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# 数据科学代写|复杂网络代写Complex Network代考|Graph Theoretical Notation

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## 数据科学代写|复杂网络代写Complex Network代考|Graph Theoretical Notation

Mathematically, a network is represented as a graph $\mathcal{G}(V, E)$, i.e., an object that consists of a set of nodes or vertices $V$ representing the objects or agents in the network and a set $E$ of edges or links or connections representing the interactions or relations of the nodes. The cardinality of these sets, i.e, the number of nodes and edges, is generally denoted by $N$ and $M$, respectively. One may assign different values $w_{i j}$ to the links between nodes $i$ and $j$ in $E$, rendering an edge weighted or otherwise non-weighted $\left(w_{i j}=1\right.$ by convention, if one is only interested in the presence or absence of the relation). The number of connections of node $i$ is denoted by its degree $k_i$. One can represent the set of edges conveniently in an $N \times N$ matrix $A_{i j}$, called the adjacency matrix. $A_{i j}=w_{i j}$ if an edge between node $i$ and $j$ is present and zero otherwise. Relations may be directed, in which case $A_{i j}$ is non-symmetric $\left(A_{i j} \neq A_{j i}\right)$ or undirected in which case $A_{i j}$ is symmetric. Here, we are mainly concerned with networks in which self-links are absent $\left(A_{i i}=0, \forall i \in V\right)$. In case of a directed network, $A_{i j}$ denotes an outgoing edge from $i$ to $j$. Hence, the outgoing links of node $i$ are found in row $i$, while the incoming links to $i$ are found in column $i$. For undirected networks, it is clear that $\sum_{j=1}^N A_{i j}=k_i$. For directed networks, $\sum_{j=1} A_{i j}=k_i^{\text {out }}$ is the out-degree and equivalently $\sum_{j=1} A_{j i}=k_i^{i n}$ is the in-degree of node $i$. It is understood that in undirected networks, the sum of degrees of all nodes in the network equals twice the number of edges $\sum_{i=1}^N k_i=2 M$. The distribution of the number of connections per node is called degree distribution $P(k)$ and denotes the probability that a randomly chosen node from the network has degree $k$. The average degree in the network is denoted by $\langle k\rangle$ and one has $N\langle k\rangle=2 M$. One can define a probability $p=2 M / N(N-1)=\langle k\rangle /(N-1)$ as the probability that an edge exists between two randomly chosen nodes from the network.
An (induced) subgraph is a subset of nodes $v \subseteq V$ with $n$ nodes and edges $e \subseteq E$ connecting only the nodes in $v$. A path is a sequence of nodes, subsequent nodes in the sequence being connected by edges from $E$. A node $i$ is called reachable from node $j$ if there exists a path from $j$ to $i$. A subgraph is said to be connected if every node in the subgraph is reachable from every other. The number of steps (links) in the shortest path between two nodes $i$ and $j$ is called the geodesic distance $d(i, j)$ between nodes $i$ and $j$. A network is generally not connected, but may consist of several connected components. The largest of the shortest path distances between any pair of nodes in a connected component is called the diameter of a connected component. The analysis in this monograph shall be restricted to connected components only since it can be repeated on every single one of the connected components of a network. More details on graph theory may be found in the book by Bollobás.
With these notations and terms in mind, let us now turn to a brief overview of some aspects of physicists research on networks.

## 数据科学代写|复杂网络代写Complex Network代考|Random Graphs

For the study of the topology of the interactions of a complex system it is of central importance to have proper random null models of networks, i.e., models of how a graph arises from a random process. Such models are needed for comparison with real world data. When analyzing the structure of real world networks, the null hypothesis shall always be that the link structure is due to chance alone. This null hypothesis may only be rejected if the link structure found differs significantly from an expectation value obtained from a random model. Any deviation from the random null model must be explained by non-random processes.
The most important model of a random graph is due to Erdös and Rényi (ER) [12]. They consider the following two ensembles of random graphs: $\mathcal{G}(N, M)$ and $\mathcal{G}(N, p)$. The first is the ensemble of all graphs with $N$ nodes and exactly $M$ edges. A graph from this ensemble is created by placing the $M$ edges randomly between the $N(N-1) / 2$ possible pairs of nodes. The second ensemble is that of all graphs in which a link between two arbitrarily chosen nodes is present with probability $p$. The expectation value for the number of links of a graph from this ensemble is $\langle M\rangle=p N(N-1) / 2$. In the limit of $N \rightarrow \infty$, the two ensembles are equivalent with $p=2 M / N(N-1)$. The typical graph from these ensembles has a Poissonian degree distribution
$$P(k)=e^{-\langle k\rangle} \frac{\langle k\rangle^k}{k !} .$$
Here, $\langle k\rangle=p(N-1)=2 M / N$ denotes the average degree in the network.
The properties of ER random graphs have been studied for considerable time and an overview of results can be found in the book by Bollobás [13]. Note that the equivalence of the two ensembles is a remarkable result. If all networks with a given number of nodes and links are taken to be equally probable, then the typical graph from this ensemble will have a Poissonian degree distribution. To draw a graph with a non-Poissonian degree distribution from this ensemble is highly improbable, unless there is a mechanism which leads to a different degree distribution. This issue will be discussed below in more detail.
Another aspect of random networks is worth mentioning: the average shortest path between any pair of nodes scales only as the logarithm of the system size. This is easily seen: Starting from a randomly chosen node, we can visit $\langle k\rangle$ neighbors with a single step. How many nodes can we explore with the second step? Coming from node $i$ to node $j$ via a link between them, we now have $d_j=k_j-1$ options to proceed. Since we have $k_j$ possible ways to arrive at node $j$, the average number of second step neighbors is hence $\langle d\rangle=\sum_{k=2}^{\infty}(k-1) k P(k) /\left(\sum_k^{\infty} k P(k)\right)=\left\langle k^2\right\rangle /\langle k\rangle-1$. Hence, after two steps we may explore $\langle k\rangle\langle d\rangle$ nodes and after $m$ steps $\langle k\rangle\langle d\rangle^{m-1}$ nodes which means that the entire network may be explored in $m \approx \log N$ steps. This also shows that even in very large random networks, all nodes may be reached with relatively few steps. The number $d=k-1$ of possible ways to exit from a node after entering it via one of its links is also called the “excess degree” of a node. Its distribution $q(d)=(d+1) P(d+1) /\langle k\rangle$ is called the “excess degree distribution” and plays a central role in the analysis of many dynamical phenomena on networks. Note that our estimate is based on the assumption that in every new step we explore $\langle d\rangle$ nodes which we have not seen before! For ER networks, though, this is a reasonable assumption. However, consider a regular lattice as a counterexample. There, the average shortest distance between any pair of nodes scales linearly with the size of the lattice.

## 数据科学代写|复杂网络代写Complex Network代考|Random Graphs

$$P(k)=e^{-\langle k\rangle} \frac{\langle k\rangle^k}{k !} .$$

ER随机图的性质已经被研究了相当长的时间，研究结果的概述可以在Bollobás[13]的书中找到。注意，两个集合的等效性是一个显著的结果。如果所有具有给定数量节点和链接的网络都是等概率的，那么这个集合的典型图将具有泊松度分布。要从这个集合中画出一个非泊松度分布的图是极不可能的，除非有一种机制导致不同的度分布。下面将更详细地讨论这个问题。

## 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。