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# 物理代写|统计物理代写Statistical Physics of Matter代考|PHYC40650 Degree Distribution

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## 物理代写|统计物理代写Statistical Physics of Matter代考|Degree Distribution

Networks can be characterized by their degree distribution $P(k)$. For a square lattice, the degree distribution is $P(k)=\delta_{k 4}$ (i.e., each node has degree $k=4$ ). If the distribution follows a power law
$$P(k) \propto k^{-\gamma},$$
the network is called scale-free [91]. Examples of networks that are approximately scale-free are the Internet and some social networks [92, 93]. One mechanism that produces scale-free networks is preferential attachment where each new node is more likely to attach to existing nodes with high degree. Mathematically, each new node will be attached to $m \leq m_{0}$ existing nodes and the attachment probability is proportional to the number of edges of the existing nodes. Here, $m_{0}$ is the initial number of nodes. Networks that result from this type of preferential attachment are called BarabásiAlbert networks and their degree distribution is $P(k) \propto k^{-3}$. We show an example of a Barabási-Albert network in Figure 2.44.

Networks that are generated by a random process are also called random-graph models [94, 95]. Another example of a random-graph model are Erdős-Rényi networks [96]. To generate them, we start with $N$ isolated nodes and add new edges between two uniformly at random selected nodes with probability $p$. The resulting degree distribution is binomial:
$$P(k)=\left(\begin{array}{c} N-1 \ k \end{array}\right) p^{k}(1-p)^{N-1-k},$$
where $N$ is the number of nodes.
Watts-Strogatz networks [97] also belong to the class of random-graph models and are generated by arranging nodes in a ring and connecting each node to its $K$ nearest neighbors. The $K / 2$ rightmost edges of each node are then rewired (i.e., reconnected) with probability $p$ (see Figure 2.45). The degree distribution of a Watts-Strogatz network is [98]
$$P(k)=\sum_{l=0}^{f(k, K)}\left(\begin{array}{c} K / 2 \ l \end{array}\right)(1-p)^{l} p^{K / 2-l} \frac{(p K / 2)^{k-K / 2-l}}{(k-K / 2-l) !} e^{-p K / 2}$$
where $f(k, K)=\min (k-K / 2, K / 2)$.

## 物理代写|统计物理代写Statistical Physics of Matter代考|Dijkstra’s Algorithm

In the previous section, we defined the shortest path between two points on an arbitrary network $G(V, E)$, where $V$ is a set of nodes and $E$ is a set of edges. With the burning method (see Section 2.3.1), we have already seen an example of an algorithm that identifies the shortest-path length (“burning time”) on a percolation cluster. For a general graph $G(V, E)$, Edsger Dijkstra (see Figure 2.46) proposed a greedy algorithm ${ }^{7}$ in $1959 .$

Dijkstra’s Algorithm

1. We assign to every node a temporary distance value (zero for our initial node and infinity for all other nodes).
2. We set the initial node as “burned” and mark all other nodes as “unburned.” All unburned nodes are stored in a corresponding set.
3. For all unburned neighbors of the current node, we compute the corresponding distances. If the computed distance is smaller than the current one, we assign the smaller value. Otherwise, we keep the current value.
4. After having considered all of the neighbors of the current node, we mark the current node as “burned” and remove it from the set of unburned nodes. We will not consider a burned node again.
5. We stop the algorithm if we reach the destination node or if the smallest tentative distance among the nodes in the unvisited set is infinity.
6. Otherwise, we select the “unburned” node that is marked with the smallest distance, consider it as current node, and return to step $3 .$

When combined with additional optimization techniques, the run time of Dijkstra’s algorithm is proportional to $|E|+|V| \log |V|$. In addition to Dijkstra’s algorithm, there also exists the Bellman-Ford-Moore algorithm that can also handle negative weights [104].

## 物理代写|统计物理代与写Statistical Physics of Matter代考|Degree Distribution

$$P(k) \propto k^{-\gamma},$$

$$P(k)=(N-1 k) p^{k}(1-p)^{N-1-k},$$

Watts-Strogatz 网络 [97] 也属于随机图模型类，是通过将节点排列成一个环并将每个节点连接到它的 $K$ 最近的邻居。这 $K / 2$ 然后以概率重新连接 (即重新连接) 每个节点的最右边的边缬 $p$ (见图 2.45)。Watts-Strogatz网絡的度数分布为 [98]
$$P(k)=\sum_{l=0}^{f(k, K)}(K / 2 l)(1-p)^{l} p^{K / 2-l} \frac{(p K / 2)^{k-K / 2-l}}{(k-K / 2-l) !} e^{-p K / 2}$$

## 物理代写|统计物理代写Statistical Physics of Matter代考|Dijkstra’s Algorithm

Dijkstra 算法

1. 我们为每个节点分配一个临时距离值（初始节点为零，所有其他节点为无穷大）。
2. 我们将初始节点设置为“已烧毁”，并将所有其他节点标记为“末烧录”。所有末销毁的节点都存储在相应的集合中。
3. 对于当前节点的所有末肤峣邻居，我们计算相应的距离。如果计算的距离小于当前距离，我们分配较小的值。否则，我们保 留当前值。 的节点。
4. 如果我们到达目标节点或者末访问集中的节点之间的最小暂定距离为无穷大，我们将停止算法。
当与其他优化技术结合使用时，Dijkstra 算法的运行时间与 $|E|+|V| \log |V|$. 除了 Dijkstra 算法之外，还有可以处理负权重的 Bellman-Ford-Moore 算法[104]。

## 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。