# 计算机代写|复杂网络代写complex network代考|CS7280

#### Doug I. Jones

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## 计算机代写|复杂网络代写complex network代考|Comparison with Other Definitions of Communities

In Sect. 4.1.1 the term community was defined as a set of nodes having properties (i) through (iii). Compared with the many definitions of community in the sociological literature [6], this definition is most similar to that of an “LS set”. Recall, LS set is a set of nodes $S$ in a network such that each of its proper subsets has more links to its complement in $S$ than to the rest of the network [7]. Note, however, that the problem in the definition of an LS set mentioned in Sect. 2.3.1 does not occur.

Previously, Radicchi et al. [8] had given a definition of community “in a strong sense” as a set of nodes $V$ with the condition $k_i^{i n}>k_i^{\text {out }}, \forall i \in V$, i.e., every node in the group has more links to other members of the group than to the rest of the network. In the same manner, they define a community in a “weak sense” as a set of nodes $V$ for which $\sum_{i \in V} k_i^{i n}>\sum_{i \in V} k_i^{\text {out }}$, i.e., the total number of internal links is larger than half of the number of the external links, since the sum of $k_i^{i n}$ is twice the number of internal edges. The similarity with properties (1) and (2) of the new definition is evident, but instead of comparing absolute numbers for single nodes, the new definition compares absolute numbers to expectation values for these quantities in the form of the coefficients of cohesion and adhesion not only for single nodes but also for sets of nodes. As already discussed in Sect. 2.3.2, one of the consequences of Radicchi et al.’s definitions is that every union of two communities is also a community. This leads to the strange situation that a community in the “strong” or “weak” sense can also be an ensemble of disjoint groups of nodes. This paradox may only be resolved if one assumes a priori that there exists a hierarchy of communities. The following considerations and examples will show that hierarchies in community structures are possible, but cannot be taken for granted. The representation of community structures by dendrograms, therefore, cannot always capture the true community structure and hence all hierarchical community detection algorithms should be used with caution.

## 计算机代写|复杂网络代写complex network代考|Hierarchy and Overlap of Community Assignments

Even though hierarchical community structures cannot be taken for granted and hence should not be enforced by using hierarchical community detection algorithms, they still form an important organizational principle in networks which shall be investigated directly from the adjacency matrix. When ordering the rows and columns according to the assignment of nodes into communities, the link density in the adjacency matrix is directly transformed into point density and hence into gray levels. Since the inner link density of a community is higher than the external, one can distinguish communities as square blocks of darker gray. Different orderings may be combined into a consensus ordering. That is, starting from a super-ordering given, the nodes within each community are reordered according to a second given sub-ordering, i.e., one only changes the internal order of the nodes within communities of the superordering. This leads to the formation of new blocks of those nodes that are assigned together in one community in both orderings. One can then repeat the procedure to obtain further iterative consensus orderings.

First, an example of a completely hierarchical network is given very similar to that used in Ref. [11]. Here, hierarchy implies that all communities found at a value of $\gamma_2>\gamma_1$ are proper sub-communities of the communities found at $\gamma_1$. In the example, a network made of four large communities of 128 nodes each was constructed. Each of these nodes has an average of 7.5 links to the 127 other members of their community and 5 links to the remaining 384 nodes in the network. Each of these 4 communities is composed of 4 sub-communities of 32 nodes each. Each node has an additional 10 links to the 31 other nodes in its sub-community. Figure 4.3 shows the adjacency matrix of this network in different orderings.

# 复杂网络代写

## 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。

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