### 数学代写|随机图论代写Random Graph代考|Math572

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等概率论
• Advanced Mathematical Statistics 高等数理统计学
• (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 $O(\log n)$.

Mancini、Mei、Panconesi 和 Radhakrishnan [259]、
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。

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## MATLAB代写

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