### 统计代写|复杂网络代写complex networks代考| A First Principles Approach to Block

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。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等楖率论
• 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

## 统计代写|复杂网络代写complex networks代考|Mapping the Problem

Common to all of the before-mentioned approaches is their attempt to discover patterns in the link structure of networks. Patterns were either block structures in the adjacency matrix or-more specifically-cohesive subgroups. We will try to define a quality function for block structure in networks and optimize the ordering of rows and columns of the matrix as to maximize the quality of the blocking. The search for cohesive subgroups will prove to be a special case of this treatment. It makes sense to require that our quality function will be independent of the order of rows and columns within one block. It will depend only on the assignment of nodes, i.e., rows and columns, into blocks. Finding a good assignment into blocks is hence a combinatorial optimization problem. In many cases, it is possible to map such a combinatorial optimization problem onto minimizing the energy of a spin system [1]. This approach had been suggested for the first time by Fu and Anderson in 1986 [2] in the context of bi-partitioning of graphs and it has been applied successfully to other problems such as vertex cover [3], k-sat [4], the traveling salesmen [5] and many others as well.

Before introducing such a quality function, it is instructive to leave the field of networks for a moment and take a detour into the dimensionality reduction of multivariate data.

## 统计代写|复杂网络代写complex networks代考|Dimensionality Reduction with Minimal Squared Error

Suppose we are given a set of real valued measurements of some objects. As an example, for all boats in a marina, we measure length over all, width, height of the mast, the area of the sail, power of the engine, length of the waterline, and so forth. Let $N$ be the number of measurements, i.e., the number of boats in the marina, and let the measurements be vectors of dimension $d$, i.e., the number of things we have measured. We compile our measurements into a

data matrix $\mathbf{A} \in \mathbb{R}^{N \times d}$, i.e., we write the individual measurement vectors as the rows of matrix A. Let us further assume that we have already subtracted the mean across all measurements from each individual sample such that the columns of A sum to zero, i.e., we have centered our data.

Now we see that $\mathbf{A}^{T} \mathbf{A}$ is a $d \times d$ matrix describing the covariance of the individual dimensions in which we measured our data.

We now ask if we can drop some of the $d$ dimensions and still describe our data well. Naturally, we want to drop those dimensions in which our data do not vary much or we would like to replace two dimensions which are correlated by a single dimension. We can discard the unnecessary dimensions by projecting our data from the $d$-dimensional original space in a lower dimensional space of dimension $q<d$. Such a projection can be achieved by a matrix $\mathbf{V} \in \mathbb{R}^{d \times q}$. Taking measurement $\mathbf{a}{\mathbf{i}} \in \mathbb{R}^{d}$ from row $i$ of $\mathbf{A}$. we find the coordinates in the new space to be $\mathbf{b}{\mathrm{i}}=\mathbf{a}{\mathrm{i}} \mathbf{V}$ with $\mathbf{b}{\mathrm{i}} \in \mathbb{R}^{q}$. We can also use the transpose of $\mathbf{V}$ to project back into the original space of dimension $d$ via $\mathbf{a}{\mathrm{i}}^{\prime}=\mathbf{b}{\mathrm{i}} \mathbf{V}^{T}$. Since in the two projections we have visited a lower dimensional space, we find that generally the reconstructed data point does not coincide with the original datum $\mathbf{a}{\mathbf{i}} \mathbf{V V}^{T}=\mathbf{a}{\mathbf{i}}^{\prime} \neq \mathbf{a}_{\mathbf{i}}$.

However, if we would have first started in the $q$-dimensional space with $\mathbf{b}{\mathrm{i}}$ and projected it into the $d$-dimensional space via $\mathbf{V}^{T}$ and then back again via $V$ we require that our projection does not lose any information and hence $\mathbf{b}{\mathbf{i}} \mathbf{V}^{T} \mathbf{V}=\mathbf{b}_{\mathbf{i}}$. This means that we require $\mathbf{V}^{T} \mathbf{V}=\mathbb{1}$ or in other words we require that our projection matrix $V$ be unitary.

The natural question is now how to find a unitary matrix such that it minimizes some kind of reconstruction error. Using the mean square error, we could write
\begin{aligned} E \propto \sum_{i}^{N} \sum_{j}^{d}\left(\mathbf{A}-\mathbf{A}^{\prime}\right){i j}^{2} &=\sum{i}^{N} \sum_{j}^{d}\left(\mathbf{A}-\mathbf{A V V ^ { \mathbf { T } } ) _ { i j } ^ { 2 }}\right.\ &=\operatorname{Tr}\left(\mathbf{A}-\mathbf{A V V ^ { T }}\right)^{T}\left(\mathbf{A}-\mathbf{A V V ^ { T }}\right) \end{aligned}
The new coordinates that we project our data onto are called “principal components” of the data set and the technique of finding them is known as “principal component analysis” or PCA for short. Already at this point, we can mention that the $q$ columns of $V$ must be made of the eigenvectors belonging to the largest $q$ eigenvalues of $\mathbf{A}^{T} \mathbf{A}$. To show this, we discuss a slightly different problem, solve it and then show that it is equivalent to the above.
Consider the singular value decomposition (SVD) of a matrix of $\mathbf{A} \in \mathbb{R}^{N \times d}$ into a unitary matrix $\mathbf{U} \in \mathbb{R}^{N \times N}$, a diagonal matrix $\mathbf{S} \in \mathbb{R}^{N \times d}$ (in case $N \neq d$ there are maximally $\min (N, d)$ non-zero entries, the number of nonzero entries in $\mathbf{S}$ is the rank of $\mathbf{A})$ and another unitary matrix $\mathbf{V} \in \mathbb{R}^{d \times d}$ such that $\mathbf{A}=\mathbf{U S V}^{T}$ and $\mathbf{S}=\mathbf{U}^{T} \mathbf{A V}$. The entries on the diagonal of $\mathbf{S}$ are called singular values. We will assume that they are ordered decreasing in absolute value. It is straightforward to see some of the properties of this $\mathbf{S V D}: \mathbf{U}^{T} \mathbf{A}=\mathbf{S V}^{T}$ and $\mathbf{A V}=\mathbf{U S}$ follow from the $\mathbf{U}$ and $\mathbf{V}$ being unitary.

## 统计代写|复杂网络代写complex networks代考|Squared Error for Multivariate Data and Networks

Let us consider in the following the reconstruction of the adjacency matrix of a network $\mathbf{A} \in{0,1}^{N \times N}$ of rank $r$ by another adjacency matrix $\mathbf{B} \in{0,1}^{N \times N}$ possibly of lower rank $q<r$ as before. For the squared error we have
$$E=\sum_{i j}(\mathbf{A}-\mathbf{B})_{i j}^{2}$$
Then, there are only four different cases we need to consider in Table 3.1. The squared error gives equal value to the mismatch on the edges and missing edges in A. We could say it weighs every error by its own magnitude. While this is a perfectly legitimate approach for multivariate data, it is, however, highly problematic for networks. The first reason is that many networks are sparse. The fraction of non-zero entries in $\mathbf{A}$ is generally very, very small compared to the fraction of zero entries. A low rank approximation under the squared error will retain this sparsity to the point that $\mathrm{B}$ may be completely zero. Furthermore, we have seen that real networks tend to have a very heterogeneous degree distribution, i.e., the distribution of zeros and ones per row and column in $\mathbf{A}$ is also very heterogeneous. Why give every entry the same weight in the error function? Most importantly, for multivariate data, all entries of $\mathbf{A}_{i j}$ are equally important measurements in principle. For networks this is not the case: the edges are in principle more important than the missing edges. There are fewer of them and they should hence be given more importance than missing edges. Taken all of these arguments together, we see that our first goal will have to be the derivation of an error function specifically tailored for networks that does not suffer from these deficiencies.

## 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。