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# CS代写|计算机网络代写Computer Networking代考|CS589 Assessing SSC

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## CS代写|计算机网络代写Computer Networking代考|Estimating $\rho\left(V_q\right)$

To compute $\rho\left(V_q\right)$, we need to compute $\tilde{h}\left(v_i, V_q \backslash\left{v_i\right}\right)$ for all $v_i \in V_q$. However, for large-scale graphs, $V_q$ may also have a large size, posing a challenge for the efficient computation of $\rho\left(V_q\right)$. Although these $\tilde{h}$ ‘s are dependent on each other, they form a finite population. We can still use sampling techniques to efficiently estimate $\rho\left(V_q\right)$ by applying Hoeffding’s inequality for finite populations (Hoeffding, 1963). Specifically, we randomly select $c^{\prime}$ nodes from $V_q$, denoted by $v_1, \ldots, v_c$, to estimate their DHTs to the remaining nodes and take the average $\overline{\rho\left(V_q\right)}$ as an estimate for $\rho\left(V_q\right)$. Here, we can use either Iterative-alg or Sampling-alg for estimating each $\tilde{h}\left(v_i, V_q \backslash\left{v_i\right)\right)$. If Iterative-alg is used, from Theorem 2.1, we obtain bounds for each $\tilde{h}\left(v_i, V_q \backslash\left(v_i\right)\right)$ in the sample set. Aggregating those bounds, we can get bounds for $\overline{\rho\left(V_q\right)}$.
Following the same manner for the proof of Theorem $2.3$ and applying Hoeffding’s inequality for finite populations (Hoeffding, 1963), we can obtain the lower bound for $c^{\prime}$ in order to obtain an $e^{\prime}$-correct answer. We omit the details due to the space limitation. When Samplingalg is used, we provide the lower bound for $c^{\prime}$ in the following theorem.

Theorem $2.4$
Suppose we randomly select the $c^{\prime}$ nodes from $V_q$ to estimate their DHTs to the remaining nodes and take the average $\overline{\rho\left(V_q\right)}$ as an estimate of $\rho\left(V_q\right)$. For the sake of clarity, let $B_i=V_q \backslash\left{v_i\right}$. Suppose, we have used Sampling-alg to obtain an $\epsilon$-correct answer for each $\tilde{h}\left(v_i, B_i\right) \quad\left(i=1, \ldots, c^{\prime}\right)$ with respect to $\left[\tilde{h}{i B_i}^{\prime}, \bar{h}{i B_i}^{\prime \prime}\right]$, then, for any $\epsilon^{\prime}>0$ and $\delta^{\prime}>0$, in order to obtain
$$\operatorname{Pr}\left(\frac{\sum_{i=1}^{c^{\prime}} \tilde{h}{i B_i}^{\prime}}{c^{\prime}}-\epsilon-\epsilon^{\prime} \leq \rho\left(V_q\right) \leq \frac{\sum{i=1}^{c^{\prime}} \tilde{h}{i B_i}^{\prime \prime}}{c^{\prime}}+\epsilon+\epsilon^{\prime}\right) \geq 1-\delta^{\prime},$$ $c^{\prime}$ should satisfy $(1-\delta)^{c^{\prime}}\left(1-2 e^{-2 e^{\prime} \epsilon^n}\right) \geq 1-\delta^{\prime}$. Proof From the conditions, we have $$\operatorname{Pr}\left(\tilde{h}{i B_i}^{\prime}-\epsilon \leq \tilde{h}\left(v_i, B_i\right) \leq \tilde{h}{i B_i}^{\prime \prime}+\epsilon\right) \geq 1-\delta, \quad i=1, \ldots, c^{\prime} .$$ Notice $\overline{\rho\left(V_q\right)}=\sum{i=1}^{c^{\prime}} \tilde{h}\left(v_i, B_i\right) / c^{\prime}$. Since $\tilde{h}^{\prime}$ ‘s are estimated independently, multiplying those probability inequalities together we obtain
$$\operatorname{Pr}\left(\frac{\sum_{i=1}^{c^{\prime}} \tilde{h}{i B_i}^{\prime}}{c^{\prime}}-\epsilon \leq \overline{\rho\left(V_q\right)} \leq \frac{\sum{i=1}^{c^{\prime}} \tilde{h}_{i B_i}^{\prime \prime}}{c^{\prime}}+\epsilon\right) \geq(1-\delta)^{c^{\prime}} .$$

## CS代写|计算机网络代写Computer Networking代考|Estimating the Significance of

After obtaining the estimate of $\rho\left(V_q\right)$, we need to measure the deviation of $\rho\left(V_q\right)$ from the expected $\rho$ value of $\widehat{V}m$ (i.e., a set of randomly selected $m$ nodes from the graph), in order to distinguish SSC from random results. In particular, we have $$E\left[\rho\left(\hat{V}_m\right)\right]=\frac{\sum{V_m \subseteq V} \rho\left(V_m\right)}{C_n^m},$$
where $V_m$ is any set of $m$ nodes. The ideal solution is to obtain the distribution of $\rho\left(\hat{V}_m\right)$ and use the ratio between the number of node sets with size $m$ whose $\rho$ values are greater than or equal to $\rho\left(V_q\right)$ and $C_n^m$ as the significance score for $q$. However, for a large-scale graph, it is very hard to get the distribution since $C_n^m$ is very large. Here, we propose an approximation method. Notice $\rho\left(\hat{V}_m\right)$ is defined as the average of $\tilde{h}\left(v_i, \hat{V}_m \backslash\left{v_i\right}\right)$ where $v_i \in \hat{V}_m$. If we assume these $\tilde{h}$ ‘s are independent, according to Central Limit Theorem, $\rho\left(\hat{V}_m\right)$ can be approximated by a normal distribution, where $\operatorname{Var}\left[\rho\left(\hat{V}_m\right)\right]=\operatorname{Var}\left[\tilde{h}\left(v_i, \hat{V}_m \backslash\left{v_i\right}\right)\right] / m$. If we obtain $E\left[\rho\left(\hat{V}_m\right)\right]$ and $\operatorname{Var}\left[\rho\left(\hat{V}_m\right)\right]$, we can calculate the adjusted SSC $\tilde{\rho}$ for $q$ as follows:
$$\tilde{\rho}\left(V_q\right)=\frac{\rho\left(V_q\right)-E\left[\rho\left(\hat{V}_m\right)\right]}{\sqrt{\operatorname{Var}\left[\rho\left(\hat{V}_m\right)\right]}} .$$

## CS代写|计算机网络代写Computer Networking代考| estimated $\rho\left(V_q\right)$

. CS

$$\operatorname{Pr}\left(\frac{\sum_{i=1}^{c^{\prime}} \tilde{h}{i B_i}^{\prime}}{c^{\prime}}-\epsilon-\epsilon^{\prime} \leq \rho\left(V_q\right) \leq \frac{\sum{i=1}^{c^{\prime}} \tilde{h}{i B_i}^{\prime \prime}}{c^{\prime}}+\epsilon+\epsilon^{\prime}\right) \geq 1-\delta^{\prime},$$$c^{\prime}$应该满足$(1-\delta)^{c^{\prime}}\left(1-2 e^{-2 e^{\prime} \epsilon^n}\right) \geq 1-\delta^{\prime}$。从条件来看，我们有$$\operatorname{Pr}\left(\tilde{h}{i B_i}^{\prime}-\epsilon \leq \tilde{h}\left(v_i, B_i\right) \leq \tilde{h}{i B_i}^{\prime \prime}+\epsilon\right) \geq 1-\delta, \quad i=1, \ldots, c^{\prime} .$$通知$\overline{\rho\left(V_q\right)}=\sum{i=1}^{c^{\prime}} \tilde{h}\left(v_i, B_i\right) / c^{\prime}$。由于$\tilde{h}^{\prime}$是独立估计的，将这些概率不等式相乘，我们得到
$$\operatorname{Pr}\left(\frac{\sum_{i=1}^{c^{\prime}} \tilde{h}{i B_i}^{\prime}}{c^{\prime}}-\epsilon \leq \overline{\rho\left(V_q\right)} \leq \frac{\sum{i=1}^{c^{\prime}} \tilde{h}_{i B_i}^{\prime \prime}}{c^{\prime}}+\epsilon\right) \geq(1-\delta)^{c^{\prime}} .$$

## CS代写|计算机网络代写Computer Networking代考|估算

. CS的重要性

，其中$V_m$是$m$节点的任意集合。理想的解决方案是获得$\rho\left(\hat{V}_m\right)$的分布，并使用规模为$m$的节点集的数量之间的比率(其$\rho$值大于或等于$\rho\left(V_q\right)$和$C_n^m$)作为$q$的显著性评分。但是对于一个大规模的图来说，$C_n^m$非常大，很难得到分布。在这里，我们提出了一种近似方法。注意$\rho\left(\hat{V}_m\right)$被定义为$\tilde{h}\left(v_i, \hat{V}_m \backslash\left{v_i\right}\right)$的平均值，其中$v_i \in \hat{V}_m$。如果我们假设这些$\tilde{h}$是独立的，根据中心极限定理，$\rho\left(\hat{V}_m\right)$可以近似为正态分布，其中$\operatorname{Var}\left[\rho\left(\hat{V}_m\right)\right]=\operatorname{Var}\left[\tilde{h}\left(v_i, \hat{V}_m \backslash\left{v_i\right}\right)\right] / m$。如果我们得到$E\left[\rho\left(\hat{V}_m\right)\right]$和$\operatorname{Var}\left[\rho\left(\hat{V}_m\right)\right]$，我们可以计算$q$调整后的SSC $\tilde{\rho}$:
$$\tilde{\rho}\left(V_q\right)=\frac{\rho\left(V_q\right)-E\left[\rho\left(\hat{V}_m\right)\right]}{\sqrt{\operatorname{Var}\left[\rho\left(\hat{V}_m\right)\right]}} .$$

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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。