数学代写|随机图论代写Random Graph代考|CSL866

statistics-lab™ 为您的留学生涯保驾护航 在代写随机图论Random Graph方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写随机图论Random Graph代写方面经验极为丰富，各种代写随机图论Random Graph相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

数学代写|随机图论代写Random Graph代考|Generalized Random Intersection Graphs

Godehardt and Jaworski [ 389 ] introduced a model which generalizes both the binomial and uniform models of random intersection graphs. Let $P$ be a probability measure on the set ${0,1,2, \ldots, m}$. Let $V={1,2, \ldots, n}$ be the vertex set. Let $M=$ ${1,2, \ldots, m}$ be the set of attributes. Let $S_1, S_7, \ldots, S_n$ be independent random subsets of $M$ such that for any $v \in V$ and $S \subseteq M$ we have $\mathbb{P}\left(S_v=S\right)=P(|S|) /\left(\begin{array}{l}m \ |S|\end{array}\right)$. If we put an edge between any pair of vertices $i$ and $j$ when $S_i \cap S_j \neq \emptyset$, then we denote such a random intersection graph as $G(n, m, P)$, while if the edge is inserted if $\left|S_i \cap S_i\right| \geq s, s \geq 1$, the respective graph is denoted as $G_s(n, m, P)$. Bloznelis [99] extends these definitions to random intersection digraphs.
The study of the degree distribution of a typical vertex of $G(n, m, P)$ is given in [464], [242] and [97], see also [465]. Bloznelis ( see [98] and [100]) shows that the order of the largest component $L_1$ of $G(n, m, P)$ is asymptotically equal to $n \rho$, where $\rho$ denotes the non-extinction probability of a related multi-type Poisson branching process. Kurauskas and Bloznelis [541] study the asymptotic order of the clique number of the sparse random intersection graph $G_s(n, m, P)$.
Finally, a dynamic approach to random intersection graphs is studied by Barbour and Reinert [63], Bloznelis and Karoński [107], Bloznelis and Goetze [104] and Britton, Deijfen, Lageras and Lindholm [166].
One should also notice that some of the results on the connectivity of random intersection graphs can be derived from the corresponding results for random hyperghraphs, see for example [519], [707] and [390].

数学代写|随机图论代写Random Graph代考|Random Geometric Graphs

McDiarmid and Müller [588] gives the leading constant for the chromatic number when the average degree is $\Theta(\log n)$. The paper also shows a “surprising” phase change for the relation between $\chi$ and $\omega$. Also the paper extends the setting to arbitrary dimensions. Müller [618] proves a two-point concentration for the clique number and chromatic number when $n r^2=o(\log n)$.
Blackwell, Edmonson-Jones and Jordan [96] studied the spectral properties of the adjacency matrix of a random geometric graph (RGG). Rai [672] studied the spectral measure of the transition matrix of a simple random walk. Preciado and Jadbabaie [665] studied the spectrum of RGG’s in the context of the spreading of viruses.
Sharp thresholds for monotone properties of RGG’s were shown by McColm [582] in the case $d=1$ viz. a graph defined by the intersection of random subintervals. And for all $d \geq 1$ by Goel, Rai and Krishnamachari [391].
First order expressible properties of random points
$\mathscr{X}=\left{X_1, X_2, \ldots, X_n\right}$ on a unit circle were studied by McColm [581]. The graph has vertex set $\mathscr{X}$ and vertices are joined by an edge if and only if their angular distance is less than some parameter $d$. He showed among other things that for each fixed $d$, the set of a.s. FO sentences in this model is a complete noncategorical theory. McColm’s results were anticipated in a more precise paper [388] by Godehardt and Jaworski, where the case $d=1$, i.e., the evolution a random interval graph, was studied.
Diaz, Penrose, Petit and Serna [256] study the approximability of several layout problems on a family of RGG’s. The layout problems that they consider are bandwidth, minimum linear arrangement, minimum cut width, minimum sum cut, vertex separation, and edge bisection. Diaz, Grandoni and Marchetti-Spaccemela [255] derive a constant expected approximation algorithm for the $\beta$-balanced cut problem on random geometric graphs: find an edge cut of minimum size whose two sides contain at least $\beta n$ vertices each.
Bradonjić, Elsässer, Friedrich, Sauerwald and Stauffer [162] studied the broadcast time of RGG’s. They study a regime where there is likely to be a single giant component and show that w.h.p. their broadcast algorithm only requires $O\left(n^{1 / 2} / r+\log n\right)$ rounds to pass information from a single vertex, to every vertex of the giant. They show on the way that the diameter of the giant is $\Theta\left(n^{1 / 2} / r\right)$ w.h.p. Friedrich, Sauerwald and Stauffer [334] extended this to higher dimensions.

随机图论代写

数学代写|随机图论代写Random Graph代考|Generalized Random Intersection Graphs

Godehardt 和 Jaworski [389] 引入了一个模型，该模型概括了随机相交图的二项式模型和均匀模型。让P是集合上的概率测度0,1,2,…,米. 让在=1,2,…,n是顶点集。让米= 1,2,…,米是属性集。让小号1,小号7,…,小号n是独立的随机子集米这样对于任何在∈在和小号⊆米我们有P(小号在=小号)=P(|小号|)/(米 |小号|). 如果我们在任意一对顶点之间放置一条边一世和j什么时候小号一世∩小号j≠∅，那么我们将这样的随机交叉图表示为G(n,米,P), 而如果边被插入如果|小号一世∩小号一世|≥秒,秒≥1，相应的图表示为G秒(n,米,P). Bloznelis [99] 将这些定义扩展到随机相交有向图。

数学代写|随机图论代写Random Graph代考|Random Geometric Graphs

McDiarmid 和 Müller [588] 给出了当平均度数为日(日志⁡n). 该论文还显示了两者之间关系的“令人惊讶”的相变H和哦. 该论文还将设置扩展到任意维度。Müller [618] 证明了团数和色数的两点集中，当nr2=欧(日志⁡n).
Blackwell、Edmonson-Jones 和 Jordan [96] 研究了随机几何图 (RGG) 的邻接矩阵的谱特性。Rai [672] 研究了简单随机游走的转移矩阵的谱测度。Preciado 和 Jadbabaie [665] 在病毒传播的背景下研究了 RGG 的谱。
McColm [582] 在案例中展示了 RGG 的单调特性的尖锐阈值d=1即。由随机子区间的交集定义的图。对于所有人d≥1Goel、Rai 和 Krishnamachari [391]。

\mathscr{X}=\left{X_1, X_2, \ldots, X_n\right}\mathscr{X}=\left{X_1, X_2, \ldots, X_n\right}McColm [581] 在单位圆上进行了研究。该图有顶点集X当且仅当它们的角距离小于某个参数时，顶点才由边连接d. 他表明，除其他外，对于每个固定的d, 该模型中的 as FO 句子集是一个完整的非分类理论。McColm 的结果在 Godehardt 和 Jaworski 的更精确的论文 [388] 中得到了预期，其中案例d=1，即随机区间图的演化，进行了研究。
Diaz、Penrose、Petit 和 Serna [256] 研究了 RGG 族上几个布局问题的近似性。他们考虑的布局问题是带宽、最小线性排列、最小切割宽度、最小和切割、顶点分离和边缘平分。Diaz、Grandoni 和 Marchetti-Spaccemela [255] 为b- 随机几何图上的平衡切割问题：找到最小尺寸的边切割，其两侧至少包含bn每个顶点。
Bradonjić、Elsässer、Friedrich、Sauerwald 和 Stauffer [162] 研究了 RGG 的广播时间。他们研究了一个可能存在单个巨大组件的机制，并表明他们的广播算法只需要欧(n1/2/r+日志⁡n)rounds 将信息从单个顶点传递到巨人的每个顶点。他们在路上表明巨人的直径是日(n1/2/r)whp Friedrich、Sauerwald 和 Stauffer [334] 将其扩展到更高的维度。

有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

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。

数学代写|随机图论代写Random Graph代考|Math572

statistics-lab™ 为您的留学生涯保驾护航 在代写随机图论Random Graph方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写随机图论Random Graph代写方面经验极为丰富，各种代写随机图论Random Graph相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

数学代写|随机图论代写Random Graph代考|Binomial Random Intersection Graphs

For $G(n, m, p)$ with $m=n^\alpha, \alpha$ constant, Rybarczyk and Stark [694] provided a condition, called strictly $\alpha$-balanced for the Poisson convergence for the number of induced copies of a fixed subgraph, thus complementing the results of Theorem $12.5$ and generalising Theorem 12.7. (Thresholds for small subgraphs in a related model of random intersection digraph are studied by Kurauskas [540]).
Rybarczyk [696] introduced a coupling method to find thresholds for many properties of the binomial random intersection graph. The method is used to establish sharp threshold functions for $k$-connectivity, the existence of a perfect matching and the existence of a Hamilton cycle.
Stark [725] determined the distribution of the degree of a typical vertex of $G(n, m, p)$, $m=n^\alpha$ and showed that it changes sharply between $\alpha<1, \alpha=1$ and $\alpha>1$.
Behrisch [70] studied the evolution of the order of the largest component in $G(n, m, p)$, $m=n^\alpha$ when $\alpha \neq 1$. He showed that when $\alpha>1$ the random graph $G(n, m, p)$ behaves like $\mathbb{G}_{n, p}$ in that a giant component of size order $n$ appears w.h.p. when the expected vertex degree exceeds one. This is not the case when $\alpha<1$. There is a jump in the order of size of the largest component, but not to one of linear size. Further study of the component structure of $G(n, m, p)$ for $\alpha=1$ is due to Lageras and Lindholm in [542].
Behrisch, Taraz and Ueckerdt [71] study the evolution of the chromatic number of a random intersection graph and showed that, in a certain range of parameters, these random graphs can be colored optimally with high probability using various greedy algorithms.

数学代写|随机图论代写Random Graph代考|Uniform Random Intersection Graphs

Uniform random intersection graphs differ from the binomial random intersection graph in the way a subset of the set $M$ is defined for each vertex of $V$. Now for every $k=1,2, \ldots, n$, each $S_k$ has fixed size $r$ and is randomly chosen from the set $M$. We use the notation $G(n, m, r)$ for an $r$-uniform random intersection graph. This version of a random intersection graph was introduced by Eschenauer and Gligor [296] and, independently, by Godehardt and Jaworski [389].
Bloznelis, Jaworski and Rybarczyk [106] determined the emergence of the giant component in $G(n, m, r)$ when $n(\log n)^2=o(m)$. A precise study of the phase transition of $G(n, m, r)$ is due to Rybarczyk [697]. She proved that if $c>0$ is a constant, $r=r(n) \geq 2$ and $r(r-1) n / m \approx c$, then if $c<1$ then w.h.p. the largest component of $G(n, m, r)$ is of size $O(\log n)$, while if $c>1$ w.h.p. there is a single giant component containing a constant fraction of all vertices, while the second largest component is of size $O(\log n)$.
The connectivity of $G(n, m, r)$ was studied by various authors, among them by Eschenauer and Gligor [296] followed by DiPietro, Mancini, Mei, Panconesi and Radhakrishnan [259],
Blackbourn and Gerke [95] and Yagan and Makowski [766]. Finally, Rybarczyk [697] determined the sharp threshold for this property. She proved that if $c>0$ is a constant, $\omega(n) \rightarrow \infty$ as $n \rightarrow \infty$ and $r^2 n / m=\log n+\omega(n)$, then similarly as in $\mathbb{G}_{n, p}$, the uniform random intersection graph $G(n, m, r)$ is disconnected w.h.p. if $\omega(n) \rightarrow \infty$, is connected w.h.p. if $\omega(n) \rightarrow \infty$, while the probability that $G(n, m, r)$ is connected tends to $e^{-e^{-c}}$ if $\omega(n) \rightarrow c$. The Hamiltonicity of $G(n, m, r)$ was studied in [109] and by Nicoletseas, Raptopoulos and Spirakis [636].
If in the uniform model we require $\left|S_i \cap S_j\right|>s$ to connect vertices $i$ and $j$ by an edge, then we denote this random intersection graph by $G_s(n, m, r)$. Bloznelis, Jaworski and Rybarczyk [106] studied phase transition in $G_s(n, m, r)$. Bloznelis and Łuczak [108] proved that w.h.p. for even $n$ the threshold for the property that $G_s(n, m, r)$ contains a perfect matching is the same as that for $G_s(n, m, r)$ being connected. Bloznelis and Rybarczyk [110] show that w.h.p. the edge density threshold for the property that each vertex of $G_s(n, m, r)$ has degree at least $k$ is the same as that for $G_s(n, m, r)$ being $k$-connected (for related results see [771]).

随机图论代写

数学代写|随机图论代写Random Graph代考|Binomial Random Intersection Graphs

Rybarczyk [696] 引入了一种耦合方法来寻找二项式随机交集图的许多属性的阈值。该方法用于建立尖锐 的阈值函数 $k$-连通性，完美匹配的存在性和哈密顿循环的存在。
Stark [725] 确定了典型顶点的度数分布 $G(n, m, p), m=n^\alpha$ 并表明它在之间急剧变化 $\alpha<1, \alpha=1$ 和 $\alpha>1$
Behrisch [70] 研究了最大分量阶数的演变 $G(n, m, p), m=n^\alpha$ 什么时候 $\alpha \neq 1$. 他表明，当 $\alpha>1$ 随机 图 $G(n, m, p)$ 表现得像 $G_{n, p}$ 那是一个巨大的尺寸订单组成部分 $n$ 当预期的顶点度数超过 1 时出现 whp。 情况并非如此 $\alpha<1$. 最大组件的大小顺序有一个跳跃，但不是线性大小之一。进一步研究的组件结构 $G(n, m, p)$ 为了 $\alpha=1$ 归功于 [542] 中的 Lageras 和 Lindholm。
Behrisch、Taraz 和 Ueckerdt [71] 研究了随机相交图的色数的演变，并表明，在一定的参数范围内，可 以使用各种贪心算法以高概率对这些随机图进行最佳着色。

数学代写|随机图论代写Random Graph代考|Uniform Random Intersection Graphs

Bloznelis、Jaworski 和 Rybarczyk [106] 确定了巨大成分在 $G(n, m, r)$ 什么时候 $n(\log n)^2=o(m)$. 相 $r(r-1) n / m \approx c$ ，那么如果 $c<1$ 然后whp最大的组成部分 $G(n, m, r)$ 是大小 $O(\log n)$, 而如果 $c>1$ $w h p$ 有一个巨大的组件包含所有顶点的常数部分，而第二大组件的大小 $O(\log n)$.

Blackbourn 和 Gerke [95] 以及 Yagan 和 Makowski [766]。最后，Rybarczyk [697] 确定了该属性的尖锐 阈值。她证明了如果 $c>0$ 是常数， $\omega(n) \rightarrow \infty$ 作为 $n \rightarrow \infty$ 和 $r^2 n / m=\log n+\omega(n)$, 然后类似于 $\mathbb{G}_{n, p}$, 均匀随机交集图 $G(n, m, r)$ 断开 whp 如果 $\omega(n) \rightarrow \infty$ ， 连接 $w h p$ 如果 $\omega(n) \rightarrow \infty$ ，而概率 $G(n, m, r)$ 连接趋向于 $e^{-e^{-c}}$ 如果 $\omega(n) \rightarrow c$. 的哈密顿性 $G(n, m, r)$ 在 [109] 以及 Nicoletseas、 Raptopoulos 和 Spirakis [636] 中进行了研究。

有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

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。

数学代写|随机图论代写Random Graph代考|MS-E1603

statistics-lab™ 为您的留学生涯保驾护航 在代写随机图论Random Graph方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写随机图论Random Graph代写方面经验极为丰富，各种代写随机图论Random Graph相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

数学代写|随机图论代写Random Graph代考|RANDOM GEOMETRIC GRAPHS

(c) If $B$ is within distance $100 r$ of two boundary edges of $D$ then $B$ contains no bad cells.

Proof. (a) There are less than $\ell_0^2<n$ such blocks. Furthermore, the probability that a fixed block contains $k_0$ or more bad cells is at most
$$\begin{array}{r} \left(\begin{array}{c} K^2 \ k_0 \end{array}\right)\left(\sum_{i=0}^{i_0}\left(\begin{array}{c} n \ i \end{array}\right)\left(\eta^2 r^2\right)^i\left(1-\eta^2 r^2\right)^{n-i}\right)^{k_0} \ \leq\left(\frac{K^2 e}{k_0}\right)^{k_0}\left(2\left(\frac{n e}{i_0}\right)^{i_0}\left(\eta^2 r^2\right)^{i_0} e^{-\eta^2 r^2\left(n-i_0\right)}\right)^{k_0} \end{array}$$
Here we have used Corollary $23.4$ to obtain the LHS of (12.12).
Now
\begin{aligned} &\left(\frac{n e}{i_0}\right)^{i_0}\left(\eta^2 r^2\right)^{i_0} e^{-\eta^2 r^2\left(n-i_0\right)} \ & \leq n^{O\left(\eta^3 \log (1 / \eta)-\eta^2(1+\varepsilon-o(1)) / \pi\right.} \leq n^{-\eta^2(1+\varepsilon / 2) / \pi}, \end{aligned}
for $\eta$ sufficiently small. So we can bound the RHS of (12.12) by
$$\left(\frac{2 K^2 e n^{-\eta^2(1+\varepsilon / 2) / \pi}}{(1-\varepsilon / 10) \pi / \eta^2}\right)^{(1-\varepsilon / 10) \pi / \eta^2} \leq n^{-1-\varepsilon / 3} .$$
Part (a) follows after inflating the RHS of (12.14) by $n$ to account for the number of choices of block.
(b) Replacing $k_0$ by $k_0 / 2$ replaces the LHS of (12.14) by
$$\left(\frac{4 K^2 e n^{-\eta^2(1+\varepsilon / 2) / \pi}}{(1-\varepsilon / 10) \pi / 2 \eta^2}\right)^{(1-\varepsilon / 10) \pi / 2 \eta^2} \leq n^{-1 / 2-\varepsilon / 6}$$
Observe now that the number of choices of block is $O\left(\ell_0\right)=o\left(n^{1 / 2}\right)$ and then Part (b) follows after inflating the RHS of (12.15) by $o\left(n^{1 / 2}\right)$ to account for the number of choices of block.
(c) Equation (12.13) bounds the probability that a single cell is bad. The number of cells in question in this case is $O(1)$ and (c) follows.
We now do a simple geometric computation in order to place a lower bound on the number of cells within a ball $B(X, r)$.

数学代写|随机图论代写Random Graph代考|Chromatic number

We look at the chromatic number of $G_{\mathscr{X}, r}$ in a limited range. Suppose that $n \pi r^2=$ $\frac{\log n}{\omega_r}$ where $\omega_r \rightarrow \infty, \omega_r=O(\log n)$. We are below the threshold for connectivity here. We will show that w.h.p.
$$\chi(G \mathscr{X}, r) \approx \Delta(G \mathscr{X}, r) \approx c l(G \mathscr{X}, r)$$
where will use $c l$ to denote the size of the largest clique. This is a special case of a result of McDiarmid [587].
We first bound the maximum degree.
Lemma 12.13.
$$\Delta\left(G_{\mathscr{X}, r}\right) \approx \frac{\log n}{\log \omega_r} \text { w.h.p. }$$
Proof. Let $Z_k$ denote the number of vertices of degree $k$ and let $Z_{\geq k}$ denote the number of vertices of degree at least $k$. Let $k_0=\frac{\log n}{\omega_d}$ where $\omega_d \rightarrow \infty$ and $\omega_d=$ $o\left(\omega_r\right)$. Then
$$\mathbb{E}\left(Z_{\geq k_0}\right) \leq n\left(\begin{array}{c} n \ k_0 \end{array}\right)\left(\pi r^2\right)^{k_0} \leq n\left(\frac{n e \omega_d \log n}{n \omega_r \log n}\right)^{\frac{\log n}{\omega_d}}=n\left(\frac{e \omega_d}{\omega_r}\right)^{\frac{\log n}{\Phi_d}} .$$
So,
$$\log \left(\mathbb{E}\left(Z_{\geq k_0}\right)\right) \leq \frac{\log n}{\omega_d}\left(\omega_d+1+\log \omega_d-\log \omega_r\right)$$

随机图论代写

数学代写|随机图论代写Random Graph代考|RANDOM GEOMETRIC GRAPHS

(c) 如果 $B$ 在距离之内 $100 r$ 的两个边界边缘 $D$ 然后 $B$ 不含坏细胞。

$$\left(K^2 k_0\right)\left(\sum_{i=0}^{i_0}(n i)\left(\eta^2 r^2\right)^i\left(1-\eta^2 r^2\right)^{n-i}\right)^{k_0} \leq\left(\frac{K^2 e}{k_0}\right)^{k_0}\left(2\left(\frac{n e}{i_0}\right)^{i_0}\left(\eta^2 r^2\right)^{i_0} e^{-\eta^2 r^2\left(n-i_0\right)}\right)^{k_0}$$

$$\left(\frac{n e}{i_0}\right)^{i_0}\left(\eta^2 r^2\right)^{i_0} e^{-\eta^2 r^2\left(n-i_0\right)} \quad \leq n^{O\left(\eta^3 \log (1 / \eta)-\eta^2(1+\varepsilon-o(1)) / \pi\right.} \leq n^{-\eta^2(1+\varepsilon / 2) / \pi},$$

$$\left(\frac{2 K^2 e n^{-\eta^2(1+\varepsilon / 2) / \pi}}{(1-\varepsilon / 10) \pi / \eta^2}\right)^{(1-\varepsilon / 10) \pi / \eta^2} \leq n^{-1-\varepsilon / 3}$$
(a) 部分是在将 (12.14) 的 RHS 膨胀后的 $n$ 考虑块的选择数量。
(b) 更换 $k_0$ 经过 $k_0 / 2$ 将 (12.14) 的 LHS 替换为
$$\left(\frac{4 K^2 e n^{-\eta^2(1+\varepsilon / 2) / \pi}}{(1-\varepsilon / 10) \pi / 2 \eta^2}\right)^{(1-\varepsilon / 10) \pi / 2 \eta^2} \leq n^{-1 / 2-\varepsilon / 6}$$

(c) 等式 (12.13) 界定了单个电池坏的概率。在这种情况下，有问题的细胞数量是 $O(1)(\mathrm{c})$ 如下。 我们现在做一个简单的几何计算，以便为球内的细胞数量设置一个下限 $B(X, r)$.

数学代写|随机图论代写Random Graph代考|Chromatic number

$$\chi(G \mathscr{X}, r) \approx \Delta(G \mathscr{X}, r) \approx \operatorname{cl}(G \mathscr{X}, r)$$

$$\Delta\left(G_{\mathscr{X}, r}\right) \approx \frac{\log n}{\log \omega_r} \text { w.h.p. }$$

$$\mathbb{E}\left(Z_{\geq k_0}\right) \leq n\left(n k_0\right)\left(\pi r^2\right)^{k_0} \leq n\left(\frac{n e \omega_d \log n}{n \omega_r \log n}\right)^{\frac{\log n}{\omega_d}}=n\left(\frac{e \omega_d}{\omega_r}\right)^{\frac{\log n}{\Phi_d}} .$$

$$\log \left(\mathbb{E}\left(Z_{\geq k_0}\right)\right) \leq \frac{\log n}{\omega_d}\left(\omega_d+1+\log \omega_d-\log \omega_r\right)$$

有限元方法代写

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MATLAB代写

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