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


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
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
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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