广义线性模型代考

广义线性模型（GLM）归属统计学领域，是一种应用灵活的线性回归模型。该模型允许因变量的偏差分布有除了正态分布之外的其它分布。

机器学习代写

多元统计分析代考

基础数据: $N$ 个样本， $P$ 个变量数的单样本，组成的横列的数据表
变量定性: 分类和顺序；变量定量：数值
数学公式的角度分为: 因变量与自变量

时间序列分析代写

回归分析代写

MATLAB代写

R语言代写	问卷设计与分析代写
PYTHON代写	回归分析与线性模型代写
MATLAB代写	方差分析与试验设计代写
STATA代写	机器学习/统计学习代写
SPSS代写	计量经济学代写
EVIEWS代写	时间序列分析代写
EXCEL代写	深度学习代写
SQL代写	各种数据建模与可视化代写

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

Posted on 2023年8月29日2023年8月29日 by statistics-lab

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
$$
\mathrm{E}[Y \mid \mathcal{H}]=\mathrm{E}[\mathrm{E}[Y \mid \mathcal{G}] \mid \mathcal{H}] .
$$

复杂网络代写

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

在线性规划理论中，标准原问题被定义为最优化问题;
$$
\begin{array}{lrll}
\max & c^{\mathrm{T}} x & & \
\text { s.t. } & \sum_j a_{i j} x_j & \leq b_i, & \text { for } i=1, \ldots, m ; \
& x_j & \geq 0, & \text { for } j=1, \ldots, n .
\end{array}
$$
它的对偶是线性规划，
$$
\begin{array}{lrl}
\text { min } b^{\mathrm{T}} w & \
\text { s.t. } & \sum_j a_{j i} w_j \geq c_i, & \text { for } i=1, \ldots, n ; \
& w_j \geq 0, & \text { for } j=1, \ldots, m .
\end{array}
$$
原数通常用矩阵符号表示，$\max c^{\mathrm{T}} x$以$A x \leq b, x \geq 0$为准;而双重为$\min b^{\mathrm{T}} w$服从$A^{\mathrm{T}} w \geq c, w \geq 0$。

任何线性规划问题都可以用标准形式(1.8)表示。例如，通过改变目标函数的符号，可以将最小化问题重新表述为最大化问题。等式约束$y=b$可以表示为两个不等式约束$y \leq b$和$-y \leq-b$。在结果对偶中，我们发现两个对应的变量可以被一个符号不受限制的变量所取代。

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

直到第三部分，这本书需要很少的知识，在概率的高级主题。在这里概述一些高级材料是有用的，因为例如，大数定律和鞅和更新过程的中心极限定理是第一部分和第二部分中开发的理想化网络模型的动机。
概率论的起点是概率空间，定义为三重$(\Omega, \mathcal{F}, \mathrm{P})$，其中$\Omega$是点的抽象集合，$\mathcal{F}$是$\Omega$的子集的$\sigma$ -字段，$\mathrm{P}$是$\mathcal{F}$的概率度量。映射$X: \Omega \rightarrow \mathrm{X}$称为随机变量
$$
X^{-1}{B}:={\omega: X(\omega) \in B} \in \mathcal{F}
$$
对于所有集合$B \in \mathcal{B}(\mathrm{X})$:也就是说，如果$X$是从$\Omega$到$\mathrm{X}$的可测量映射。
给定概率空间$(\Omega, \mathcal{F}, \mathrm{P})$上的一个随机变量$X$，我们定义由$X$生成的$\sigma$域(记为$\sigma{X} \subseteq \mathcal{F}$)为$X$可测量的最小$\sigma$域。

如果$X$是从概率空间$(\Omega, \mathcal{F}, \mathrm{P})$到一般可测空间$(\mathrm{X}, \mathcal{B}(\mathrm{X}))$的一个随机变量，$h$是从$(\mathrm{X}, \mathcal{B}(\mathrm{X}))$到实线$(\mathbb{R}, \mathcal{B}(\mathbb{R}))$的一个实值可测映射，那么复合函数$h(X)$就是$(\Omega, \mathcal{F}, \mathrm{P})$上的一个实值随机变量:注意，有些作者为这种实值映射保留了“随机变量”这个术语。对于这样的函数，我们定义期望为
$$
\mathrm{E}[h(X)]=\int_{\Omega} h(X(\omega)) \mathrm{P}(d w)
$$
将期望定义良好且有限的实值随机变量集$Y$记为$L^1(\Omega, \mathcal{F}, \mathrm{P})$。类似地，我们用$L^{\infty}(\Omega, \mathcal{F}, \mathrm{P})$表示本质上有界的实值随机变量的集合$Y$;也就是说，有一个界$M$和一个集$A_M \subset \mathcal{F}$，其中$\mathrm{P}\left(A_M\right)=0$使得${\omega:|Y(\omega)|>M} \subseteq A_M$。
假设$Y \in L^1(\Omega, \mathcal{F}, \mathrm{P})$和$\mathcal{G} \subset \mathcal{F}$是$\mathcal{F}$的子$\sigma$字段。如果$\hat{Y} \in$$L^1(\Omega, \mathcal{G}, \mathrm{P})$和满足
$$
\mathrm{E}[Y Z]=\mathrm{E}[\hat{Y} Z] \quad \text { for all } Z \in L_{\infty}(\Omega, \mathcal{G}, \mathrm{P})
$$
那么$\hat{Y}$被称为$Y$给定$\mathcal{G}$的条件期望，记为$\mathrm{E}[Y \mid \mathcal{G}]$。以这种方式定义的条件期望存在，并且对于任何$Y \in L^1(\Omega, \mathcal{F}, \mathrm{P})$和任何子$\sigma$ -字段$\mathcal{G}$都是唯一的(模P-null集)。
假设现在我们有另一个$\sigma$ -字段$\mathcal{H} \subset \mathcal{G} \subset \mathcal{F}$。然后
$$
\mathrm{E}[Y \mid \mathcal{H}]=\mathrm{E}[\mathrm{E}[Y \mid \mathcal{G}] \mid \mathcal{H}] .
$$

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

随机过程代考

贝叶斯方法代考

广义线性模型代考

广义线性模型（GLM）归属统计学领域，是一种应用灵活的线性回归模型。该模型允许因变量的偏差分布有除了正态分布之外的其它分布。

机器学习代写

多元统计分析代考

基础数据: $N$ 个样本， $P$ 个变量数的单样本，组成的横列的数据表
变量定性: 分类和顺序；变量定量：数值
数学公式的角度分为: 因变量与自变量

时间序列分析代写

回归分析代写

MATLAB代写

R语言代写	问卷设计与分析代写
PYTHON代写	回归分析与线性模型代写
MATLAB代写	方差分析与试验设计代写
STATA代写	机器学习/统计学习代写
SPSS代写	计量经济学代写
EVIEWS代写	时间序列分析代写
EXCEL代写	深度学习代写
SQL代写	各种数据建模与可视化代写

数学代写|复杂网络代写complex networks代考|CS7280

Posted on 2023年8月15日2023年8月28日 by statistics-lab

数学代写|复杂网络代写complex networks代考|User Networks

From the original data a number of market networks can be constructed. The most natural one would of course be the network of users connected by actual transactions. Another would be the network of sellers that are connected if they have sold to the same user. Then, the links in the network would represent a possible competition or a possibility for cooperation, depending on the portfolio of articles offered by these sellers. This situation is also known as “co-opetition”. Similarly, such a co-opetition network could also be constructed based on the fact that different sellers have received bids from the same user. Further, the construction of customer groups based on the fact that they have bought from the same seller is possible which again would define a co-opetition situation for the sellers that join these customers. One could also study the relations of articles or categories based on joining them to networks when they have received bids from the same users. This would then resemble a frequent item or frequent category analysis.

Here, the focus lies on the bidder network based on single articles. Two bidders are linked if they have competed in an auction. Since all users who bid in a single auction are connected, this network results from overlaying fully connected cliques of bidders that result from each auction. Such graphs are also known as affiliation networks [29-31]. Note that one could also assign weights to links between bidders according to the number of times they have met or according to some function of the amount of money they have bid.
Prior to a block modeling analysis in this bidder network, we study its general statistical properties looking for indications of block structure [32]. We compare the results to a randomized null model (RNM) obtained from reshuffling the original data, i.e., keeping the attractiveness of each auction and the activity of each bidder constant, but randomizing which bidders take part in which auction. If the presence of clusters of users with a common interest has an influence on the statistical parameters of the network, it should be detectable by comparison with such a random null model. Figure 7.6 shows a comparison between the empirical data and the RNM in terms of the cumulative degree distribution, the cumulative distribution of the link weights as well as the clustering coefficient $c(k)$ as a function of the degree $k$.

数学代写|复杂网络代写complex networks代考|Network Clustering

The analysis of the user interests in the eBay market is based on the bidder network as constructed in the previous section. The links in this network represent articles the connected bidders (nodes) have a common interest in. The network is reduced to only those bidders that have taken part in at least two auctions and only auctions with a final price below 1,000 Euro are considered, thereby focussing on consumer goods. See Table 7.4 for the basic parameters of this reduced network.

If one now finds groups of users (clusters or communities [34-36]) with a high density of links among themselves and a low density of links to the rest of the network, the total set of links within such a group of users can be interpreted as a unifying common interest of this group. We fit a diagonal block model as introduced in Chap. 4. Recall the quality function $Q$ used:
$$
Q=\sum_s \underbrace{\left(m_{s s}-\gamma\left[m_{s s}\right]\right)}{c{s s}}=-\sum_{s<r} \underbrace{\left(m_{r s}-\gamma\left[m_{r s}\right]\right)}{a{r s}} .
$$
Note that any assignment of bidders into groups which maximizes $Q$ will be characterized by both maximum cohesion of groups and minimal adhesion between groups. If $Q$ is maximal, every node is classified in that group to which it has the largest adhesion. Compare Chap. 4 again for examples and further details of this quality function. Maximally 500 different groups of bidders were allowed in the analysis which gives a sufficient level of detail.
Figure 7.7 compares the results obtained with $\gamma=0.5$ and $\gamma=1$. Shown are the adjacency matrices $A_{i j}$ of the largest connected component of the bidder network. A black pixel at position $(i, j)$ and $(j, i)$ is shown on an $889,828 \times 889,828$ square if bidders $i$ and $j$ have competed in an auction and hence $A_{i j}=1$, otherwise the pixel is left white corresponding to $A_{i j}=0$. The rows and columns are ordered such that bidders who are classified as being in the same group are next to each other. The internal order of bidders within groups is random. The order of the groups was chosen to optimally show the correspondence between the ordering resulting from the $\gamma=0.5$ and the $\gamma=1$ ordering. In this representation, link densities correspond to pixel densities and thus to gray levels in the figure. Information about the exact size and link density contrast of the clusters is given in Table 7.5. Note the high contrast between internal and external link density.

数学代写|复杂网络代写complex networks代考|User Networks

从原始数据可以构造出许多市场网络。最自然的一种当然是由实际交易连接起来的用户网络。另一个是卖家网络，如果他们向同一用户销售产品，他们就会联系在一起。然后，网络中的链接将代表可能的竞争或合作的可能性，这取决于这些卖家提供的产品组合。这种情况也被称为“合作竞争”。同样，这种合作竞争网络也可以基于不同的卖家收到来自同一用户的出价这一事实来构建。此外，基于他们从同一卖家那里购买的事实来构建客户群体是可能的，这将再次定义加入这些客户的卖家的合作竞争情况。人们还可以研究文章或类别之间的关系，当它们收到来自同一用户的出价时，将它们加入网络。这将类似于频繁项目或频繁类别分析。

这里的重点是基于单篇文章的竞价网络。如果两个竞标者在拍卖中有过竞争，他们就会被联系在一起。由于在一次拍卖中出价的所有用户都是相互连接的，因此该网络是通过覆盖每次拍卖产生的完全连接的竞标者集团而形成的。这种图也被称为隶属网络[29-31]。注意，我们也可以根据竞标者见面的次数或根据他们出价的金额的某种函数来为竞标者之间的联系分配权重。
在对该投标人网络进行区块建模分析之前，我们研究了其一般统计特性，寻找区块结构的迹象[32]。我们将结果与通过重新洗牌原始数据获得的随机零模型(RNM)进行比较，即保持每次拍卖的吸引力和每个竞标者的活动不变，但随机选择竞标者参加哪种拍卖。如果具有共同兴趣的用户群的存在对网络的统计参数有影响，则应该通过与这种随机零模型进行比较来检测。图7.6显示了经验数据与RNM在累积度分布、链路权重累积分布以及聚类系数$c(k)$作为程度的函数$k$方面的比较。

数学代写|复杂网络代写complex networks代考|Network Clustering

对eBay市场中用户利益的分析是基于上一节所构建的投标人网络。该网络中的链接表示所连接的投标人(节点)有共同兴趣的物品。该网络只考虑那些参加过至少两次拍卖的投标人，并且只考虑最终价格低于1000欧元的拍卖，从而专注于消费品。该简化网络的基本参数见表7.4。

如果现在发现用户群(集群或社区[34-36])之间的链接密度很高，而与网络其他部分的链接密度很低，那么这样一组用户内的总链接集可以被解释为该组的统一共同利益。我们拟合了第4章中介绍的对角块模型。召回质量功能 $Q$ 使用:
$$
Q=\sum_s \underbrace{\left(m_{s s}-\gamma\left[m_{s s}\right]\right)}{c{s s}}=-\sum_{s<r} \underbrace{\left(m_{r s}-\gamma\left[m_{r s}\right]\right)}{a{r s}} .
$$
请注意，任何投标人分配到最大限度的组 $Q$ 将以群体之间最大的凝聚力和群体之间最小的粘连为特征。如果 $Q$ 如果是最大值，则每个节点都被分类到与其粘附最大的组中。再次比较第4章的例子和这个质量函数的进一步细节。最多500个不同的投标人组被允许在分析，给出了足够的细节水平。
图7.7将得到的结果与 $\gamma=0.5$ 和 $\gamma=1$．所示为邻接矩阵 $A_{i j}$ 投标人网络中最大的连接组件。位置上的黑色像素 $(i, j)$ 和 $(j, i)$ 显示在 $889,828 \times 889,828$ 对竞标者进行清算 $i$ 和 $j$ 参加过拍卖，然后呢 $A_{i j}=1$，否则像素为左白对应 $A_{i j}=0$．行和列的顺序是这样的竞标者被归类为在同一组是彼此相邻。组内竞标者的内部顺序是随机的。所选择的组的顺序是为了最佳地显示由的排序之间的对应关系 $\gamma=0.5$ 还有 $\gamma=1$ 订购。在这种表示中，链路密度对应于像素密度，从而对应于图中的灰度级。表7.5给出了关于集群的确切大小和链接密度对比的信息。注意内部和外部链接密度之间的高对比度。

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

随机过程代考

贝叶斯方法代考

广义线性模型代考

广义线性模型（GLM）归属统计学领域，是一种应用灵活的线性回归模型。该模型允许因变量的偏差分布有除了正态分布之外的其它分布。

机器学习代写

多元统计分析代考

基础数据: $N$ 个样本， $P$ 个变量数的单样本，组成的横列的数据表
变量定性: 分类和顺序；变量定量：数值
数学公式的角度分为: 因变量与自变量

时间序列分析代写

回归分析代写

MATLAB代写

R语言代写	问卷设计与分析代写
PYTHON代写	回归分析与线性模型代写
MATLAB代写	方差分析与试验设计代写
STATA代写	机器学习/统计学习代写
SPSS代写	计量经济学代写
EVIEWS代写	时间序列分析代写
EXCEL代写	深度学习代写
SQL代写	各种数据建模与可视化代写

数学代写|复杂网络代写complex networks代考|TSKS33

Posted on 2023年8月15日2023年8月28日 by statistics-lab

数学代写|复杂网络代写complex networks代考|Graphs with More than Two Clusters

After we have dealt with graphs containing only two clusters, we will now turn to the problem of networks with more than two clusters. While in the case of two clusters we could write the cavity equations for an arbitrary degree distribution, we were able to find a simple formulation for more than two clusters only for one special topology, the random Bethe lattice with three links per node, such that a cavity field is composed of combining only two messages. For graphs with a different topology, we will have to resort to a population dynamics algorithm to be described in the next section.

Let us first consider how to calculate the $\eta_{c w}^{i n}$ for general $q$. A message with $\tau=c+w$ non-zero entries will have one correct entry and $w$ wrong entries for $\tau$ types of nodes, while for $q-\tau$ types of nodes, it will have no correct entry but $w+1$ wrong entries. Hence we can write straightforwardly
$$
\begin{aligned}
& \eta_{1 w}^{i n}=p_{i n} \eta_{1 w}+\frac{\left(1-p_{i n}\right)}{q-1}\left(w \eta_{1 w}+(q-1-w) \eta_{0, w+1}\right), \
& \eta_{0 w}^{i n}=p_{i n} \eta_{0 w}+\frac{\left(1-p_{i n}\right)}{q-1}\left(w \eta_{1, w-1}+(q-1-w) \eta_{0, w}\right) .
\end{aligned}
$$
We see that this recovers the unclustered case if $\eta_{1 w}=\eta_{0, w+1}=\eta_\tau$.

For a Bethe lattice with exactly three links per node, we can write the cavity equations for the order parameters $\eta_{c w}$ for an arbitrary number of clusters as
$$
\begin{aligned}
& \eta_{1 w}=\sum_{c=0}^1 \sum_{\alpha=1-c}^{w-c}\left(\begin{array}{l}
w \
\alpha
\end{array}\right) \eta_{c, \alpha}^{i n} \eta_{1-c, w-\alpha}^{i n}+\left(\eta_{1 w}^{i n}\right)^2+2 \eta_{1 w}^{i n} \sum_{\alpha=1}^{q-1-w}\left(\begin{array}{c}
q-1-w \
\alpha
\end{array}\right) \eta_{1, w+\alpha}^{i n} \
& +\sum_{\alpha=1}^{q-1-w} \sum_{\beta=1}^{q-1-w-\alpha}\left(\begin{array}{c}
q-1-w \
\alpha
\end{array}\right)\left(\begin{array}{c}
q-1-w-\alpha \
\beta
\end{array}\right) \eta_{1, w+\alpha}^{i n} \eta_{1, w+\beta}^{i n}, \
& \eta_{0 w}=\sum_{\alpha=1}^{w-1}\left(\begin{array}{c}
w \
\alpha
\end{array}\right) \eta_{0, \alpha}^{i n} \eta_{0, w-\alpha}^{i n}+\left(\eta_{0 w}^{i n}\right)^2+2 \eta_{1 w}^{i n} \eta_{0 w}^{i n} \
& +2 \sum_{\alpha=1}^{q-1-w}\left(\begin{array}{c}
q-1-w \
\alpha
\end{array}\right)\left(\eta_{0 w}^{i n} \eta_{1, w+\alpha}^{i n}+\eta_{1 w}^{i n} \eta_{0, w+\alpha}^{i n}+\eta_{0 w}^{i n} \eta_{0, w+\alpha}^{i n}\right) \
& +\sum_{\alpha=1}^{q-1-w} \sum_{\beta=1}^{q-1-w-\alpha}\left(\begin{array}{c}
q-1-w \
\alpha
\end{array}\right)\left(\begin{array}{c}
q-1-w-\alpha \
\beta
\end{array}\right) \
& \times\left(\eta_{0, w+\alpha}^{i n} \eta_{1, w+\beta}^{i n}+\eta_{1, w+\alpha}^{i n} \eta_{0, w+\beta}^{i n}+\eta_{0, w+\alpha}^{i n} \eta_{0, w+\beta}^{i n}\right) . \
&
\end{aligned}
$$

数学代写|复杂网络代写complex networks代考|Bethe Lattice with k = 3

Figure 6.6 shows two examples for a three and a four partitioning of the Bethe lattice with three links per node. Note the similarity also to Fig. 6.4 and that $p_{i n}^c$ decreases slightly for increasing $q$. The qualitative behavior of the order parameters is the same across all degree distributions. Note how $\eta_{1, w-1}=\eta_{0, w}$ for all $p_{i n} \leq p_{i n}^c$ and $\eta_{1, w-1}>\eta_{0, w}$ for all $p_{i n}>p_{i n}^c$. Also note the different scales of the graphs, $\eta_{10}$ quickly starts to dominate all other order parameters.

数学代写|复杂网络代写complex networks代考|Graphs with More than Two Clusters

在处理了仅包含两个簇的图之后，我们现在将转向包含两个以上簇的网络问题。虽然在两个簇的情况下，我们可以为任意度分布编写空腔方程，但我们能够为两个以上的簇找到一个简单的公式，只有一个特殊的拓扑，每个节点有三个链接的随机Bethe晶格，这样一个空腔场只由两个消息组合而成。对于具有不同拓扑结构的图，我们将不得不求助于种群动态算法，该算法将在下一节中描述。

让我们首先考虑如何计算一般的$q$的$\eta_{c w}^{i n}$。对于$\tau$类型的节点，具有$\tau=c+w$非零条目的消息将有一个正确条目和$w$错误条目，而对于$q-\tau$类型的节点，它将没有正确条目但有$w+1$错误条目。因此我们可以直接写出来
$$
\begin{aligned}
& \eta_{1 w}^{i n}=p_{i n} \eta_{1 w}+\frac{\left(1-p_{i n}\right)}{q-1}\left(w \eta_{1 w}+(q-1-w) \eta_{0, w+1}\right), \
& \eta_{0 w}^{i n}=p_{i n} \eta_{0 w}+\frac{\left(1-p_{i n}\right)}{q-1}\left(w \eta_{1, w-1}+(q-1-w) \eta_{0, w}\right) .
\end{aligned}
$$
我们看到，如果$\eta_{1 w}=\eta_{0, w+1}=\eta_\tau$，这将恢复未聚集的情况。

对于每个节点恰好有三个链接的贝特晶格，我们可以将任意数量簇的序参数$\eta_{c w}$的空腔方程写成
$$
\begin{aligned}
& \eta_{1 w}=\sum_{c=0}^1 \sum_{\alpha=1-c}^{w-c}\left(\begin{array}{l}
w \
\alpha
\end{array}\right) \eta_{c, \alpha}^{i n} \eta_{1-c, w-\alpha}^{i n}+\left(\eta_{1 w}^{i n}\right)^2+2 \eta_{1 w}^{i n} \sum_{\alpha=1}^{q-1-w}\left(\begin{array}{c}
q-1-w \
\alpha
\end{array}\right) \eta_{1, w+\alpha}^{i n} \
& +\sum_{\alpha=1}^{q-1-w} \sum_{\beta=1}^{q-1-w-\alpha}\left(\begin{array}{c}
q-1-w \
\alpha
\end{array}\right)\left(\begin{array}{c}
q-1-w-\alpha \
\beta
\end{array}\right) \eta_{1, w+\alpha}^{i n} \eta_{1, w+\beta}^{i n}, \
& \eta_{0 w}=\sum_{\alpha=1}^{w-1}\left(\begin{array}{c}
w \
\alpha
\end{array}\right) \eta_{0, \alpha}^{i n} \eta_{0, w-\alpha}^{i n}+\left(\eta_{0 w}^{i n}\right)^2+2 \eta_{1 w}^{i n} \eta_{0 w}^{i n} \
& +2 \sum_{\alpha=1}^{q-1-w}\left(\begin{array}{c}
q-1-w \
\alpha
\end{array}\right)\left(\eta_{0 w}^{i n} \eta_{1, w+\alpha}^{i n}+\eta_{1 w}^{i n} \eta_{0, w+\alpha}^{i n}+\eta_{0 w}^{i n} \eta_{0, w+\alpha}^{i n}\right) \
& +\sum_{\alpha=1}^{q-1-w} \sum_{\beta=1}^{q-1-w-\alpha}\left(\begin{array}{c}
q-1-w \
\alpha
\end{array}\right)\left(\begin{array}{c}
q-1-w-\alpha \
\beta
\end{array}\right) \
& \times\left(\eta_{0, w+\alpha}^{i n} \eta_{1, w+\beta}^{i n}+\eta_{1, w+\alpha}^{i n} \eta_{0, w+\beta}^{i n}+\eta_{0, w+\alpha}^{i n} \eta_{0, w+\beta}^{i n}\right) . \
&
\end{aligned}
$$

数学代写|复杂网络代写complex networks代考|Bethe Lattice with k = 3

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

随机过程代考

贝叶斯方法代考

广义线性模型代考

广义线性模型（GLM）归属统计学领域，是一种应用灵活的线性回归模型。该模型允许因变量的偏差分布有除了正态分布之外的其它分布。

机器学习代写

多元统计分析代考

基础数据: $N$ 个样本， $P$ 个变量数的单样本，组成的横列的数据表
变量定性: 分类和顺序；变量定量：数值
数学公式的角度分为: 因变量与自变量

时间序列分析代写

回归分析代写

MATLAB代写

R语言代写	问卷设计与分析代写
PYTHON代写	回归分析与线性模型代写
MATLAB代写	方差分析与试验设计代写
STATA代写	机器学习/统计学习代写
SPSS代写	计量经济学代写
EVIEWS代写	时间序列分析代写
EXCEL代写	深度学习代写
SQL代写	各种数据建模与可视化代写

数学代写|复杂网络代写complex networks代考|NIT1104

Posted on 2023年8月15日2023年8月28日 by statistics-lab

数学代写|复杂网络代写complex networks代考|Limit of Dense Graphs with Poissonian Degree Distribution

The special form of the Poissonian degree distribution has led to numerous simplifications so far. Two crucial simplifications are the cancellations of the term $\left(n_0+2 n\right)$ ! which decouples the sums in (6.28) and the fact that for this distribution the degree and excess degree distributions are indeed the same which simplifies the calculation of the energy per node in (6.33).

These simplifications allow us to investigate the scaling of the ground state energies of the bi-partitioning problem for Poissonian graphs in the limit of large average degree. We will show that this allows us to recover the results of the replica calculations of Fu and Anderson [15] plus correction terms. The Bessel functions in (6.28) can be approximated for large arguments $x \gg n$ and fixed $n$ as
$$
I_1(n, x) \approx \frac{e^x}{\sqrt{2 \pi x}}
$$
Using this approximation we obtain for the order parameter $\eta_1$ the following equation:
$$
2 \eta_1 \approx 1-\left(4 \pi \lambda \eta_1\right)^{-1 / 2} .
$$
Equation (6.35) is approximated using (6.38) and (6.39) as
$$
X_\lambda \approx \frac{1}{2} \sqrt{\frac{1}{\pi \lambda}} \sqrt{\eta_1}=\eta_1-2 \eta_1^2 .
$$
Now we expand the solution of (6.39) in powers of $1 / \lambda$ which leads to an approximation for $\eta_1$ and hence $\eta_1^2$ as
$$
\begin{aligned}
\eta_1 & \approx \frac{1}{2}-\frac{1}{4} \sqrt{\frac{2}{\pi \lambda}}-\frac{1}{8 \pi \lambda}+\mathcal{O}\left(\lambda^{-3 / 2}\right) \text { and } \
\eta_1^2 & \approx \frac{1}{4}-\frac{1}{4} \sqrt{\frac{2}{\pi \lambda}}+\mathcal{O}\left(\lambda^{-3 / 2}\right) .
\end{aligned}
$$

数学代写|复杂网络代写complex networks代考|q-Partitioning of a Bethe Lattice with three Links per Node

Thus far we have dealt with bi-partitions of graphs with arbitrary degree distribution as one of the cases where we can write the field equations as a system of coupled polynomials. The other special case for which this can be done is a Bethe lattice where every node has exactly $k=3$ neighbors. Then, every edge leads to a node with excess degree $d=2$ and we can write for the order parameters $\eta_\tau$ the following equation:
$$
\begin{aligned}
\eta_\tau= & \sum_{\alpha=1}^{\tau-1}\left(\begin{array}{c}
\tau \
\alpha
\end{array}\right) \eta_\alpha \eta_{\tau-\alpha}+\eta_\tau^2+2 \eta_\tau \sum_{\alpha=1}^{q-\tau}\left(\begin{array}{c}
q-\tau \
\alpha
\end{array}\right) \eta_{\tau+\alpha} \
& +\sum_{\alpha=1}^{q-\tau} \sum_{\beta=1}^{q-\tau-\alpha}\left(\begin{array}{c}
q-\tau \
\alpha
\end{array}\right)\left(\begin{array}{c}
q-\tau-\alpha \
\beta
\end{array}\right) \eta_{\tau+\alpha} \eta_{\tau+\beta} .
\end{aligned}
$$
This is easily interpreted. A message with $\tau$ non-zero entries can be formed by combining two messages, one with $\alpha<\tau$ and one with $\tau-\alpha$ non-zero entries which do not overlap as in the first term. Then, two messages with exactly $\tau$ non-zero entries may overlap as in the second term. The third term denotes the possibility of combining one message with $\tau$ and one with $\alpha>\tau$ non-zero entries, while the last stands for the possibility of having an overlap of exactly $\tau$ non-zero entries when combining two messages which both have more than $\tau$ non-zero entries.

With the order parameters at hand, we can write the energy per link directly using (6.20). For the energy per node, unfortunately, we cannot write a simple expression for all numbers of parts $q$ and have to calculate $\Delta E_1$ by using Monte Carlo methods. It is interesting to study the ground state energy and modularity as a function of the number of parts $q$. Naturally, the absolute value of the ground state energy decreases as we divide the random lattice into more and more parts. However, when looking at the modularity, we see that the term $1 / q$ which we have to subtract from the negative value of the energy in (6.3) decreases for larger numbers of $Q$. Plotting $E_{g s}$ vs. $q^{-1 / 2}$ in Fig. 6.3 we observe a linear dependence which together with (6.3) suggests the existence of an optimal number of $q$ which maximizes the modularity $Q_q$. Empirically, we find by fitting our data
$$
E_g(q)=E_{\infty}-\frac{B}{\sqrt{q}},
$$
with $E_{\infty}=-1.141$ and $B=0.3496$. This is a remarkable result as it shows that even for large numbers of $q$ we can still satisfy 2.3 of the 3 connections per node on average. This is not much less than the 2.78 links per node which can be satisfied when partitioning in only two parts. This also means that practically every node has two or more links into its own community, which again means that every random Bethe lattice of connectivity $d+1=3$ has a community structure if the definitions of Radicchi et al. are applied. Plugging (6.45) into (6.3) we can find the number of parts which maximizes $Q_q$ as
$$
q^*=\frac{k^2}{B^2} \approx 74 .
$$

数学代写|复杂网络代写complex networks代考|Limit of Dense Graphs with Poissonian Degree Distribution

泊松度分布的特殊形式迄今已导致了许多简化。两个重要的简化是消去了$\left(n_0+2 n\right)$ !它解耦了(6.28)中的和，并且对于这个分布，度分布和多余度分布确实是相同的，这简化了(6.33)中每个节点能量的计算。

这些简化使我们能够研究大平均度极限下泊松图双分划问题的基态能量的标度问题。我们将表明，这使我们能够恢复Fu和Anderson[15]的副本计算结果加上校正项。(6.28)中的贝塞尔函数可以近似为大参数$x \gg n$和固定的$n$ as
$$
I_1(n, x) \approx \frac{e^x}{\sqrt{2 \pi x}}
$$
利用这个近似，我们可以得到阶参量$\eta_1$的下式:
$$
2 \eta_1 \approx 1-\left(4 \pi \lambda \eta_1\right)^{-1 / 2} .
$$
式(6.35)用式(6.38)和式(6.39)近似表示
$$
X_\lambda \approx \frac{1}{2} \sqrt{\frac{1}{\pi \lambda}} \sqrt{\eta_1}=\eta_1-2 \eta_1^2 .
$$
现在我们将(6.39)的解展开为$1 / \lambda$的幂，从而得到$\eta_1$的近似值，因此得到$\eta_1^2$ as
$$
\begin{aligned}
\eta_1 & \approx \frac{1}{2}-\frac{1}{4} \sqrt{\frac{2}{\pi \lambda}}-\frac{1}{8 \pi \lambda}+\mathcal{O}\left(\lambda^{-3 / 2}\right) \text { and } \
\eta_1^2 & \approx \frac{1}{4}-\frac{1}{4} \sqrt{\frac{2}{\pi \lambda}}+\mathcal{O}\left(\lambda^{-3 / 2}\right) .
\end{aligned}
$$

数学代写|复杂网络代写complex networks代考|q-Partitioning of a Bethe Lattice with three Links per Node

到目前为止，我们已经处理了任意度分布图的双分区，作为我们可以将场方程写成耦合多项式系统的一种情况。另一种特殊情况是贝特格，每个节点都有$k=3$个邻居。然后，每条边都指向一个具有多余度$d=2$的节点，我们可以将阶参数$\eta_\tau$写成如下公式:
$$
\begin{aligned}
\eta_\tau= & \sum_{\alpha=1}^{\tau-1}\left(\begin{array}{c}
\tau \
\alpha
\end{array}\right) \eta_\alpha \eta_{\tau-\alpha}+\eta_\tau^2+2 \eta_\tau \sum_{\alpha=1}^{q-\tau}\left(\begin{array}{c}
q-\tau \
\alpha
\end{array}\right) \eta_{\tau+\alpha} \
& +\sum_{\alpha=1}^{q-\tau} \sum_{\beta=1}^{q-\tau-\alpha}\left(\begin{array}{c}
q-\tau \
\alpha
\end{array}\right)\left(\begin{array}{c}
q-\tau-\alpha \
\beta
\end{array}\right) \eta_{\tau+\alpha} \eta_{\tau+\beta} .
\end{aligned}
$$
这很容易解释。包含$\tau$非零条目的消息可以通过组合两个消息来形成，一个包含$\alpha<\tau$，另一个包含$\tau-\alpha$非零条目，它们不像第一个项那样重叠。然后，两个完全具有$\tau$非零条目的消息可能像第二项一样重叠。第三项表示将一个消息与$\tau$和一个消息与$\alpha>\tau$非零条目组合在一起的可能性，而最后一项表示在组合两个消息时恰好有$\tau$个非零条目重叠的可能性，这两个消息都有超过$\tau$个非零条目。

有了顺序参数，我们可以直接使用式(6.20)写出每个链路的能量。对于每个节点的能量，不幸的是，我们不能写出一个简单的表达式为所有数量的部分$q$，必须通过使用蒙特卡罗方法计算$\Delta E_1$。研究基态能量和模块化作为零件数量的函数$q$是很有趣的。当我们将随机晶格分成越来越多的部分时，基态能量的绝对值自然会减小。然而，当观察模块化时，我们看到，我们必须从(6.3)中的负值中减去的项$1 / q$随着$Q$的数量增加而减少。在图6.3中绘制$E_{g s}$与$q^{-1 / 2}$，我们观察到线性依赖关系，它与(6.3)一起表明存在最优数量$q$，使模块化$Q_q$最大化。根据经验，我们通过拟合我们的数据来发现
$$
E_g(q)=E_{\infty}-\frac{B}{\sqrt{q}},
$$
有$E_{\infty}=-1.141$和$B=0.3496$。这是一个显著的结果，因为它表明，即使对于大量的$q$，我们仍然可以平均满足每个节点3个连接中的2.3个。这并不比每个节点的2.78个链接少多少，如果只划分为两个部分，就可以满足这个要求。这也意味着实际上每个节点都有两个或多个链接到自己的社区，这再次意味着如果应用Radicchi等人的定义，每个连接的随机Bethe格$d+1=3$都有一个社区结构。将(6.45)代入(6.3)，我们可以找到使$Q_q$ as最大化的零件数
$$
q^*=\frac{k^2}{B^2} \approx 74 .
$$

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

随机过程代考

贝叶斯方法代考

广义线性模型代考

广义线性模型（GLM）归属统计学领域，是一种应用灵活的线性回归模型。该模型允许因变量的偏差分布有除了正态分布之外的其它分布。

机器学习代写

多元统计分析代考

基础数据: $N$ 个样本， $P$ 个变量数的单样本，组成的横列的数据表
变量定性: 分类和顺序；变量定量：数值
数学公式的角度分为: 因变量与自变量

时间序列分析代写

回归分析代写

MATLAB代写

R语言代写	问卷设计与分析代写
PYTHON代写	回归分析与线性模型代写
MATLAB代写	方差分析与试验设计代写
STATA代写	机器学习/统计学习代写
SPSS代写	计量经济学代写
EVIEWS代写	时间序列分析代写
EXCEL代写	深度学习代写
SQL代写	各种数据建模与可视化代写

数学代写|复杂网络代写complex networks代考|CS7280

Posted on 2023年7月26日2023年8月25日 by statistics-lab

数学代写|复杂网络代写complex networks代考|Benchmarking the Algorithm

In order to benchmark the performance of the Potts model approach to community detection, it is applied to computer-generated test networks. Networks with communities of equal and different size were constructed. Those with equal size had 128 nodes, grouped into 4 communities of size 32 . Those with differently sized communities had 320 nodes, grouped into 4 communities of size $32,64,96$ and 128 . In both types of networks, each node has an average degree of $\langle k\rangle=16$. The average number of links to members of the same community $\left\langle k_{i n}\right\rangle$ and to members of different communities $\left\langle k_{\text {out }}\right\rangle$ is then varied, but always ensuring $\left\langle k_{\text {in }}\right\rangle+\left\langle k_{\text {out }}\right\rangle=\langle k\rangle$. Hence, decreasing $k_{\text {in }}$ renders the problem of community detection more difficult.

Recovering a known community structure, any algorithm has to fulfill two criteria: it has to group nodes in the same community which belong together by design and it has to group nodes apart which belong to different communities by design. The first criterion is called “sensitivity” and measures the percentage of pairs of nodes which are correctly grouped together. The second criterion is called “specificity” and measures the percentage of pairs of nodes which are correctly grouped apart.

Because of the Poisson nature of the degree distribution, a connection model of $p_{i j}=p$ was used. Figure 4.5 shows the result of this experiment in comparison with the results obtained from the algorithm of Girvan and Newman [14]. Clearly, both algorithms show high sensitivity and high specificity. However, the Potts model outperforms the GN algorithm on both types of networks in both sensitivity and specificity. When relaxing the Potts model Hamiltonian from random initial conditions at zero temperature, performance decreases, but is still as good as that of the GN algorithm.

An important aspect is the dependence of the sensitivity (specificity) of the algorithm on the number of allowed spin states $q$. Figure 4.6 shows that as long as $q \geq 4$, i.e., the actual number of communities in the network, the value of $q$ is irrelevant. This result is also independent of the strength of the community structure under investigation, i.e., independent of $k_{i n}$. Furthermore, it is necessary to study the stability of results with respect to a change in $\gamma$. As Fig. 4.6 shows, the better the community structure is defined, i.e., the greater $k_{i n}$ is with respect to $\langle k\rangle$, the more stable are the results. The maxima of the curves for all values of $k_{i n}$, however, coincide at $\gamma=1$, i.e., at the point where the contribution of missing and existing links is equal. The same statements also apply to the specificity.

数学代写|复杂网络代写complex networks代考|Community Detection and Graph Partitioning

In order to assess the statistical significance of the community structures found with any algorithm, it is necessary to compare them with expectation values for random networks. This is of course always possible by rewiring the network randomly [15], keeping the degree distribution invariant and then running a community detection algorithm again, comparing the result to the original network. This method, however, can only give an answer to what a particular community detection algorithm may find in a random network and hence depends on the very method of community detection used. It seems a much better method would be to compare the results of a community detection algorithm with a theoretical result, obtained independently of any algorithm. It was shown in previous sections that the problem of community detection can be mapped onto finding the ground state of an infinite range spin glass. A number of techniques exist to calculate expectation values for the energy and the local field distribution in the ground state of spin glasses, given that the couplings between spins are random, but with a known distribution. In the remainder of this chapter and in the following ones, we will make extensive use of these techniques.

What does the community structure of a completely random network look like? The first observation one makes is that the configuration space or the number of possible assignments into $q$ communities is largest when these groups are equal in size, i.e., contain the same number of nodes $N / q$. One can also show that the variance of the number of links within communities is largest when they are of equal size. These two facts taken together mean that in random networks, the assignment of nodes into communities with maximal modularity will with very high probability lead to equal-sized communities. Hence, we only need to calculate expectation values for the modularity of a partition of the network into equal-sized groups. In the language of spin glasses, this means we are looking for the energy of a ground state with zero magnetization. We can also say that we are looking for a partition into $q$ equal-sized groups with a maximum number of links within groups, or, equivalently, with a minimum number of links between groups. The latter is a standard problem in combinatorial optimization and is known as the graph partitioning problem. In short, the communities which form the partition of maximum modularity in a random network will correspond to a minimum cut equipartition.

数学代写|复杂网络代写complex networks代考|Benchmarking the Algorithm

为了测试Potts模型在社区检测中的性能，将其应用于计算机生成的测试网络。构建了大小不等的社区网络。大小相同的有128个节点，分成4个32个节点的社区。不同大小的群落有320个节点，分为4个群落，大小分别为$32,64,96$和128。在这两种类型的网络中，每个节点的平均度为$\langle k\rangle=16$。到同一社区成员$\left\langle k_{i n}\right\rangle$和不同社区成员$\left\langle k_{\text {out }}\right\rangle$的平均链接数是不同的，但始终确保$\left\langle k_{\text {in }}\right\rangle+\left\langle k_{\text {out }}\right\rangle=\langle k\rangle$。因此，减少$k_{\text {in }}$使得社区检测问题更加困难。

恢复一个已知的社区结构，任何算法都必须满足两个标准:它必须将同一社区中的节点按照设计分组在一起，并且必须将属于不同社区的节点按照设计分组分开。第一个标准被称为“灵敏度”，衡量正确组合在一起的节点对的百分比。第二个标准被称为“特异性”，衡量正确分组的节点对的百分比。

由于度分布的泊松性质，使用$p_{i j}=p$的连接模型。图4.5为本实验结果与Girvan和Newman[14]算法结果的对比。显然，这两种算法都具有高灵敏度和高特异性。然而，Potts模型在两种类型的网络上的敏感性和特异性都优于GN算法。当在零温度下从随机初始条件放宽Potts模型哈密顿量时，性能下降，但仍然与GN算法一样好。

一个重要的方面是该算法的灵敏度(特异性)依赖于允许的自旋态的数量$q$。从图4.6可以看出，只要$q \geq 4$即网络中社区的实际数量，$q$的值就无关紧要。该结果也独立于所调查的社区结构的强度，即独立于$k_{i n}$。此外，有必要研究结果相对于$\gamma$的变化的稳定性。如图4.6所示，群落结构定义得越好，即$k_{i n}$相对于$\langle k\rangle$越大，则结果越稳定。然而，所有$k_{i n}$值的曲线的最大值在$\gamma=1$重合，即在缺失链接和现有链接的贡献相等的点上。同样的陈述也适用于特异性。

数学代写|复杂网络代写complex networks代考|Community Detection and Graph Partitioning

为了评估用任何算法找到的群体结构的统计显著性，有必要将它们与随机网络的期望值进行比较。当然，这总是可以通过随机重新连接网络来实现[15]，保持度分布不变，然后再次运行社区检测算法，将结果与原始网络进行比较。然而，这种方法只能给出特定社区检测算法在随机网络中可能发现的答案，因此取决于所使用的社区检测方法。似乎更好的方法是将社区检测算法的结果与独立于任何算法获得的理论结果进行比较。在前面的章节中已经表明，共同体检测的问题可以映射到寻找无限范围自旋玻璃的基态。考虑到自旋玻璃之间的耦合是随机的，但有一个已知的分布，存在许多技术来计算自旋玻璃基态的能量和局部场分布的期望值。在本章的剩余部分和以下几章中，我们将广泛使用这些技术。

一个完全随机网络的社区结构是什么样的?第一个观察结果是，当这些组的大小相等时，配置空间或可能分配到$q$社区的数量最大，即包含相同数量的节点$N / q$。我们还可以证明，当社区内的链接大小相等时，它们的数量差异最大。这两个事实结合在一起意味着，在随机网络中，将节点分配到具有最大模块化的社区将有很大的概率导致等大小的社区。因此，我们只需要计算网络划分为大小相等的组的模块化的期望值。在自旋玻璃的语言中，这意味着我们正在寻找具有零磁化的基态的能量。我们也可以说，我们正在寻找一个划分为$q$大小相等的组，组内具有最大数量的链接，或者，等价地，组之间具有最小数量的链接。后者是组合优化中的一个标准问题，被称为图划分问题。简而言之，在一个随机网络中，组成最大模块划分的社团对应于一个最小切割均分。

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

随机过程代考

贝叶斯方法代考

广义线性模型代考

广义线性模型（GLM）归属统计学领域，是一种应用灵活的线性回归模型。该模型允许因变量的偏差分布有除了正态分布之外的其它分布。

机器学习代写

多元统计分析代考

基础数据: $N$ 个样本， $P$ 个变量数的单样本，组成的横列的数据表
变量定性: 分类和顺序；变量定量：数值
数学公式的角度分为: 因变量与自变量

时间序列分析代写

回归分析代写

MATLAB代写

R语言代写	问卷设计与分析代写
PYTHON代写	回归分析与线性模型代写
MATLAB代写	方差分析与试验设计代写
STATA代写	机器学习/统计学习代写
SPSS代写	计量经济学代写
EVIEWS代写	时间序列分析代写
EXCEL代写	深度学习代写
SQL代写	各种数据建模与可视化代写

统计代写|复杂网络代写complex networks代考|PCS810

Posted on 2023年7月26日2023年8月25日 by statistics-lab

统计代写|复杂网络代写complex networks代考|Choice of a Penalty Function and Null Model

We have introduced $p_{i j}$ as a penalty on the matching of missing links in $\mathbf{A}$ to links in B. As such, it can in principle take any form or value that may seem suitable. However, we have already hinted at the fact that $p_{i j}$ can also be interpreted as a probability. As such, it provides a random null model for the network under study. The quality functions (3.13), (3.13) and (3.15) then all compare distribution of links as found in the network for a given assignment of nodes into blocks to the expected link (weight) distribution if links (weight) were distributed independently of the assignment of nodes into blocks according to $p_{i j}$. Maximizing the quality functions (3.13), (3.13) and (3.15) hence means to find an assignment of nodes into blocks such that the number (weight) of edges in blocks deviates as strongly as possible from the expectation value due to the random null model.

Two exemplary choices of link distributions or random null models shall be illustrated. Both fulfill the constraint that $\sum_{i j} w_{i j} A_{i j}=\sum_{i j} p_{i j}$. The simplest choice is to assume every link equally probable with probability $p_{i j}=p$ independent from $i$ to $j$. Writing
$$
p_{i j}=p=\frac{\sum_{k l} w_{k l} A_{k l}}{N^2}
$$
leads naturally to
$$
\left[m_{r s}\right]_p=p n_r n_s,
$$
with $n_r$ and $n_s$ denoting the number of nodes in group $r$ and $s$, respectively.

统计代写|复杂网络代写complex networks代考|Cohesion and Adhesion

From the above considerations and to simplify further developments, the concepts of “cohesion” and “adhesion” are introduced. The coefficient of adhesion between groups $r$ and $s$ is defined as
$$
a_{r s}=m_{r s}-\gamma\left[m_{r s}\right]{p{i j}} .
$$
For $r=s$, we call $c_{s s}=a_{s s}$ the coefficient of “cohesion”. Two groups of nodes have a positive coefficient of adhesion, if they are connected by edges bearing more weight than expected from $p_{i j}$. We hence call a group cohesive, if its nodes are connected by edges bearing more weight than expected from $p_{i j}$. This allows for a shorthand form of (3.15) as $\mathcal{Q}=\frac{1}{2} \sum_{r s}\left|a_{r s}\right|$ and we see that the block model $\mathbf{B}$ has entries of one where $a_{r s}>0$. Remember that $a_{r s}$ depends on the global parameter $\gamma$ and the assumed penalty function $p_{i j}$. For $\gamma=1$ and the model $p_{i j}=\frac{k_i^{\text {out }} k_j^{\text {in }}}{M}$ one finds
$$
\sum_{r s} a_{r s}=\sum_r a_{r s}=\sum_s a_{r s}=0 .
$$
This means that when $\mathbf{B}$ is assigned from (3.15) there exists at least one entry of one and at least one entry of zero in every row and column of $\mathbf{B}$ (provided that the network is not complete or zero).

复杂网络代写

统计代写|复杂网络代写complex networks代考|Choice of a Penalty Function and Null Model

我们引入$p_{i j}$作为对$\mathbf{A}$中缺失链接与b中的链接匹配的惩罚。因此，原则上它可以采用任何看起来合适的形式或值。然而，我们已经暗示了$p_{i j}$也可以被解释为概率的事实。因此，它为所研究的网络提供了一个随机的零模型。然后，质量函数(3.13)，(3.13)和(3.15)都将网络中发现的给定节点分配为块的链接分布与预期的链接(权重)分布进行比较，如果链接(权重)的分布独立于根据$p_{i j}$将节点分配为块。因此，最大化质量函数(3.13)、(3.13)和(3.15)意味着将节点分配到块中，使得块中边的数量(权重)尽可能强烈地偏离由于随机零模型而产生的期望值。

将举例说明链路分布或随机零模型的两个示例性选择。都满足$\sum_{i j} w_{i j} A_{i j}=\sum_{i j} p_{i j}$。最简单的选择是假设每个链接都是等概率的，概率$p_{i j}=p$独立于$i$到$j$。写作
$$
p_{i j}=p=\frac{\sum_{k l} w_{k l} A_{k l}}{N^2}
$$
自然会导致
$$
\left[m_{r s}\right]_p=p n_r n_s,
$$
其中$n_r$和$n_s$分别表示$r$和$s$组中的节点数。

统计代写|复杂网络代写complex networks代考|Cohesion and Adhesion

从上述考虑出发，为了简化进一步的发展，我们引入了“内聚”和“粘附”的概念。定义组间的粘附系数$r$和$s$为
$$
a_{r s}=m_{r s}-\gamma\left[m_{r s}\right]{p{i j}} .
$$
对于$r=s$，我们称$c_{s s}=a_{s s}$为“内聚”系数。如果两组节点通过比$p_{i j}$所期望的权重更大的边连接，则它们的粘附系数为正。因此，我们称一个群为内聚的，如果它的节点由比$p_{i j}$所期望的权重更大的边连接。这允许(3.15)的简写形式为$\mathcal{Q}=\frac{1}{2} \sum_{r s}\left|a_{r s}\right|$，我们看到块模型$\mathbf{B}$有一个条目，其中$a_{r s}>0$。请记住，$a_{r s}$依赖于全局参数$\gamma$和假定的惩罚函数$p_{i j}$。对于$\gamma=1$和模型$p_{i j}=\frac{k_i^{\text {out }} k_j^{\text {in }}}{M}$，可以找到
$$
\sum_{r s} a_{r s}=\sum_r a_{r s}=\sum_s a_{r s}=0 .
$$
这意味着当从(3.15)中分配$\mathbf{B}$时，在$\mathbf{B}$的每一行和每一列中至少存在一个1和至少一个0的条目(前提是网络不完整或为零)。

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

随机过程代考

贝叶斯方法代考

广义线性模型代考

广义线性模型（GLM）归属统计学领域，是一种应用灵活的线性回归模型。该模型允许因变量的偏差分布有除了正态分布之外的其它分布。

机器学习代写

多元统计分析代考

基础数据: $N$ 个样本， $P$ 个变量数的单样本，组成的横列的数据表
变量定性: 分类和顺序；变量定量：数值
数学公式的角度分为: 因变量与自变量

时间序列分析代写

回归分析代写

MATLAB代写

R语言代写	问卷设计与分析代写
PYTHON代写	回归分析与线性模型代写
MATLAB代写	方差分析与试验设计代写
STATA代写	机器学习/统计学习代写
SPSS代写	计量经济学代写
EVIEWS代写	时间序列分析代写
EXCEL代写	深度学习代写
SQL代写	各种数据建模与可视化代写

统计代写|复杂网络代写complex networks代考|CS60078

Posted on 2023年7月26日2023年7月26日 by statistics-lab

统计代写|复杂网络代写complex networks代考|Non-hierarchical

The non-hierarchical methods approach the problem from a different perspective. In principle, they intend to calculate a full distance matrix for the nodes of the network. This can then be treated by conventional techniques.

One of the earliest approaches to community detection is due to Eriksen et al. $[41,42]$. They study a diffusion process on a network and analyze the decay of the modes of the following diffusive system with discrete time:

$$
\rho_i(t+1)-\rho_i(t)=\sum_j\left(T_{i j}-\delta_{i j}\right) \rho_j(t) .
$$
Here $T_{i j}$ represents the adjacency matrix of the network such that $T_{i j}=1 / k_j$ for $A_{i j}=1$ and zero otherwise. Hence $T_{i j}$ represents the probability of a random walker to go from $j$ to $i$. The decay of a random initial configuration $\rho(t=0)$ toward the steady state is characterized by the eigenmodes of the transition matrix $T_{i j}$. The eigenvectors corresponding to the largest eigenvalues can then be used to define a distance between nodes which helps in identifying communities. To do this, the eigenvectors belonging to the largest non-trivial positive eigenvalues are plotted against each other. This diffusion approach is very similar in spirit to other algorithms based on the idea of using flow simulations for community detection as suggested by van Dongen [43] under the name of “Markov clustering” (MCL).

The method presented by Zhou [44-46] first converts the sparse adjacency matrix of the graph into a full distance matrix by calculating the average time a Brownian particle needs to move from node $i$ to $j$. Then this distance matrix is clustered using ordinary hierarchical clustering algorithms. This approach is based on the observation that a random walker has shorter traveling time between two nodes if many (short) alternative paths exist.

Another spectral approach has been taken by Muños and Donetti [47]. They work with the Laplacian matrix of the network. The Laplacian is defined as
$$
L_{i j}=k_i \delta_{i j}-A_{i j} .
$$

统计代写|复杂网络代写complex networks代考|Optimization Based

A different approach which is reminiscent of the parametric clustering procedures known in computer science is the idea of searching for partitions with maximum modularity $Q$ using combinatorial optimization techniques [48]. This approach has been adopted by Guimera et al. in Refs. [2, 49] or Massen et al. [50] using simulated annealing [51] or Duch and Arenas using extremal optimization [52].

Though this approach will be the preferred one for the remainder of this book, a number of issues remain. For the hierarchical algorithms, a community was to be understood as whatever the algorithm outputs. Now, it is not the algorithm that defines what a community is, but the quality function, i.e., the modularity $Q$ in this case. Also, the modularity $Q$ as defined by Newman $[23]$ is parameter free and an understanding for hierarchical and overlapping structures needs to be developed.

Block structure in networks is a very common and well-studied phenomenon. The concepts of structural and regular equivalence as well as the types of blocks defined for generalized block modeling are well defined but appear too rigid to be of practical use for large and noisy data sets. Diagonal block models or modular structures have received particular attention in the literature and have developed into an almost independent concept of cohesive subgroups or communities. The comparison of many different community definitions from various fields has shown that the concept of module or community in a network is only vaguely defined. The diversity of algorithms published is only a consequence of this vague definition. None of the algorithms could be called “ideal” in the sense that it combines the features of computational efficiency, accuracy, flexibility and adaptability with regard to the network and easy interpretation of the results. More importantly, none of the above-cited publications allows an estimation to which degree the community structure found is a reality of the network or a product of the clustering process itself. The following chapters are addressing these issues and present a framework in which community detection is viewed again as a special case of a general procedure for detecting block structure in networks.

复杂网络代写

统计代写|复杂网络代写complex networks代考|Non-hierarchical

非分层方法从不同的角度处理问题。原则上，他们打算计算网络节点的全距离矩阵。然后可以用常规技术来处理。

最早的社区检测方法之一来自Eriksen等人$[41,42]$。他们研究了网络上的扩散过程，并分析了以下扩散系统的模态随离散时间的衰减:

$$
\rho_i(t+1)-\rho_i(t)=\sum_j\left(T_{i j}-\delta_{i j}\right) \rho_j(t) .
$$
这里$T_{i j}$表示网络的邻接矩阵，使得$A_{i j}=1$为$T_{i j}=1 / k_j$，否则为零。因此$T_{i j}$表示随机步行者从$j$到$i$的概率。随机初始构型$\rho(t=0)$向稳态的衰减由跃迁矩阵$T_{i j}$的特征模态表征。与最大特征值相对应的特征向量可以用来定义节点之间的距离，这有助于识别社区。为了做到这一点，属于最大非平凡正特征值的特征向量被相互绘制。这种扩散方法在精神上与van Dongen[43]以“马尔可夫聚类”(Markov clustering, MCL)的名义提出的基于使用流模拟进行社区检测的思想的其他算法非常相似。

Zhou[44-46]提出的方法首先通过计算布朗粒子从节点$i$移动到$j$所需的平均时间，将图的稀疏邻接矩阵转换为全距离矩阵。然后使用普通的层次聚类算法对该距离矩阵进行聚类。这种方法是基于这样的观察:如果存在许多(短)可选路径，则随机行走器在两个节点之间的行走时间会更短。

Muños和Donetti[47]采用了另一种光谱方法。他们使用网络的拉普拉斯矩阵。拉普拉斯式定义为
$$
L_{i j}=k_i \delta_{i j}-A_{i j} .
$$

统计代写|复杂网络代写complex networks代考|Optimization Based

另一种与计算机科学中已知的参数聚类过程类似的方法是使用组合优化技术搜索具有最大模块化$Q$的分区[48]。Guimera等人在参考文献中采用了这种方法。[2,49]或Massen等[50]使用模拟退火[51]或Duch和Arenas使用极值优化[52]。

尽管这种方法将是本书其余部分的首选方法，但仍然存在一些问题。对于分层算法，社区被理解为算法的输出。现在，定义社区的不是算法，而是质量函数，即本例中的模块化Q。此外，Newman $[23]定义的模块化$Q$是无参数的，需要发展对分层和重叠结构的理解。

网络中的块结构是一种非常普遍且被充分研究的现象。结构等价和规则等价的概念以及为广义块建模定义的块的类型都有很好的定义，但对于大型和有噪声的数据集来说，似乎过于严格而无法实际使用。对角块模型或模块化结构在文献中受到了特别的关注，并已发展成为一个几乎独立的凝聚力子群体或社区的概念。通过比较不同领域对社区的定义，可以发现网络中模块或社区的定义是模糊的。发表的算法的多样性只是这种模糊定义的结果。没有一种算法可以被称为“理想”，因为它结合了计算效率、准确性、灵活性和网络适应性以及易于解释结果的特点。更重要的是，上述引用的出版物都不允许估计发现的社区结构在多大程度上是网络的现实或聚类过程本身的产物。下面的章节将讨论这些问题，并提出一个框架，在这个框架中，社区检测再次被视为检测网络中块结构的一般程序的特殊情况。

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

随机过程代考

贝叶斯方法代考

广义线性模型代考

广义线性模型（GLM）归属统计学领域，是一种应用灵活的线性回归模型。该模型允许因变量的偏差分布有除了正态分布之外的其它分布。

机器学习代写

多元统计分析代考

基础数据: $N$ 个样本， $P$ 个变量数的单样本，组成的横列的数据表
变量定性: 分类和顺序；变量定量：数值
数学公式的角度分为: 因变量与自变量

时间序列分析代写

回归分析代写

MATLAB代写

R语言代写	问卷设计与分析代写
PYTHON代写	回归分析与线性模型代写
MATLAB代写	方差分析与试验设计代写
STATA代写	机器学习/统计学习代写
SPSS代写	计量经济学代写
EVIEWS代写	时间序列分析代写
EXCEL代写	深度学习代写
SQL代写	各种数据建模与可视化代写

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

Posted on 2023年6月28日2023年6月28日 by statistics-lab

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在网络理论的背景下，复杂网络是具有非微观拓扑特征的图（网络）这些特征在格子或随机图等简单网络中不出现，但在代表真实系统的网络中经常出现。

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我们提供的复杂网络complex network及其相关学科的代写，服务范围广, 其中包括但不限于:

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
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

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

大量的对角块模型或模块化结构使得模块化成为一个非常重要的概念，以至于它经常在块建模的一般框架之外进行研究。一种解释可能是，在社交网络中，它甚至可能是主要的屏蔽结构。原因可能是同质性[16]，即倾向于与与自己相似的主体形成联系，是社会网络形成的主要机制。然而，回想一下，网络中功能角色的概念比单纯的内聚性要广泛得多，因为它特别关注节点组之间的相互依赖关系。强调节点组之间不存在依赖关系的模块化或社区结构只是一种特殊情况。也可能是模块化的概念对物理学家特别有吸引力，因为它让人想起了将系统分解成更小的子系统的还原论方法，这种方法在自然科学中非常成功。

然而，在文献中，对于社区或模块究竟是什么，并没有一个普遍接受的定义。存在各种各样的定义，它们都暗示社区成员之间的联系比与网络其他成员的联系更紧密。有两种方法可以解决这个问题。一种方法是从社区的定义开始，然后搜索与该定义匹配的节点集。或者可以使用启发式方法，通过设计一个算法并将社区定义为该算法的输出。这两种方法有一个根本的区别:当从社区的定义开始时，经常会出现网络中的一些节点不属于任何社区的情况。另一方面，算法方法通常会对顶点集进行划分，以便在某个社区中找到所有节点。是否需要将所有节点分配到一个社区中需要由研究者决定，并且可能决定哪些定义和方法在分析实际数据时是有用的。考虑到这些因素，我们将简要回顾文献中采用的方法。

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

社区结构的研究在社会学领域有着悠久的传统，毫不奇怪，引发物理学家对该领域兴趣的例子是社会学的$[17,18]$。与社区相比，术语内聚子组通常用于包含强调问题不同方面的许多定义。这些可以根据可达性、节点度或内外链接的比较来分组定义[11]。

团是完全子图，因此每个成员都与团中的其他成员相连。n-团是一个极大子图，使得任意两个成员$i, j$之间的测地线距离$d(i, j)$小于或等于$n$。很自然，小集团就是1人小集团。注意，最短路径也可能穿过不属于n-团的节点，因此n-团的直径可能大于$n$。n族表示直径小于或等于$n$的n系。自然，所有的n氏族也都是n集团。另外，$n$ -club是直径$n$的最大子图。

这些定义在几个方面存在问题。在成员节点中，小团体的规模永远不会超过最小的度，这就限制了这些团体在有限度的大型网络中通常非常小。如果网络具有小世界性质，那么依赖于距离的其他定义就有问题。这些社区的重叠部分通常与一个典型群体一样大。

另一组定义是基于社区成员的程度。$k$ -plex是$n$节点的最大子图，使得每个成员与k-plex中的其他节点至少有$n-k$个连接。这个定义没有集团的定义那么严格，因为它允许丢失一些链接。同时，k-plex只包含最小度$d \geq(n-k)$的节点。$k$ -core表示最大子图，使得每个节点与k-core的其他成员至少有$k$个连接。

这里，k-plex的大小再次受到节点度的限制。k核也是有问题的，因为它们忽略了所有度小于$k$的节点，即使它们与该核内的节点有所有连接。

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金融工程代写

金融工程是使用数学技术来解决金融问题。金融工程使用计算机科学、统计学、经济学和应用数学领域的工具和知识来解决当前的金融问题，以及设计新的和创新的金融产品。

非参数统计代写

非参数统计指的是一种统计方法，其中不假设数据来自于由少数参数决定的规定模型；这种模型的例子包括正态分布模型和线性回归模型。

广义线性模型代考

广义线性模型（GLM）归属统计学领域，是一种应用灵活的线性回归模型。该模型允许因变量的偏差分布有除了正态分布之外的其它分布。

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）。

有限元方法代写

有限元方法（FEM）是一种流行的方法，用于数值解决工程和数学建模中出现的微分方程。典型的问题领域包括结构分析、传热、流体流动、质量运输和电磁势等传统领域。

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。

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