### 数学代写|计算复杂度理论代写Computational complexity theory代考| Complex Networks: A Very Short Overview

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|计算复杂度理论代写Computational complexity theory代考|Complex Networks: A Very Short Overview

Nowadays, Complex Networks represent a vibrant and independent research field that attracted the attention of scientists coming from different areas. The underlying reason is that many natural and man-made complex systems, as biological neural networks, social networks, and infrastructural networks, have a nontrivial topology that strongly influences the dynamics among the related agents (i.e., users of social networks, neurons of neural networks, and so on). An increasing amount of

investigations is demonstrating the relevance of the interaction structure in a wide amount of systems, and, even in the case of EGT, complex networks allow to obtain very interesting results. For instance, as recalled in Chap. 1, Santos and Pacheco showed the role of heterogeneity in the emergence of cooperation, modeling their system with scale-free networks. The latter, as well as others famous models, often is used as toy model both in EGT and in many other contexts as social dynamics, ecological networks, etc. Thus, in this section, we provide a very short overview on the main network properties, and on three different models that can be used for generating a complex network with a known topology. Readers interested in this topic are warmly encouraged to read the wide literature on complex networks. So, first of all, modern network theory has its basis in the classical theory of graphs. In particular, a preliminary definition of complex network can be “a graph with a nontrivial topology.” In general, a graph is a mathematical object that allows to represent relations among a collection of items, named nodes. More formally, a graph $G$ is defined as $G=(N, E)$, with $N$ set of nodes/vertices and $E$ set of edges/links (or bonds). Nodes can be described by a label and represent the elements of a system, e.g., users of a social network, websites of the WEB, and so on. In turn, the edges represent the connections among nodes, and map relations as friendship, physical links, etc. A graph can be “directed” or “undirected,” i.e., the relation can be symmetrical (e.g., friendship) or not (e.g., a one way road), and can be “weighted” or “unweighted.” The former allows to introduce some coarseness in the relations, e.g., in a transportation network the weights might refer to the actual geographical distance between two locations. The information related to the connections in a network is saved in a $N \times N$ matrix, with $N$ number of nodes, defined “adjacency matrix.” Numerical analysis on the adjacency matrix allow to investigate the properties of a network. For instance, the adjacency matrix $A$ of an unweighted graph can have the following form:
$$a_{i j}= \begin{cases}1 & \text { if } e_{i j} \text { is defined } \ 0 & \text { if } e_{i j} \text { is not defined }\end{cases}$$
On the other hand, in the case of weighted networks, the inner values of the adjacency matrix are real. Among the properties of a complex network, the degree distribution is one of the most relevant. Notably, this “centrality measure” constitutes a kind of signature for classifying the nature of a network (e.g., scalefree), where the term “degree” means amount of connections (i.e., edges) of a node. So, indicating with $k$ the degree of nodes, the distribution $P(k)$ of a network represents the probability to randomly select a node with a degree equal to $k$, i.e., a node with $k$ connections. A second network property is called clustering coefficient, and it allows to know if nodes of a network tend to cluster together. Actually, this phenomenon is common in many real networks as social networks, where it is possible to identify circles of friends, or acquaintances in which every person knows all the others. For the sake of clarity, considering a social network, if the user

$a$ is connected to the user $b$, and the latter is connected to the user $c$, there is a high probability that $a$ be connected to $c$. The clustering coefficient can be computed as
$$C=\frac{3 \times T n}{T p}$$
with $T n$ number of triangles in a network, and $T p$ number of connected triples of nodes. A connected triple is a single node with links running to an unordered pair of others. This coefficient has a range that spans the interval $0 \leq C \leq 1$. A further mathematical definition of the clustering coefficient reads
$$C_{i}=\frac{T n_{i}}{T p_{i}}$$
with $T n_{i}$ number of triangles connected to node $i$, and $T p_{i}$ number of triples centered on node $i$. The main difference between the two definitions is that the second one is local, so that to obtain a global value one has to compute the following parameter
$$C=\frac{1}{n} \sum_{i} C_{i}$$

## 数学代写|计算复杂度理论代写Computational complexity theory代考|Classical Random Networks

One of the early works on random networks has been developed by Paul Erdös and Alfred Renyi. Their model, usually called E-R model/graph, considers a graph with $N$ nodes and a probability $p$ to generate each edge. Accordingly, an E-R graph contains about $p \cdot \frac{N(N-1)}{2}$ edges, and it has a binomial degree distribution
$$P(k)=\left(\begin{array}{c} N-1 \ k \end{array}\right) p^{k}(1-p)^{n-1-k}$$
for $N \rightarrow$ inf and $n p=$ const, the degree distribution converges to a Poissonian distribution
$$P(k) \sim e^{-p n} \cdot \frac{(p n)^{k}}{k !}$$
To generate this kind of networks, one can implement the following simple algorithm:

1. Define the number of $N$ of nodes and the probability $p$ for each edge
2. Draw each potential-link with probability $p$
Figure $2.4$ illustrates the $P(k)$ for an E-R graph with $N=25,000$ and $p=4 \cdot 10^{-4}$.

## 数学代写|计算复杂度理论代写Computational complexity theory代考|Scale-Free Networks

Scale-free networks are characterized by the presence of few nodes (called hubs) that have many connections (i.e., a high degree), while the majority of nodes has a low degree. Therefore, these networks constitute a classical example of heterogeneous networks. The related degree distribution follows a power-law function
$$P(k) \sim c \cdot k^{-\gamma}$$
with $c$ normalizing constant and $\gamma$ scaling parameter of the distribution. A famous model for generating scale-free networks is the Barabasi-Albert model (BA model hereinafter) that considers two parameters: $N$ nodes and $m$ minimum number of edges drawn for each node. The BA model can be summarized as follows:

1. Define $N$ number of nodes and $m$ minimum number of edges drawn for each node
2. Add a new node and link it with other $m$ pre-existing nodes. Pre-existing nodes are selected according to the following equation:
$$\Pi\left(k_{i}\right)=\frac{k_{i}}{\sum_{j} k_{j}}$$
with $\Pi\left(k_{i}\right)$ probability that the new node generates a link with the $i$-th node (having a $k_{i}$ degree).

Figure $2.5$ illustrates the $P(k)$ for a scale-free network with $N=25,000$ and $m=5$.

C=3×吨n吨p

C一世=吨n一世吨p一世

C=1n∑一世C一世

## 数学代写|计算复杂度理论代写Computational complexity theory代考|Classical Random Networks

Paul Erdös 和 Alfred Renyi 开发了随机网络的早期作品之一。他们的模型，通常称为 ER 模型/图，考虑了一个图ñ节点和概率p生成每条边。因此，ER 图包含大约p⋅ñ(ñ−1)2边缘，并且它具有二项式度数分布

1. 定义数量ñ节点数和概率p对于每条边
2. 用概率绘制每个潜在链接p
数字2.4说明了磷(ķ)对于 ER 图ñ=25,000和p=4⋅10−4.

## 数学代写|计算复杂度理论代写Computational complexity theory代考|Scale-Free Networks

1. 定义ñ节点数和米为每个节点绘制的最小边数
2. 添加一个新节点并将其与其他节点链接米预先存在的节点。根据以下等式选择预先存在的节点：
圆周率(ķ一世)=ķ一世∑jķj
和圆周率(ķ一世)新节点生成链接的概率一世-th 节点（具有ķ一世程度）。

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## 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。