### 统计代写|复杂网络代写complex networks代考| Patterns of Link Structure

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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

## 统计代写|复杂网络代写complex networks代考|Patterns of Link Structure

The above discussion has shown the importance of investigating the link structure in real world networks. One can view this problem as a kind of pattern detection. Patterns are generally viewed as expressions of some kind of regularity. What such a regularity may be, however, remains often a vague concept. It might be sensible to define everything as regular which is not random.
The structure this monograph is concerned with is a particular type of non-random structure in complex networks which is closely related to the aforementioned correlations. The section about correlations has shown that if the different types of nodes in a network are known, the link structure of the network may show a particular signature. In the majority of cases, however, the presence of different types of nodes is only hypothesized and the type of each node is unknown. The purpose of this work is to develop methods to detect the presence of different types of nodes in networks and to find the putative type of each node. A number of possible applications from various fields shall motivate the problem again.

Suppose we are given a communication network of an enterprise. Nodes are employees and links represent communication, e.g., via e-mail, between them. We may then search for “communities of practice” – employees who are particularly well connected among each other, i.e., with highly enriched in-group communication. It is then possible to compare these communities of practice to the organizational structure of the enterprise and possibly use the results in the assembly of teams for future projects. A study in this direction has been performed by Tyler et al. [26].

Novel experimental techniques from biology allow the automatic extraction of all proteins produced by an organism. Proteins are the central building blocks of biological function, but generally, proteins do not function alone but bind to one another to form complexes which in turn are capable of performing a particular function, such as initiating the transcription of a particular piece of DNA. It is now possible to study the pairwise binding interactions of a large number of proteins in an automated way $[27]$. The result of such a study is a protein interaction network in which the links represent pairwise interactions between proteins. Protein function should be mirrored in such a network. For instance, proteins forming part of a complex should now be detectable as densely interlinked groups of nodes in such a network [28]. An analysis of the structure of a protein interaction or other biological network

created by automated experiments hence presents a first step in planning future experiments [29].

The collection of scientific articles represent a strongly fragmented repository of scientific knowledge $[30,31]$. Online databases make it possible to study these in an automated way, e.g., in form of co-authorship networks or citation networks. In the former, nodes are researchers while links represent co-authorship of one or more articles. Analysis of the structure of this network may give valuable information about the cooperation between various scientists and aid in the evaluation of funding policy or influence future funding decisions. In the latter, nodes in the network are scientific articles and links denote the citation of one from the other. Analysis of the structure of this network may yield insight into the different research areas of a particular field of science.

## 统计代写|复杂网络代写complex networks代考|Positions, Roles and Equivalences

By investigating data from a wide range of sources encompassing the life sciences, ecology, information and social sciences as well as economics, researchers have shown that an intimate relation between the topology of a network and the function of the nodes in that network indeed exists [1-9]. A central idea is that nodes with a similar pattern of connectivity will perform a similar function. Understanding the topology of a network will be a first step in understanding the function of individual nodes and eventually the dynamics of any network.

As before, we can base our analysis on the work done in the social sciences. In the context of social networks, the idea that the pattern of connectivity is related to the function of an agent in the network is known as playing a “role” or assuming a “position” $[10,11]$. Here, we will endorse this idea.

The nodes in a network may be grouped into equivalence classes according to the role they play. Two basic concepts have been developed to formalize the assignments of roles individuals play in social networks: structural and regular equivalence. Both are illustrated in Fig. 2.1. Two nodes are called structurally equivalent if they have the exact same neighbors [12]. This means that in the adjacency matrix of the network, the rows and columns of the corresponding nodes are exactly equal. The idea behind this type of equivalence is that two nodes which have the exact same interaction partners can only perform the exact same function in the network. In Fig. 2.1, only nodes $A$ and $B$ are structurally equivalent while all other nodes are structurally equivalent only to themselves.

To relax this very strict criterion, regular equivalence was introduced [13, 14]. Two nodes are regularly equivalent if they are connected in the same way to equivalent others. Clearly, all nodes which are structurally equivalent must also be regularly equivalent, but not vice versa. The seemingly circular definition of regular equivalence is most easily understood in the following way: Every class of regularly equivalent nodes is represented by a single node in an “image graph”. The nodes in the image graph are connected (disconnected), if connections between nodes in the respective classes exist (are absent) in the original network. In Fig. $2.1$, nodes $A$ and $B, C$ and $D$ as well as $E$ and $F$ form three classes of regular equivalence. If the network in Fig. $2.1$ represents the trade interactions on a market, we may interpret these three classes as producers, retailers and consumers, respectively. Producers sell to retailers, while retailers may sell to other retailers and consumers, which in turn only buy from retailers. The image graph (also “block” or “role model”) hence gives a bird’s-eye view of the network by concentrating on the roles, i.e., the functions, only. Note that no two nodes in the image graph may be structurally equivalent, otherwise the image graph is redundant.

## 统计代写|复杂网络代写complex networks代考|Block Modeling

Let us consider the larger example from Fig. 2.2. The network consists of 18 nodes in 4 designed roles. Nodes of type A connect to other nodes of type $A$ and to nodes of type $B$. Those of type $B$ connect to nodes of type $A$ and $C$, Fig. 2.2. An example network and two possible block models. The nodes in this network can be grouped into four classes of regular equivalence ( $A, B, C$ and D). Ordering the rows and columns of the adjacency matrix according to the four classes of regular equivalence makes a block structure apparent (there are 16 blocks from the 4 classes), which is efficiently represented by an image graph.

acting as a kind of intermediary. Nodes of type C have connections to nodes of type $B$, others of type $C$ and of type $D$. Finally, nodes of type $D$ form a kind of periphery to nodes of type C. An ordering of rows and columns according to the types of nodes makes a block structure in the adjacency matrix apparent. Hence the name “block model”. The image graph effectively represents the 4 roles present in the original network and the 16 blocks in the adjacency matrix. Every edge present in the network is represented by an edge in the image graph and all edges absent in the image graph are also absent in the original network.

Regular equivalence, though a looser concept than structural equivalence, is still very strict as it requires the nodes to play their roles exactly, i.e., each node must have at least one of the connections required and may not have any connection forbidden by the role model. In Fig. 2.2, a link between two nodes of type A may be removed without changing the image graph, but an additional link from a node of type $A$ to a node of type $C$ would change the role model completely. Clearly, this is unsatisfactory in situations where the data are noisy or only approximate role models are desired for a very large data set.

## 广义线性模型代考

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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。