# 数学代写|数学建模代写math modelling代考|Communication Networks

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## 数学代写|数学建模代写math modelling代考|Communication Networks

A directed graph can serve as a model for a communication network. Thus, consider the network given in Figure 7.10. If an edge is directed from $a$ to $b$, it means that $a$ can communicate with $b$. In the given network $e$ can communicate directly with $b$, but $b$ can communicate with $e$ only indirectly through $c$ and $d$. However every individual can communicate with every other individual.

Our problem is to determine the importance of each individual in this network. The importance can be measured by the fraction of the messages on average that pass through him. In the absence of any other knowledge, we can assume that if an individual can send a message direct to $n$ individuals, he will send a message to any one of them with probability $1 / n$. In the present example, the communication probability matrix is:

No individual is to send a message to himself and so all diagonal elements are zero. Since all elements of the matrix are nonnegative and the sum of elements of every row is unity, the matrix is a stochastic matrix and one of its eigenvalues is unity. The corresponding normalized eigenvector is [11/45, 13/45, 3/10, 1/10, 1/15]. In the long run, these fractions of messages will pass through $a$, $b, c, d, e$ respectively. Thus we can conclude that in this network, $c$ is the most important person.
If in a network, an individual cannot communicate with every other individual either directly or indirectly, the Markov chain is not ergodic and the process of finding the importance of each individual breaks down.

## 数学代写|数学建模代写math modelling代考|Matrices Associated with a Directed Graph

For a directed graph with $n$ vertices, we define the $n \times n$ matrix $A=\left(a_{i j}\right)$ by $a_{i j}=1$ if there is an edge directed from $i$ and $j$ and $a_{i j}=0$ if there is no edge directed from $i$ to $j$. Thus the matrix associated with the graph of Figure 7.11 is given by

We note that (i) the diagonal elements of the matrix are all zero, (ii) the number of nonzero elements is equal to the number of edges, (iii) the number of nonzero elements in any row is equal to the local outward degree of the vertex corresponding to the row, and $(i v)$ the number of nonzero elements in a column is equal to the local inward degree of the vertex corresponding to the column. The element $a_{i j}^{(2)}$ gives the number of 2 -chains from $i$ to $j$. Thus, from vertex 2 to vertex 1 , there are two 2 -chains viz. via vertex 3 and vertex 4 . We can generalize this result in the form of a theorem viz. “The element $a_{i j}^{(2)}$ of $A^2$ gives the number of 2 -chains, i.e., the number of paths with two edges from vertex $i$ to vertex $j$.”

The theorem can be further generalized to “The element $a_{i j}^{(m)}$ of $A^m$ gives the number of $m$-chains, i.e., the number of paths with $m$ edges from vertex $i$ to vertex $j$.” It is also easily seen that “The $i$ th diagonal element of $A^2$ gives the number of vertices with which $i$ has a symmetric relationship.”

From matrix $A$ of a graph, a symmetric matrix $S$ can be generated by taking the elementwise product of $\mathrm{A}$ with its transpose so that in our case
$$S=A \times A^T=\left[\begin{array}{llll} 0 & 1 & 1 & 0 \ 1 & 0 & 1 & 0 \ 1 & 1 & 0 & 0 \ 1 & 0 & 1 & 0 \end{array}\right] \times\left[\begin{array}{llll} 0 & 1 & 1 & 1 \ 1 & 0 & 1 & 0 \ 1 & 1 & 0 & 1 \ 0 & 0 & 0 & 0 \end{array}\right]=\left[\begin{array}{llll} 0 & 1 & 1 & 0 \ & 0 & 1 & 0 \ 1 & 1 & 0 & 0 \ 0 & 0 & 0 & 0 \end{array}\right]$$
$S$ obviously is the matrix of the graph from which all unreciprocated connections have been eliminated. In the matrix $S$ (as well as in $S^{2, s_3}$ the elements in the row and column corresponding to a vertex which has no symmetric relation with any other vertex are all zero.

# 数学建模代写

## 数学代写|数学建模代写math modelling代考|Matrices Associated with a Directed Graph

$$S=A \times A^T=\left[\begin{array}{llll} 0 & 1 & 1 & 0 \ 1 & 0 & 1 & 0 \ 1 & 1 & 0 & 0 \ 1 & 0 & 1 & 0 \end{array}\right] \times\left[\begin{array}{llll} 0 & 1 & 1 & 1 \ 1 & 0 & 1 & 0 \ 1 & 1 & 0 & 1 \ 0 & 0 & 0 & 0 \end{array}\right]=\left[\begin{array}{llll} 0 & 1 & 1 & 0 \ & 0 & 1 & 0 \ 1 & 1 & 0 & 0 \ 0 & 0 & 0 & 0 \end{array}\right]$$
$S$显然是图的矩阵，从中消除了所有的非往复连接。在矩阵$S$(以及$S^{2, s_3}$)中，与任何其他顶点没有对称关系的顶点对应的行和列中的元素都是零。

## 有限元方法代写

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## 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。

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