### 统计代写|复杂网络代写complex networks代考| Maximum Value of the Fit Score

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
• (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|Maximum Value of the Fit Score

The function (3.15) is monotonously increasing with the number of possible roles $q$ until it reaches its maximum value $Q_{\max }$
$$\mathrm{Q}{\max }=\sum{i j}\left(w_{i j} A_{i j}-\gamma p_{i j}\right) A_{i j} .$$
This value can be achieved when $q$ equals the number of structural equivalence classes in the network, i.e., the number of rows/columns which are genuine in A. The optimal assignment of roles ${\sigma}$ is then simply an assignment into the structural equivalence classes. For fewer roles, this allows us to compare $Q / Q_{\max }$ for the actual data and a randomized version and to use this comparison as a basis for the selection of the optimal number of roles in the image graph in order to avoid overfitting the data.A comparison of the image graphs and role assignments found independently for different numbers of roles may also allow for the detection of possible hierarchical or overlapping organization of the role structure in the network.

## 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
$$p_{i j}=p=\frac{\sum_{k l} w_{k l} A_{k l}}{N^{2}}$$
$$\left[m_{r s}\right]{p}=p n{r} n_{s},$$
with $n_{r}$ and $n_{s}$ denoting the number of nodes in group $r$ and $s$, respectively.
A second choice for $p_{i j}$ may take into account that the network does exhibit a particular degree distribution. Since links are in principle more likely between nodes of high degree, matching links between high-degree nodes should get a lower reward and mismatching them a higher penalty. One may write
$$p_{i j}=\frac{\left(\sum_{k} w_{i k} A_{i k}\right)\left(\sum_{l} w_{l j} A_{l j}\right)}{\sum_{k l} w_{k l} A_{k l}}=\frac{k_{i}^{\text {out }} k_{j}^{i n}}{M},$$
which takes this fact and the degree distribution into account. In this form, the penalty $p_{i j}$ is proportional to the product of the row and column sums of the weight matrix w. The number (weight) of outgoing links of node $i$ is given by $k_{i}^{\text {out }}$ and the number (weight) of incoming links of node $j$ is given by $k_{j}^{i n}$. With these expressions one can write
$$\left[m_{r s}\right]{p{i j}}=\frac{1}{M} K_{r}^{o u t} K_{s}^{\text {in }}$$

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
$$a_{\mathrm{Ts}}=m_{r s}-\gamma\left[m_{r s}\right]{p{i j}}$$
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 $Q=\frac{1}{2} \sum_{r s}\left|a_{r s}\right|$ 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 $p_{i j}=\frac{k_{i}^{\text {out }} k_{j}^{i n}}{M}$ one finds
$$\sum_{r s} a_{T s}=\sum_{r} a_{T s}=\sum_{s} a_{T s}=0 .$$
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).

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

p一世j=p=∑ķl在ķl一种ķlñ2

[米rs]p=pnrns,

p一世j=(∑ķ在一世ķ一种一世ķ)(∑l在lj一种lj)∑ķl在ķl一种ķl=ķ一世出去 ķj一世n米,

[米rs]p一世j=1米ķr这在吨ķs在

$$a_{\mathrm{Ts}}=m_{rs}-\gamma\left[m_{rs}\right] {p {ij}} F这rr=s,在和C一种llCss=一种ss吨H和C这和FF一世C一世和n吨这F“C这H和s一世这n”.吨在这Gr这在ps这Fn这d和sH一种在和一种p这s一世吨一世在和C这和FF一世C一世和n吨这F一种dH和s一世这n,一世F吨H和是一种r和C这nn和C吨和db是和dG和sb和一种r一世nG米这r和在和一世GH吨吨H一种n和Xp和C吨和dFr这米p一世j.在和H和nC和C一种ll一种Gr这在pC这H和s一世在和,一世F一世吨sn这d和s一种r和C这nn和C吨和db是和dG和sb和一种r一世nG米这r和在和一世GH吨吨H一种n和Xp和C吨和dFr这米p一世j.吨H一世s一种ll这在sF这r一种sH这r吨H一种ndF这r米这F(3.15)一种s问=12∑rs|一种rs|一种nd在和s和和吨H一种吨吨H和bl这Cķ米这d和l乙H一种s和n吨r一世和s这F这n和在H和r和一种rs>0.R和米和米b和r吨H一种吨一种rsd和p和nds这n吨H和Gl这b一种lp一种r一种米和吨和rC一种nd吨H和一种ss在米和dp和n一种l吨是F在nC吨一世这np一世j.F这rC=1一种nd吨H和米这d和lp一世j=ķ一世出去 ķj一世n米这n和F一世nds \sum_{rs} a_{T s}=\sum_{r} a_{T s}=\sum_{s} a_{T s}=0 。$$

## 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。