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# 数据科学代写|复杂网络代写Complex Network代考|COMP5313 Deterministic graphs

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## 数据科学代写|复杂网络代写Complex Network代考|Deterministic graphs

This book mostly concerns random systems, but, interestingly, a number of deterministic graphs demonstrate many features resembling those of random growing networks. The resemblance can be so close that deterministic graphs were even used for interpretation of empirical observations (Ravasz, Somera, Mongru, Oltvai, and Barabási, 2002; Ravasz and Barabási, 2003). As examples, we indicate a few deterministic graphs that can be naturally treated as simplicial complexes: Farey-sequence based graphs (Hardy and Wright, 1979), pseudo-fractals (Dorogovtsev, Goltsev, and Mendes, 2002b), and Apollonian graphs (Andrade Jr, Herrmann, Andrade, and Da Silva, 2005) (Figure 5.5).

These three graphs are small worlds with the ‘average’ separation between their vertices $\langle\ell\rangle \sim N$. Their Hausdorff dimension is infinite. In particular, considering the relative number of vertices $N(q)$ of degree $q$, one can easily get $N(q) \propto 2^{-q / 2}$ for the Farey graph, where $q=2,4,6, \ldots$, which resembles the exponential degree distribution of random recursive graphs. In the spectra of degrees of the pseudofractals and Apollonian graphs, the gaps between consecutive degrees exponentially increase with $q$, so once should consider the cumulative number of vertices of degree equal or larger than $q$. In both cases, this number has the asymptotics

$\sum_{k \geq q} N(k) \sim q^{-\gamma+1}$ corresponding to the degree distribution with exponent $\gamma=1+\ln 3 / \ln 2=2.584 \ldots$. In this sense, these two deterministic graphs are ‘scale-free’.

In contrast to the infinite Hausdorff dimension of these networks, their spectral dimensions are finite. Bianconi and Dorogovstev (2020) applied the renormalization group techniques of Hwang, Yun, Lee, Kahng, and Kim (2010) to obtain the spectral dimensions ${ }^{28}$
$$D_s=\left{\begin{array}{cl} 2, & \text { Farey graph, } \ 2 \frac{\ln 3}{\ln 2}=3.169 \ldots, & \text { pseudofractal } \ 2 \frac{\ln 3}{\ln 9 / 5}=3.738 \ldots, & \text { Apollonian graph. } \end{array}\right.$$

## 数据科学代写|复杂网络代写Complex Network代考|Giant connected component

Let us return to the configuration model of uncorrelated networks with a given degree distribution $P(q)$ and exploit its local tree-likeness to explore basic structural features of these networks. The elements of the techniques that we use here were first implemented in graph theory (Pittel, 1990; Molloy and Reed, 1995, 1998), and physicists developed it into a convenient and powerful mathematical apparatus (Newman, Strogatz, and Watts, 2001) applicable to various locally tree-like networks, including directed, multipartite, and correlated ones, and many others. For the sake of simplicity, we first consider uncorrelated undirected networks. The generating functions techniques is ideally suited for random trees and tree-like structures (Appendix C). The generating function for the degree distribution is defined as
$$G(z) \equiv \sum_{q=0}^{\infty} P(q) z^q,$$
hence its first moment $\langle q\rangle=G^{\prime}(1)$. Then
$$G_1(z) \equiv \frac{G^{\prime}(z)}{G^{\prime}(1)}=\sum_{q=0}^{\infty} \frac{q P(q)}{\langle q\rangle} z^{q-1}=\sum_{k=0}^{\infty} \frac{(k+1) P(k+1)}{\langle q\rangle} z^k$$
is the generating function of the branching of a randomly chosen edge in a locally tree-like network. In other words, if you choose a vertex uniformly at random and follow one of its edges to the second its end, then the distribution of the number of outgoing edges of the end vertex (degree of the vertex minus 1) has the generating function $G_1(z)$. Thus $G_1(1)=G(1)=1$, and $G_1^{\prime}(1)$ equals the average branching,
$$G_1^{\prime}(1)=\frac{\left\langle q^2\right\rangle-\langle q\rangle}{\langle q\rangle}=\langle b\rangle .$$

## 数据科学代写|复杂网络代写Complex Network代考|Accelerated growth and densification

17读者将在 Dorogovtsev、Mendes 和 Samukhin (2001a) 中找到这些计算的详细信息。
18Broder、Kumar、Maghoul、Raghavan、Rajagopalan、Stata、Tomkins 和 Wiener (2000) 于 1999 年 5 月首次测量了 WWW 中的连接密度，发现顶点的平均入度和出度相等，为 7.22。当他们在 1999 年 10 月重复测量时，平均入度和出度已经7.85.

## 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。