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物理代写|复杂网络代写Complex Network代考|MATH3002 Annealed networks and graphons

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物理代写|复杂网络代写Complex Network代考|Annealed networks and graphons

At this point, we mention a useful deterministic construction known as the annealed network approximation based on relations visually similar to formulas in the previous section. This approximation is actually applicable not only to equilibrium or evolving random networks but even to nonrandom graphs. Let us consider a large graph whose vertices have degrees $q_i$, $i=1,2, \ldots, N$. (For the sake of brevity, we assume here that this graph is simple, which is actually not necessary.) This graph is approximated by the ‘annealed network’ that is the fully connected weighted graph of $N$ vertices, whose edges have the weights ${ }^{14}$
$$w_{i j}\left(q_i, q_j\right)=\frac{q_i q_j}{N\langle q\rangle} .$$
Then the sum of the weights of the edges of a vertex coincides with the degree of the original graph,
$$\sum_j w_{i j}=q_i$$
Compare Eqs. (4.18) and (4.20), (4.17) and (4.21). Even if the original graph is random, the annealed network will be deterministic in the limit $N \rightarrow \infty$, which makes it one of the widely used simple instruments in the theory of complex networks. ${ }^{15}$

物理代写|复杂网络代写Complex Network代考|Clustering, cycles, and cliques

Let us look closer at the local tree-likeness of large sparse uncorrelated networks. To do this, we inspect the average number $\mathcal{N}{\text {cycles }}(\ell)$ of cycles of a given length $\ell$ in undirected networks of this kind. Here we consider the configuration model and adapt the derivation from Bianconi and Marsili (2005). The number $\mathcal{N}{\text {cycles }}(\ell)$ can be written as the following product:
$$\mathcal{N}_{\text {cycles }}(\ell)=R(N, \ell,{P(q)}) G(\ell) W(N, \ell) .$$
Here the first term $R(N, \ell,{P(q)})$, is the number of ways to select pairs of edges connected to $\ell$ vertices from the edges of these vertices. This number depends on the form of the degree distribution $P(q)$. We easily get
$$R(N, \ell,{P(q)})=\frac{[N\langle q(q-1)\rangle]^{\ell}}{\ell !}$$
The second term in the product on the right-hand side of Eq. (4.23), $G(\ell)$, is the number of ways to connect $\ell$ given vertices in a cycle. Equally easily, we have
$$G(\ell)=\frac{\ell !}{2 \ell}$$

物理代写|复杂网络代写Complex Network代考|Annealed networks and graphons

$$w_{i j}\left(q_i, q_j\right)=\frac{q_i q_j}{N\langle q\rangle} .$$

$$\sum_j w_{i j}=q_i$$

物理代写|复杂网络代写Complex Network代考|Clustering, cycles, and cliques 中。

$$\mathcal{N}_{\text {cycles }}(\ell)=R(N, \ell, P(q)) G(\ell) W(N, \ell) .$$

$$R(N, \ell, P(q))=\frac{[N\langle q(q-1)\rangle]^{\ell}}{\ell !}$$
$$G(\ell)=\frac{\ell !}{2 \ell}$$

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。