Posted on Categories:Complex Network, 复杂网络, 数据科学代写, 统计代写, 统计代考

# 数据科学代写|复杂网络代写Complex Network代考|CS60078 Spectra of normalized Laplacian matrices

avatest™

## avatest™帮您通过考试

avatest™的各个学科专家已帮了学生顺利通过达上千场考试。我们保证您快速准时完成各时长和类型的考试，包括in class、take home、online、proctor。写手整理各样的资源来或按照您学校的资料教您，创造模拟试题，提供所有的问题例子，以保证您在真实考试中取得的通过率是85%以上。如果您有即将到来的每周、季考、期中或期末考试，我们都能帮助您！

•最快12小时交付

•200+ 英语母语导师

•70分以下全额退款

## 数据科学代写|复杂网络代写Complex Network代考|Spectra of normalized Laplacian matrices

If the Laplacian matrix $L$ is about flows and diffusion, the following two matrices are about random walks. The symmetric normalized Laplacian matrix (also called Chung’s Laplacian) is ${ }^{37}$
$$\mathcal{L} \equiv D^{-1 / 2} L D^{-1 / 2}=I-D^{-1 / 2} A D^{-1 / 2},$$
so its entries are
$$\mathcal{L}{i j}=\left{\begin{array}{cl} 1 & \text { if } i=j \text { and } q{i} \neq 0 \ -\frac{A_{i j}}{\sqrt{q_{i} q_{j}}} & \text { if } i \neq j \ 0 & \text { otherwise. } \end{array}\right.$$
${ }^{36}$ The minor $M_{i j}$ of the matrix $L$ is the latter matrix with row $i$ and column $j$ removed. ${ }^{37}$ For Laplacian matrices in weighted graphs, see Butler and Chung (2006).

The random walk Laplacian matrix (normalised Laplacian matrix) is
$$\widetilde{\mathcal{L}} \equiv L D^{-1}=I-A D^{-1},$$
where $A D^{-1}=\mathcal{P}$ is the transition matrix for a random walk, giving the rate at which a random walker hops from vertex $j$ to $i$ :
$$\mathcal{P}{i j}=\frac{A{i j}}{q_{j}} .$$
The entries of the matrix $\widetilde{\mathcal{L}}$ are
$$\widetilde{\mathcal{L}}{i j}=\left{\begin{array}{cl} 1 & \text { if } i=j \text { and } q{i} \neq 0 \ -\frac{A_{i j}}{q_{j}} & \text { if } i \neq j \ 0 & \text { otherwise } \end{array}\right.$$

## 数据科学代写|复杂网络代写Complex Network代考|Classical Random Graphs

Here we give an insight into two basic models of equilibrium random networks. Often they are both called the Erdős-Rényi random graph, although, strictly speaking, this name is only for the second model.
Let us consider a set of $N$ vertices, each two connected by an edge with probability $p$. This is the $G(N, p)$ model, which is also called the Bernoulli, binomial, Poisson random graph, the Gilbert model, or the Erdős-Rényi random graph. This is a statistical ensemble containing
$$Z_{N}=2^{N(N-1) / 2}$$
labelled simple graphs of $N$ vertices, in which each of these graphs, $G$, has the probability of realization determined by the number of edges $E$ in this graph,
$$\mathcal{P}(G)=p^{E}(1-p)^{N(N-1) / 2-E}, \quad \sum_{G} \mathcal{P}(G)=1 .$$
Since each edge of a vertex is present with probability $p$, the degree distribution equals
$$P(q)=\left(\begin{array}{c} N-1 \ q \end{array}\right) p^{q}(1-p)^{N-1-q},$$
and so the average degree of a vertex is
$$\langle q\rangle=p(N-1) .$$
In the limit of an infinite sparse network, $N \rightarrow \infty,\langle q\rangle \rightarrow$ const $<\infty$, this binomial distribution approaches the Poisson
$$P(q)=e^{-\langle q\rangle} \frac{\langle q\rangle^{q}}{q !} .$$
Clearly, these networks are uncorrelated in the sense that their vertices ‘know’ nothing about the properties of even their nearest neighbours, let alone other vertices.

## 数据科学代写|复杂网络代写Complex Network代考|Spectra of normalized Laplacian matrices

$$\mathcal{L} \equiv D^{-1 / 2} L D^{-1 / 2}=I-D^{-1 / 2} A D^{-1 / 2}$$

$\$ \$$\backslash mathcal {L}{ ii }=\mid left { 1 if i=j and q i \neq 0-\frac{A_{i j}}{\sqrt{q i \eta_{j}}} \quad if i \neq j 0 \quad otherwise. \正确的。 \ \$$
${ }^{36}$ 末成年人 $M_{i j}$ 矩阵的 $L$ 是后一个矩阵与行 $i$ 和列j删除。 ${ }^{37}$ 有关加权图中的拉普拉斯矩阵，请参见 Butler 和 Chung (2006)。

$$\widetilde{\mathcal{L}} \equiv L D^{-1}=I-A D^{-1},$$

$$\mathcal{P}{i j}=\frac{A i j}{q{j}} .$$

$\$ \$$\backslash widetilde {\backslash mathcal {\mathrm{L}}}{\mathrm{ij}}=\backslash left { 1 if i=j and q i \neq 0-\frac{A_{i j}}{q_{j}} if i \neq j 0 otherwise 【正确的。 \ \$$

## 数据科学代写|复杂网络代写Complex Network代考|Classical Random Graphs

$$Z_{N}=2^{N(N-1) / 2}$$

$$\mathcal{P}(G)=p^{E}(1-p)^{N(N-1) / 2-E}, \quad \sum_{G} \mathcal{P}(G)=1 .$$

$$P(q)=(N-1 q) p^{q}(1-p)^{N-1-q},$$

$$\langle q\rangle=p(N-1) .$$

$$P(q)=e^{-\langle q\rangle} \frac{\langle q\rangle^{q}}{q !} .$$

## 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。