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数据科学代写|复杂网络代写Complex Network代考|COMP5313 Using message passing

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数据科学代写|复杂网络代写Complex Network代考|Using message passing

Message-passing, in particular, belief-propagation, algorithms are widely used for a rapidly expanding range of problems: treatment of various probabilistic graphical models, spreading processes including disease spreading (Altarelli, Braunstein, Dall’Asta, Lage-Castellanos, and Zecchina, 2014a; Altarelli, Braunstein, Dall’Asta, Wakeling, and Zecchina, 2014b), cooperative systems – Bethe-Peierls approximation, cavity method, and others. One may ask when these techniques are computationally more efficient than direct simulations and when they are less efficient. For percolation problems, the update equations of the message-passing algorithms contain $p$, and so these algorithms enable one to escape averaging over different configurations of removed vertices or edges, in contrast to simulations, saving time. This is a significant advantage. Let us, however, estimate the number of operations required to solve the message-passing equations by iterations (computational complexity). Roughly, this number is of the order of $N\langle q\rangle \times\langle\ell\rangle(N) \sim N \ln N$ (see, e.g., Timár, da Costa, Dorogovtsev, and Mendes, $2017 a$ for details). On the other hand, to determine in simulations whether a vertex belongs to the giant (actually, the largest) connected component or not, we must explore the neighbourhood of this vertex. For that, we must visit a number of vertices exceeding the typical size of a finite connected component by a factor determined by the precision with which we want to find the size of the giant component. The typical cluster size depends on the distance from the critical point but not on the system size. Consequently, at a given distance from the critical point, for a given desired accuracy, the size of the giant component can be obtained in simulations in a constant time independent of $N$. Thus, at least for large network sizes, direct simulations are more computationally effective than message-passing algorithms.

数据科学代写|复杂网络代写Complex Network代考|Beyond tree-likeness

Section $6.3$ explained that the application of the message passing algorithm to a finite graph can be treated as replacing of the original loopy graph by its infinite non-backtracking expansion (Figure 6.6b). This figure allows us to grasp the major drawback of this approximation falsely giving the infinite diameter for any finite graph if it is only not a tree. Due to neglecting finite cycles, the non-backtracking expansion contains numerous (actually infinite number) replicas of the vertices of the original loopy graph. Hence the update equations of message-passing algorithms, in particular, Eq. (6.48), overestimate sizes of finite sets of vertices to which edges lead. Karrer, Newman, and Zdeborová (2014) showed that this overestimation leads to the following inequality valid for any network:
$$p_c^{(\text {exact })} \geq \frac{1}{\lambda_1},$$
where $p_c^{(\text {exact })}$ is the exact value of the percolation threshold for a network, and $\lambda_1$ is the largest eigenvalue of the non-backtracking matrix. ${ }^{12}$ In other words, the message-passing approximation provides the lower bound for the percolation threshold. ${ }^{13}$

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。