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# 统计代写|统计计算代写STATISTICAL COMPUTING代写|STAT250 Network Representation

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## 统计代写|统计计算代写STATISTICAL COMPUTING代写|Network Representation

In investigating network problems, the method for expressing network structure is important. Usually, the adjacency matrix 25 is employed. An adjacency matrix is a square matrix used to represent a finite graph. The elements of the matrix indicate whether pairs of vertices (arcs) are adjacent or not in the graph. Figure 1 gives an example of node-oriented AM expression, where the entry in the matrix indicates the number of links between ‘from’ node and ‘to’ node. Such entry implicitly indicates the direction of the links between two nodes. Thus, multiple same direction links between two nodes are allowed in such AM expression. Figure 2 gives another arc-oriented AM expression. However, only 1 or 0 are allowed in this expression. Since both nodes-oriented and arc-oriented AM are equivalent in nature, we will take node-oriented AM in the following discussion.

A network is a graph except that the network has source nodes and sink nodes. Therefore, using AM to solve network problems normally faces difficulties in handling source nodes and sink nodes. Furthermore, in the view points of computer science, AM is not the best choice for program implementation. Specially, in solving network reliability, the linked path structure (LPS) [10] is now more and more popular in the literature. A LPS is a hybrid form of node-oriented and arc-oriented representation of a network. In LPS, entry 0 indicates the source nodes, and the negative entry values denote the sink nodes. The other entry values in LPS are vectors representing nodes with outbound arcs as their values in the vector. The index of LPS is the arc no. pointing the node (i.e. the entry value). Therefore, a LPS gives all the necessary information for networks in the applications. Figure 3 shows a LPS example. This LPS has a source node ${1,2}$ with outbound arc 1 and 2 . Arc 1 is connected with node ${3,5}$ with outbound arc 3 and 5 . Arc 2 is connected with node ${4,6}$ with outbound $\operatorname{arc} 4$ and 6 . Arc 3 is connected with node ${4,6}$ with outbound $\operatorname{arc} 4$ and 6 . Arc 4 is connected with node ${3,5}$ with outbound arc 3 and 5 . Arc 5 is connected with sink node ${-1}$. Arc 6 is connected with sink node ${-1}$. Thus, a tool for transforming AM to LPS is required.

## 统计代写|统计计算代写STATISTICAL COMPUTING代写|The Proposed Approach

Given a demand $d$, the reliability denoted by $\omega_{d}$ is the probability at sink node that the maximal flow in the network is no less than $d$, i.e., $\omega_{d} \equiv \operatorname{Pr}\left{X \mid \varpi_{X} \geq d\right}$. To calculate $\omega_{d}$, find the lower boundary vectors directly in the set $\left{X \mid \varpi_{X} \geq d\right}$. A lower boundary vector $X$ is said to be a $d$-MP for $d$ if and only if (i) $\varpi_{X} \geq d$ and (ii) $\varpi_{W}<d$ for any other vector $W$ such that $W<X$, in which $W \leq X$ if and only if $w_{j} \leq x_{j}$ for each $j=1,2, \ldots, n$ and $W<X$ if and only if $W \leq X$ and $w_{j}<x_{j}$

for at least one $j$. Suppose there are totally $q d$-MPs for $d: X_{1}, X_{2}, \ldots, X_{q}$. The reliability is equal to
$$\omega_{d}=\operatorname{Pr}\left{\bigcup_{k=1}^{q}\left{X \mid X \geq X_{k}\right}\right}$$
which can be calculated by reduced recursive inclusion-exclusion principle (RRIEP) [5].

Let $L$ be a LPS for the network. Then, we have the following properties.
Property $1 \exists l_{k} \in L$, then, $l_{0}$ is the set of source nodes.
Property $2 \exists l_{k} \in L$ and $k>0$, then, $l_{k}$ is a vector of arcs outbound from the node connected by $\operatorname{arc} k$.

Property $3 \exists l_{k} \in L$ and $k>0$, if $l_{k}$ is a vector of one negative integer, then, arc $k$ connected with a sink node.

## 统计代写|统计计算代写STATISTICAL COMPUTING代写IThe Proposed Approach

\left 的分隔符缺失或无法识别 计算 $\omega_{d}$, 直接在集合中找到下边界向量
\left 的分隔符缺失或无法识别 下边界向量 $X$ 据说是一个 $d-\mathrm{MP}$ 为 $d$ 当且

## 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。