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# 数据科学代写|复杂网络代写Complex Network代考|CS60078

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## 数据科学代写|复杂网络代写Complex Network代考|A New Error Function

We already said that we would like to use a statistical mechanics approach. The problem of finding a block structure which reflects the network as good as possible is then mapped onto finding the solution of a combinatorial optimization problem. Trying to approximate the adjacency matrix $\mathbf{A}$ of rank $r$ by a matrix $\mathbf{B}$ of rank $q<r$ means approximating $\mathbf{A}$ with a block model of only full and zero blocks. Formally, we can write this as $\mathbf{B}_{i j}=B\left(\sigma_i, \sigma_j\right)$ where $B(r, s)$ is a ${0,1}^{q \times q}$ matrix and $\sigma_i \in{1, \ldots, q}$ is the assignment of node $i$ from A into one of the $q$ blocks. We can view $B(r, s)$ as the adjacency matrix of the blocks in the network or as the image graph discussed in the previous chapter and its nodes represent the different equivalence classes into which the vertices of $\mathbf{A}$ may be grouped. From Table 3.1, we see that our error function can have only four different contributions. They should

reward the matching of edges in $\mathbf{A}$ to edges in $\mathbf{B}$,

penalize the matching of missing edges (non-links) in $\mathbf{A}$ to edges in $\mathbf{B}$,

penalize the matching of edges in $\mathbf{A}$ to missing edges in $\mathbf{B}$ and

reward the matching of missing edges in $\mathbf{A}$ to edges in $\mathbf{B}$

## 数据科学代写|复杂网络代写Complex Network代考|Fitting Networks to Image Graphs

The above-defined quality and error functions in principle consist of two parts. On one hand, there is the image graph $\mathbf{B}$ and on the other hand, there is the mapping of nodes of the network to nodes in the image graph, i.e., the assignment of nodes into blocks, which both determine the fit. Given a network $\mathbf{A}$ and an image graph $\mathbf{B}$, we could now proceed to optimize the assignment of nodes into groups ${\sigma}$ as to optimize (3.6) or any of the derived forms. This would correspond to “fitting” the network to the given image graph. This allows us to compare how well a particular network may be represented by a given image graph. We will see later that the search for cohesive subgroups is exactly of this type of analysis: If our image graph is made of isolated vertices which only connect to themselves, then we are searching for an assignment of nodes into groups such that nodes in the same group are as densely connected as possible and nodes in different groups as sparsely as possible. However, ultimately, we are interested also in the image graph which best fits to the network among all possible image graphs B. In principle, we could try out every possible image graph, optimize the assignment of nodes into blocks ${\sigma}$ and compare these fit scores. This quickly becomes impractical for even moderately large image graphs. In order to solve this problem, it is useful to consider the properties of the optimally fitting image graph $\mathbf{B}$ if we are given the networks plus the assignment of nodes into groups ${\sigma}$.

We have already seen that the two terms of (3.7) are extremized by the same $B\left(\sigma_i, \sigma_j\right)$. It is instructive to introduce the abbreviations
\begin{aligned} m_{r s} & =\sum_{i j} w_{i j} A_{i j} \delta\left(\sigma_i, r\right) \delta\left(\sigma_j, s\right) \text { and } \ {\left[m_{r s}\right]{p{i j}} } & =\sum_{i j} p_{i j} \delta\left(\sigma_i, r\right) \delta\left(\sigma_j, s\right), \end{aligned}
and write two equivalent formulations for our quality function:
\begin{aligned} & Q^1({\sigma}, \mathbf{B})=\sum_{r, s}\left(m_{r s}-\gamma\left[m_{r s}\right]{p{i j}}\right) B(r, s) \text { and } \ & Q^0({\sigma}, \mathbf{B})=-\sum_{r, s}\left(m_{r s}-\gamma\left[m_{r s}\right]{p{i j}}\right)(1-B(r, s)) . \end{aligned}

## 数据科学代写|复杂网络代写Complex Network代考|Fitting Networks to Image Graphs

\begin{aligned} m_{r s} & =\sum_{i j} w_{i j} A_{i j} \delta\left(\sigma_i, r\right) \delta\left(\sigma_j, s\right) \text { and } \ {\left[m_{r s}\right]{p{i j}} } & =\sum_{i j} p_{i j} \delta\left(\sigma_i, r\right) \delta\left(\sigma_j, s\right), \end{aligned}

\begin{aligned} & Q^1({\sigma}, \mathbf{B})=\sum_{r, s}\left(m_{r s}-\gamma\left[m_{r s}\right]{p{i j}}\right) B(r, s) \text { and } \ & Q^0({\sigma}, \mathbf{B})=-\sum_{r, s}\left(m_{r s}-\gamma\left[m_{r s}\right]{p{i j}}\right)(1-B(r, s)) . \end{aligned}

## 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。