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# 数据科学代写|复杂网络代写Complex Network代考|TSKS33 Cores and controllability

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## 数据科学代写|复杂网络代写Complex Network代考|Cores and controllability

A matching (independent edge set) of an undirected graph is a subset of its edges having no common vertices (Hartmann and Weigt, 2006). A vertex cover of an undirected graph is a subset of its vertices such that any edge of the graph is adjacent to at least one vertex from this subset. Finding a maximum matching (containing the largest possible number of edges) and a minimum vertex cover of a graph are among key combinatorial optimization problems. Perfect matching (a matching that matches all vertices) is related to statistics of dimers in physics, while the minimum vertex cover problem has a vivid representation as the task of hiring the minimum number of guards that succeed to watch all corridors in a museum (Weigt and Hartmann, 2000). For many important graphs, including the bipartite graphs, the maximum matching problem and the minimum vertex cover one are equivalent and can be solved rapidly, in a polynomial time. The solution can be found by the Karp-Sipser greedy leaf removal algorithm (Karp and Sipser, 1981). The leaves (vertices of degree 1) and their nearest-neighbouring vertices are progressively removed from an undirected graph in any order until it is possible. The edges of the removed dimers during this process form a maximum matching. It turns out that in sufficiently dense graphs, at some point of this leaf removal process, no leaves remain in a still not eliminated graph, the process becomes stacked, and extra efforts have to be applied to proceed further and complete the algorithm. In this situation, the algorithm slows down and it is an NP-complete problem. ${ }^{55}$ In sparse networks, this problem is treatable in the framework of statistical physics by the cavity method-belief propagation (Zhou and Ou-Yang, 2003; Zdeborová and Mézard, 2006), and in the simplest case of an Erdös-Rényi random graph, the boundary between these two regimes occurs at the mean degree $\langle q\rangle=e=2.718 \ldots$ The same two regimes and the same boundary $\langle q\rangle=e$ in the case of an Erdös-Rényi random graph were found in the minimum vertex cover problem AKA searching for the maximum number of museum guards (Weigt and Hartmann, 2000).

The subgraph remaining after the completion of the greedy leaf removal algorithm is called the core of a graph (Figure 6.32). Bauer and Golinelli (2001a) developed the theory of cores in undirected uncorrelated networks, enabled them to compute the thresholds of the core percolation, the core sizes, and their characteristics. ${ }^{56}$ Liu, Csóka, Zhou, and Pósfai (2012) extended this theory to the important case of directed uncorrelated networks. Let us outline the idea of these works in the more compact case of undirected uncorrelated networks.

## 数据科学代写|复杂网络代写Complex Network代考|Bootstrap percolation

In this chapter we mainly focus on the results of activation processes in networks and on various combinations of activation and deactivation processes.
The bootstrap percolation problem is about the basic activation process on networks, in which vertices can be in active and inactive states. A vertex becomes active when the number of its active neighbours exceeds some threshold; and once active, a vertex never becomes inactive (Adler and Aharony, 1988; Adler, 1991). This is one of the spreading processes with discontinuous phase transitions (Bizhani, Paczuski, and Grassberger, 2012). Let us define bootstrap percolation on undirected graphs in more strict terms. In the initial state, a fraction $f$ of vertices is active (seed vertices). These vertices are chosen uniformly at random. Each inactive vertex becomes active if it has at least $k_{\mathrm{b}}$ active nearest neighbours. Here we introduce the subscript ‘b’ to distinguish this threshold from a threshold in the $k$-core percolation problem. In heterogeneous bootstrap percolation, the thresholds are individual for different vertices, $k_{\mathrm{b}, i}$, where $i=1,2, \ldots, N$. The main point of the interest is the final state of the process, in which the fraction $S_{\mathrm{b}} \geq f$ of all vertices are active and, assuming that a network is infinite, the fraction $S_{G \mathrm{~b}} \leq S_{\mathrm{b}}$ of vertices with their connections form a giant active component. It is reasonable to compare bootstrap percolation with another threshold process already discussed in Chapter 6 , namely, with the heterogeneous $k$-core problem. Speaking in terms of active and inactive vertices, a $k$-core is the final result of the progressive deactivation of initially active vertices, where each vertex $i$ becomes inactive if it has less than $k_{\mathrm{c}, i}$ active nearest neighbours. These two processes are complementary if the individual thresholds $k_{\mathrm{b}, i}$ and $k_{\mathrm{c}, i}$ for each vertex $i$ in a network are related with each other in the following way:
$$k_{\mathrm{b}, i}=q_i+1-k_{\mathrm{c}, i}$$

where $q_i$ is the degree of vertex $i$ (Miller, 2016; Di Muro, Valdez, Stanley, Buldyrev, and Braunstein, 2019). One can easily obtain this relation. The point is that an active vertex at each moment of the bootstrap percolation process is inactive at the corresponding moment of the complementary $k$ core process. ${ }^1$ So at this moment, the numbers of active nearest neighbours of vertex $i, a_{\mathrm{b}, i}$ for bootstrap percolation and $a_{\mathrm{c}, i}$ for the $k$-core process, should be related, $a_{\mathrm{c}, i}=q_i-a_{\mathrm{b}, i}$. The condition of the activation of inactive vertex $i$ in the heterogeneous bootstrap percolation is $a_{\mathrm{b}, i} \geq k_{\mathrm{b}, i}$ while the condition of the deactivation of active vertex $i$ in the heterogeneous $k$-core problem is $a_{\mathrm{c}, i}=q_i-a_{\mathrm{b}, i} \leq k_{\mathrm{c}, i}-1$. These two inequalities coincide when the equality in Eq. (7.1) is true.

## 数据科学代写|复杂网络代写Complex Network代考|Bootstrap percolation

(Bizhani、Paczuski 和 Grassberger，2012 年) 。让我们用更严格的术语定义无向图上的自举 渗透。在初始状态下，分数 顶点数处于活动状态 (种子顶点) 。这些顶点是随机均匀选择的。如果 每个非活动顶点至少有 $k_{\mathrm{b}}$ 活跃的最近邻。这里我们引入下标’ $b^{\prime}$ 来区分这个阈值和 $k$-核心渗透问题。 在异构引导渗透中，阈值对于不同的顶点是独立的， $k_{\mathrm{b}, i}$ ，在哪里 $i=1,2, \ldots, N$. 感兴趣的要点 是过程的最终状态，其中分数 $S_{\mathrm{b}} \geq f$ 的所有顶点都是活动的，并且假设网络是无限的，分数 $S_G \mathrm{~b} \leq S_{\mathrm{b}}$ 顶点及其连接形成了一个巨大的活动组件。将 bootstrap 渗流与第 6 章中已经讨论过的 另一个阈值过程进行比较是合理的，即与异构 $k$-核心问题。就活动和非活动顶点而言， $k$-core 是初 始活动顶点逐斩停用的最终结果，其中每个顶点 $i$ 如果它少于 $k_{\mathrm{c}, i}$ i舌跃的最近邻。如果单独的闾值，这
$$k_{\mathrm{b}, i}=q_i+1-k_{\mathrm{c}, i}$$

## 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。