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## 数据科学代写|复杂网络代写Complex Network代考|Representations of graphs

By definition, a graph is represented by a set of pairs of labels; each pair contains two labels of the end vertices of the corresponding edge. For a directed edge, this pair is ordered. There are two other convenient ways to mathematically represent a graph: (i) an incidence matrix indicating the pairs: an edge – a vertex incident to it, and (ii) an adjacency matrix indicating the pairs of adjacent vertices.

The incidence matrix $B$ of a graph with $N$ vertices and $E$ edges has $N$ rows and $E$ columns. For an oriented graph, ${ }^{11}$ this $N \times E$ matrix has the elements: ${ }^{12}$

• $B_{i j}=-1$ if vertex $i$ is the source vertex of edge $j$;
• $B_{i j}=+1$ if vertex $i$ is the target vertex of edge $j$;
• $B_{i j}=2$ if edge $j$ is a self-loop at vertex $i$;
• the remaining entries are zero.
So, for a simple oriented graph, the sum of elements in each column of this matrix is zero, $\sum_{i} B_{i j}=0$ (an edge has one source and one target).

The adjacency matrix of an undirected graph of $N$ vertices, $A(G)$, can be constructed in the following way. Start with the $N \times N$ matrix having all its entries zeroes. For each edge between vertices $i$ and $j$ in the graph, add 1 to the entries $A_{i j}$ and $A_{j i}$ of the matrix, and for each self-loop at vertex $i$ add 2 to the entry $A_{i i}$ of the matrix. The resulting adjacency matrix of an undirected graph of $N$ vertices is a $N \times N$ symmetric matrix with the entries:

• $A_{i j}=n$ if there are $n$ edges interconnecting vertices $i$ and $j, i \neq j$
• $A_{i i}=2 m$ if vertex $i$ has $m$ self-loops;
• the remaining entries are zero.
In particular, in the adjacency matrix of a simple undirected graph, $A_{i j}=1$ if there is an edge between vertices $i$ and $j, A_{i j}=0$ if such edge is absent, and the diagonal elements are zero, $A_{i i}=0$.

The adjacency matrix of an oriented graph of $N$ vertices can be introduced in a similar way. Start with the $N \times N$ matrix having all its entries zeroes. For each edge going from vertex $i$ to vertex $j$ in the graph, add 1 to the entry $A_{i j}$ of the matrix, and for each self-loop at vertex $i$ add 1 to the entry $A_{i i}$ of the matrix. ${ }^{14}$ The resulting adjacency matrix of an oriented graph of $N$ vertices is a $N \times N$ matrix with the entries:

• $A_{i j}=n$ if there are $n$ edges going from vertex $i$ to vertex $j, i \neq j$;
• $A_{i i}=m$ if vertex $i$ has $m$ directed self-loops;
• the remaining entries are zero.

## 数据科学代写|复杂网络代写Complex Network代考|Representations of graphs

$B_{i j}=-1$ 如果顶点 $i$ 是边的源页点 $j$;

$B_{i j}=+1$ 如果顶点 $i$ 是边的目标顶点 $j$;

$B_{i j}=2$ 如果边豚 $j$ 是顶点处的自环 $i$;

$A_{i j}=n$ 如果有 $n$ 连接顶点的边 $i$ 和 $j, i \neq j$

$A_{i i}=2 m$ 如果顶点 $i$ 有 $m$ 自循环;

$A_{i j}=n$ 如果有 $n$ 从顶点开始的边 $i$ 到顶点 $j, i \neq j ;$

$A_{i i}=m$ 如果顶点 $i$ 有 $m$ 有向自环;

## 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。