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# 物理代写|统计物理代写Statistical Physics of Matter代考|PHYS6562 Complex Networks

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## 物理代写|统计物理代写Statistical Physics of Matter代考|Complex Networks

We can simulate random walks not only on regular lattices as in Section $2.5$, but also on more general networks that mimic the structure of complex physical and social systems. Examples of systems that can be mathematically treated as networks include power grids, railways, air traffic, gas pipes, highways, and the Internet. We show two illustrations of networks in Figure 2.42. In the following sections, we provide an overview of concepts from network science, an interdisciplinary research field that combines methods from graph theory, statistical physics, and computer science. For further reading on network science, there exist many excellent textbooks [85-87].

## 物理代写|统计物理代写Statistical Physics of Matter代考|Adjacency Matrix

Mathematically, a network $G(V, E)$ is an ordered pair, where $V$ and $E$ are the corresponding sets of nodes and edges. Connections between nodes are described by the adjacency matrix $A$. We set the matrix element $A_{w v}=1$ if there exists an edge $e \in E$ between the node pair $\langle u, v\rangle \in V$ and otherwise we set $A_{u v}=0$. In a directed network $A_{u v}$ can be different from $A_{v u}$, whereas $A_{u v}=A_{v u}$ for undirected networks. The adjacency matrix of the (undirected) graph that we show in Figure $2.43$ is
$$A=\left(\begin{array}{lllllll} 0 & 1 & 1 & 1 & 0 & 1 & 1 \ 1 & 0 & 1 & 0 & 1 & 1 & 0 \ 1 & 1 & 0 & 1 & 0 & 0 & 1 \ 1 & 0 & 1 & 0 & 1 & 0 & 0 \ 0 & 1 & 0 & 1 & 0 & 1 & 0 \ 1 & 1 & 0 & 0 & 1 & 0 & 1 \ 1 & 0 & 1 & 0 & 0 & 1 & 0 \end{array}\right) .$$
In this example, the adjacency matrix is sparse and it is computationally more efficient to use edge lists instead of storing the complete matrix. For complete graphs where all nodes are connected with each other, we have to store the whole adjacency matrix.

The elements $(u, v)$ of the matrix powers $A^k$ represent the number of paths of length $k$ from node $u$ to node $v$. For an undirected network, the degree of node $u$ is
$$k_u=\sum_{v \in V} A_{u v}=\sum_{v \in V} A_{v u} .$$
For regular networks, all nodes have the same degree. An example of a regular network is the square lattice with periodic boundary conditions. For a directed network, we distinguish between in-degree (number of incoming edges) and out-degree (number of outgoing edges). Mathematically, we denote the in-degree of node $u$ by
$$k_u^{\mathrm{in}}=\sum_{v \in V} A_{v u},$$
and the corresponding out-degree by
$$k_u^{\text {out }}=\sum_{v \in V} A_{u v} .$$
For undirected networks, summing over $k_u$ yields
$$\sum_{u \in V} k_u=\sum_{u \in V} \sum_{v \in V} A_{u v}=2|E|,$$
where $|E|$ denotes the number of edges of the network $G(V, E)$. This equation implies that every undirected network has an even number of nodes with an odd degree. If there would be an odd number of nodes with odd degrees, the right-hand side of eq. (2.60) would not be even (i. e., equal to $2|E|$ ). This is also known as the handshaking lemma. In a group (“network”) of handshaking people, an even number of people (“nodes”) must shake an odd number of other people’s (“neighboring nodes”) hands. For directed networks, we obtain
$$\sum_{u \in V} k_u^{\text {in }}=\sum_{u \in V} k_u^{\text {out }}=|E| .$$

## 物理代写|统计物理代写物质统计物理代考|邻接矩阵

$$A=\left(\begin{array}{lllllll} 0 & 1 & 1 & 1 & 0 & 1 & 1 \ 1 & 0 & 1 & 0 & 1 & 1 & 0 \ 1 & 1 & 0 & 1 & 0 & 0 & 1 \ 1 & 0 & 1 & 0 & 1 & 0 & 0 \ 0 & 1 & 0 & 1 & 0 & 1 & 0 \ 1 & 1 & 0 & 0 & 1 & 0 & 1 \ 1 & 0 & 1 & 0 & 0 & 1 & 0 \end{array}\right) .$$

$$k_u=\sum_{v \in V} A_{u v}=\sum_{v \in V} A_{v u} .$$

$$k_u^{\mathrm{in}}=\sum_{v \in V} A_{v u},$$

$$k_u^{\text {out }}=\sum_{v \in V} A_{u v} .$$

$$\sum_{u \in V} k_u=\sum_{u \in V} \sum_{v \in V} A_{u v}=2|E|,$$
，其中$|E|$表示网络$G(V, E)$的边数。这个方程表明，每个无向网络都有奇数次的偶数节点。如果有奇数个节点具有奇数度，eq.(2.60)的右边就不是偶数(即等于$2|E|$)。这也被称为握手引理。在一组握手的人(“网络”)中，偶数人(“节点”)必须与奇数人(“相邻节点”)握手。对于有向网络，我们得到
$$\sum_{u \in V} k_u^{\text {in }}=\sum_{u \in V} k_u^{\text {out }}=|E| .$$

## 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。