数据科学代写|复杂网络代写Complex Network代考|CS60078 Statistical mechanics view of random systems

如果你也在 怎样代写复杂网络Complex Network CS60078这个学科遇到相关的难题,请随时右上角联系我们的24/7代写客服。复杂网络Complex Network在网络理论的背景下,复杂网络是指具有非微观拓扑特征的图(网络)–这些特征在简单的网络(如格子或随机图)中不会出现,但在代表真实系统的网络中经常出现。复杂网络的研究是一个年轻而活跃的科学研究领域(自2000年以来),主要受到现实世界网络的经验发现的启发,如计算机网络、生物网络、技术网络、大脑网络、气候网络和社会网络。

复杂网络Complex Network大多数社会、生物和技术网络显示出实质性的非微观拓扑特征,其元素之间的连接模式既不是纯粹的规则也不是纯粹的随机。这些特征包括学位分布的重尾、高聚类系数、顶点之间的同态性或异态性、社区结构和层次结构。在有向网络的情况下,这些特征还包括互惠性、三联体重要性概况和其他特征。相比之下,过去研究的许多网络的数学模型,如格子和随机图,并没有显示这些特征。最复杂的结构可以由具有中等数量相互作用的网络实现。这与中等概率获得最大信息含量(熵)的事实相对应。

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数据科学代写|复杂网络代写Complex Network代考|CS60078 Statistical mechanics view of random systems

数据科学代写|复杂网络代写Complex Network代考|Statistical mechanics view of random systems

Look at the two graphs in Figure 1.1. Could you say which graph is random, left or right? The typical student’s response is: ‘Of course, the right one!’ This immediate answer is dramatically incorrect. It is actually impossible to say whether a finite graph is random or non-random (deterministic) since one can generate any finite graph by some deterministic algorithm. The example in Figure $1.2$ explains what is really a random graph. This graph has a number of different realizations (individual graphs), and each of them occurs with some associated probability. Thus a random graph (random net-work) is a statistical ensemble of individual graphs, in which each member has its probability of realization. (Note that we do not distinguish the terms ‘graph’, or its generalization-‘hypergraph’, and ‘network’.) In particular, all these probabilities may be equal, which provides a uniform ensemble. In short, a random graph is a statistical ensemble of graphs. In this picture, the result of the measurement of some characteristic of a random graph, an observable, is the average of this characteristic over the statistical ensemble accounting for the probabilities of realization of its members. ${ }^{1}$

This construction is identical to the foundational notion of statistical mechanics, which is just the statistical ensemble. The statistical ensemble is the complete set of all possible states (configurations) of a system. Each member of a statistical ensemble is accompanied by its statistical weight, a positive number, which is equal to the probability of realization of this member with some unimportant factor, equal for all members. In the 1870s, Ludwig Boltzmann explained that the statistical weights $\mathcal{W}{\alpha}$ of the members $\alpha$ of a statistical ensemble are expressed in terms of the energy $E{\alpha}$ of these members (which are states or configurations),
$$
\mathcal{W}{\alpha} \propto e^{-\beta E{\alpha}} .
$$
Here, $\beta$ is a control parameter, a positive number, $\beta=1 /\left(k_{B} T\right)$, where $k_{B}$ is the Boltzmann constant and $T$ is temperature. The first to comprehend that thermodynamics is about probability and probability distributions, that is, statistics, was James Clerk Maxwell. Maxwell first applied the term ‘statistical’ to thermodynamical systems in 1871, and in 1884, Josiah Willard Gibbs coined the term ‘statistical mechanics’; in 1902 he essentially completed the construction of equilibrium statistical physics elaborating the idea of statistical ensembles in his book Elementary Principles in Statistical Mechanics.

数据科学代写|复杂网络代写Complex Network代考|History of networks

It is Leonhard Euler’s solution of the Königsberg bridge problem in 1735 that is usually regarded as the birth point of graph theory. In Euler’s time, the connections in Königsberg were as shown in Figure 1.3a, where the vertices of the graph denote lands (the left vertex is Kneiphof Island, a remarkably perfect rectangle) and edges denote bridges between these pieces of land across the Pregel River. The question was: could a pedestrian walk around Königsberg, crossing each bridge only once? In graph theory, a walk is an alternating sequence of vertices and edges, which begins and ends with vertices. A trail is a walk that has all edges distinct (though it can have repeating vertices). A trail is closed if its initial and end vertices coincide, otherwise a trail is open. Therefore, using the language of graph theory, the question was: is there a trail (open or closed) that passes all the edges of the graph in Figure 1.3a? A closed trail visiting all edges of a graph is called an Eulerian circuit. A graph having an Eulerian circuit is called a Eulerian graph. An open trail visiting all edges of a graph is called an Eulerian trail (tour). We need one more definition: the number of the edge ends attached to a vertex is its degree. Euler’s theorem says that a connected graph is Eulerian iff each vertex has an even degree, and so such tours were impossible in Königsberg. (In a connected graph, there exists a walk between each two vertices.) In fact, Euler proved that this is the necessary condition. (The proof is quite easy, the reader can immediately reproduce it, and, of course, the real breakthrough was formulating the problem in terms of a graph.) In 1873 , Carl Hierholzer proved that this condition is also sufficient. Furthermore, a connected graph has an open trail visiting all edges if and only if only two of the vertices have odd degree. As is natural, this trail runs between these vertices.

The set of bridges in Königsberg (now Kaliningrad) evolved with time, Figure 1.3. In Euler’s time, indeed, there was no trail passing every bridge. Afterwards, at some moment, the graph became Eulerian. Today there exists an open trail passing every bridge-an Eulerian trail.

In 1759 Euler made a big advance in solving the knight’s tour problem, one of the oldest in mathematics. The problem appeared in a piece of Sanskrit poetry by Rudrata in the ninth century and it probably dates back to the sixth century when chess originated. A knight’s tour is a series of moves of a knight visiting every square of a chessboard exactly once. A knight’s tour is closed if its start coincides with its end and open otherwise. Euler found one of the closed knight’s tours on the $8 \times 8$ chessboard (Figure 1.4), which shows this tour on the knight’s graph whose vertices are the chessboard’s squares and the edges are the knight’s legal moves. ${ }^{2}$ In graph theory, a path is an open walk having no repeated vertices (and so having no repeated edges). A cycle is a closed walk having no repeated vertices (and edges). ${ }^{3}$ Graphs without cycles are called trees. A path visiting all the vertices of a graph is called a Hamiltonian path. A cycle that passes through all the vertices of a graph is called a Hamiltonian cycle, and a graph having a Hamiltonian cycle is called a Hamiltonian graph. Therefore, an open knight’s tour is a Hamiltonian path, a closed knight’s tour is a Hamiltonian cycle, and the $8 \times 8$ knight’s graph turns out to be Hamiltonian.

数据科学代写|复杂网络代写Complex Network代考|CS60078 Statistical mechanics view of random systems

复杂网络代写

数据科学代写|复杂网络代写Complex Network代考|Statistical mechanics view of random systems


请看图 $1.1$ 中的两个图表。你能说哪个图是随机的, 左边还是右边? 典型的学生的反应是: “当然是对的! ”
合, 其中每个成员都姷其实现的概率。(请注意, 我们不区分术语“图”或其泛化一-“超图”和“网络”。)特
别是, 所有这些概率可能相等, 这提供了一个统一的集合。简而言之, 随机图是图的统计集合。在这张图片
中, 随机图的某些特征 (可观察) 的测量结果是该特征在统计集合中的平均值, 该统计集合说明了其成员的
实现概率。
这种结构与统计力学的基本概念相同, 即统计系综。统计集成是系统所有可能状态 (配置) 的完整集合。统
对所有成员都相等。在 1870 年代, 路德维希·玻尔兹曼解释说, 统计权重 $\mathcal{W} \alpha$ 成员的 $\alpha$ 统计系综的能量表示
为 $E \alpha$ 这些成员中的(它们是状态或配置),
$\mathcal{W} \alpha \propto e^{-\beta E \alpha}$
这里, $\beta$ 是一个控制参数, 一个正数, $\beta=1 /\left(k_{B} T\right)$ ,在哪里 $k_{B}$ 是玻尔兹曼常数和 $T$ 是温度。James


数据科学代写复杂网络代写Complex Network代考|History of networks


1735年莱昂哈德·欧拉 (Leonhard Euler) 对柯尼斯堡桥问题的解通常被认为是图论的诞生点。在欧拉时
代, 柯尼斯堡的连接如图 $1.3 \mathrm{a}$ 所示, 图中的顶点表示陆地 (左边的顶点是克奈霍夫岛, 一个非常完美的矩
形),边表示这些陆地之间横跨普雷格尔河的桥梁. 问题是: 行人可以绕着柯尼斯堡走, 每座桥只过一次
此, 使用图论的语言, 问题是: 是否存在通过图 1.3a 中图形所有边的轨迹 (开放或封闭)? ? 访问图所有边
的闭合路径称为欧拉回路。具有欧拉回路的图称为欧拉图。访问图的所有边的开放路径称为欧拉路径(游
览)。我们还需要一个定义:连接到一个顶点的边数就是它的度数。欧拉定理说, 如果每个顶点都有一个偶
数度, 则连通图是欧拉图, 因此在柯尼斯堡是不可能的。(在连通图中, 每两个顶点之间存在一条游走。)
事实上, 欧拉证明了这是必要条件。(证明很简单, 读者可以立即复制它, 当然, 真正的突破是用图表来表
述问题。)在 1873 年, Carl Hierholzer 证明了这个条件也是充分的。此外, 当且仅当只有两个顶点具有
可数度时, 连通图具有访回所有边的开放路径。很自然,这条路径在这些顶点之间运行。
柯尼斯保(现在的加里宁格勒)的桥梁集随着时间的推移而演变, 图 1.3。事实上,在欧拉的时代, 每座桥 都没有经过的小径。之后, 在某个时刻, 图变成了欧拉图。今天, 每座桥都有一条开放的小径一一一条欧拉 小都汉有 小径。
1759 年, 欧拉在解决骑士巡回问题方面取得了重大进展, 这是数学中最古老的问题之一。这个问题出现在
9 世纪 Rudrata 的一首梵文诗歌中, 它可能可以追溯到国际象棋起源的 6 世纪。骑士之旅是一个骑士在棋
盘的每个方格中走完一次的一系列动作。如果骑士的旅程开始与结束重合, 则骑士之旅关闭, 否则打开。欧
拉友现了一个封闭的骑士之旅 $8 \times 8$ chessboard (图 1.4), ‘显示了马图上的这次旅行, 具顶点是椇盘的 图的所有顶点的圈标为哈密顿圈, 具有哈密顿圈的图秒 游是哈密顿循环, 而 $8 \times 8$ 骑士的图原来是哈密顿量。

数据科学代写|复杂网络代写Complex Network代考

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微观经济学代写

微观经济学是主流经济学的一个分支,研究个人和企业在做出有关稀缺资源分配的决策时的行为以及这些个人和企业之间的相互作用。my-assignmentexpert™ 为您的留学生涯保驾护航 在数学Mathematics作业代写方面已经树立了自己的口碑, 保证靠谱, 高质且原创的数学Mathematics代写服务。我们的专家在图论代写Graph Theory代写方面经验极为丰富,各种图论代写Graph Theory相关的作业也就用不着 说。

线性代数代写

线性代数是数学的一个分支,涉及线性方程,如:线性图,如:以及它们在向量空间和通过矩阵的表示。线性代数是几乎所有数学领域的核心。



博弈论代写

现代博弈论始于约翰-冯-诺伊曼(John von Neumann)提出的两人零和博弈中的混合策略均衡的观点及其证明。冯-诺依曼的原始证明使用了关于连续映射到紧凑凸集的布劳威尔定点定理,这成为博弈论和数学经济学的标准方法。在他的论文之后,1944年,他与奥斯卡-莫根斯特恩(Oskar Morgenstern)共同撰写了《游戏和经济行为理论》一书,该书考虑了几个参与者的合作游戏。这本书的第二版提供了预期效用的公理理论,使数理统计学家和经济学家能够处理不确定性下的决策。



微积分代写

微积分,最初被称为无穷小微积分或 “无穷小的微积分”,是对连续变化的数学研究,就像几何学是对形状的研究,而代数是对算术运算的概括研究一样。

它有两个主要分支,微分和积分;微分涉及瞬时变化率和曲线的斜率,而积分涉及数量的累积,以及曲线下或曲线之间的面积。这两个分支通过微积分的基本定理相互联系,它们利用了无限序列和无限级数收敛到一个明确定义的极限的基本概念 。



计量经济学代写

什么是计量经济学?
计量经济学是统计学和数学模型的定量应用,使用数据来发展理论或测试经济学中的现有假设,并根据历史数据预测未来趋势。它对现实世界的数据进行统计试验,然后将结果与被测试的理论进行比较和对比。

根据你是对测试现有理论感兴趣,还是对利用现有数据在这些观察的基础上提出新的假设感兴趣,计量经济学可以细分为两大类:理论和应用。那些经常从事这种实践的人通常被称为计量经济学家。



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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