### 数学代写|计算复杂度理论代写Computational complexity theory代考| Network Structures

statistics-lab™ 为您的留学生涯保驾护航 在代写计算复杂度理论Computational complexity theory方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写计算复杂度理论Computational complexity theory代写方面经验极为丰富，各种代写计算复杂度理论Computational complexity theory相关的作业也就用不着说。

• 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|Small-World Networks

A small-world network is characterized by the fact that every node can be reached from any other in a small number of hops. More formally, small-world networks have a distance $L$, between two randomly chosen nodes, equal to $L \propto \ln N$. Two main properties allow to evaluate if a network has a small-world structure, i.e., a shortest average path length and a relatively high clustering coefficient. In particular, the clustering coefficient of a small-world network is higher than that of its related E-R graph, i.e., the classical random network generated with the same set of nodes. Watts and Strogatz developed a very famous algorithm, i.e., the Watts-Strogatz model (WS hereinafter), for implementing small-world networks:

1. Define a regular ring lattice with $N$ nodes, each connected to $k$ neighbors ( $k / 2$ on each side)
2. For every node $i$ take every edge $(i, j)$ with $i \leq j$ and rewire it with probability $\beta$. Rewiring is done by replacing the edge $(i, j)$ with $(i, k)$ with $k$ chosen with uniform probability from all nodes avoiding loop and edge duplication

The WS model shows an interesting behavior studying the effect of the rewiring probability $\beta$. In particular, we can start with a regular (ring) lattice setting $\beta=0$, and we can obtain a completely disordered network by increasing the value of $\beta$ up to 1 . So, at intermediate values of $\beta$, the WS model generates networks that consist of a mixture of random and regular connections, providing the network with the small-world structure. This behavior is illustrated in Fig. 2.6. To conclude, the reader can use these algorithms for generating structured populations, whose agents play an evolutionary game. In doing so, it is possible to compare the outcomes on varying the underlying topology.

## 数学代写|计算复杂度理论代写Computational complexity theory代考|Phase Transitions in the Prisoner’s Dilemma

Now, we introduce an analytical model for studying the evolution toward equilibrium in spatial games, with “memory-aware” agents, i.e., agents that accumulate their payoff over time. In particular, we focus our attention on the PD, since as previously mentioned it constitutes an emblematic example of a game whose Nash equilibrium is defection. Previous investigations showed that, under opportune conditions, in this game, it is possible to reach an equilibrium of cooperation. In particular, it has been proved that some mechanisms, as random motion, can support an agent population to become cooperative. In the proposed model, we map agents to particles of a gas so that their motion can be related to the system temperature. In doing so, we can identify a relation between the temperature and the final equilibrium of our population, explaining how it is possible to break the classical Nash equilibrium in this game. It is worth to emphasize that the underlying condition, adopted in this investigation, is that agents are able to increase their payoff over time (thus named “memory-aware” agents). Remarkably, this condition represents the major difference with most of the evolutionary game models studied by computational approaches. On the other hand, considering “memory-aware” agents makes the problem more tractable from an analytical perspective. Finally, we introduce a formalism for studying order-disorder phase transitions in these dynamics. We remind that beyond trying to understand why the random motion supports cooperation (in this game), an important goal of this investigation is to strengthen the link between EGT and statistical physics (see also Chap. 1).

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

Here, we are interested in studying the prisoner’s dilemma by an analytical approach, for the reasons above mentioned. Before introducing our model, let us remind the general form of a payoff matrix:
$C D$
$C$
$D$$\left(\begin{array}{ll}R & S \ T & P\end{array}\right)$
where the set of strategies is $\Sigma={C, D}: C$ stands for Cooperation and $D$ for Defection. In the matrix (3.1), $R$ is the gain achieved by two interacting cooperators, $T$ represents the Temptation, i.e., the payoff that an agent receives whether it defects while its opponent cooperates, $S$ the Sucker’s payoff, i.e., the gain received by a cooperator while the opponent defects, eventually $P$ the payoff of two interacting defectors. In the case of the $\mathrm{PD}$, we can set the matrix elements of (3.1) to the following values: $R=1,0 \leq S \leq-1,1 \leq T \leq 2$, and $P=0$. As stated before, during the evolution of the system, agents can change their strategy from $C$ to $D$, and vice versa, following an updating rule, as, for instance, the one named “imitation of the best,” where they imitate the strategy of their richest (i.e., fittest) neighbor.

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

1. 定义一个规则的环格子ñ节点，每个连接到ķ邻居 （ķ/2在每一侧）
2. 对于每个节点一世抓住每一个优势(一世,j)和一世≤j并用概率重新连接它b. 通过更换边缘完成重新布线(一世,j)和(一世,ķ)和ķ从所有节点中以均匀概率选择，避免循环和边缘重复

WS 模型显示了一个有趣的行为，研究了重新布线概率的影响b. 特别是，我们可以从常规（环形）晶格设置开始b=0，我们可以通过增加b最多 1 。所以，在中间值b，WS 模型生成由随机连接和规则连接混合组成的网络，为网络提供小世界结构。这种行为如图 2.6 所示。总而言之，读者可以使用这些算法来生成结构化种群，其代理人玩进化游戏。这样做，可以比较改变底层拓扑的结果。

CD
C
D(R小号 吨磷)

## 有限元方法代写

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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。