# 统计代写|网络分析代写Network Analysis代考|CSCl5352

## 统计代写|网络分析代写Network Analysis代考|Complex graphs

Over the past few decades, large real-world systems, such as WWW, Internet, social networks, wireless networks, supply-chain networks, are often modeled as a graph for the ease of computational analysis. The interaction relationships across different entities or macromolecules in the biological systems, such as protein-protein interactions (PPI), gene regulation and association, signaling pathways, metabolic activities, neuronal connectivity of a brain, etc., are also modeled as graphs or networks. However, such graphs are not simple graphs as regular graphs. Due to their unconventional topological properties, they are often treated as complex graphs or networks. Unlike conventional graphs, realworld networks exhibit non-trivial topological properties, such as varying degree distributions, high or low clustering coefficients, degree assortativity or average path length of the network. Accordingly, networks are classified into different models.

Before discussing various network models, it is important to understand the topological characteristics of any complex network. Usually, network models are defined based on such characteristics only. Thus we discuss topological characteristics first before introducing available models.

In a network, the interconnection patterns among the nodes are termed as network topology. The varying topological properties of any complex networks make the task of network comparison and classification a challenging activity. Therefore a set of summary statistics or quantitative performance measures are important to describe and compare the complex networks. In the last few years, many quantities and measures are proposed and investigated for complex network analysis. However, among all, three measures, namely average path length $(L)$ [2], clustering coefficient (Cc) $[16,33]$, and degree distribution $\left(P_k\right)[1,3]$ play a key role in complex network analysis. Next, we discuss different topological characteristics considered for any complex networks.

## 统计代写|网络分析代写Network Analysis代考|Rich club coefficient

The rich-club coefficient, introduced by Zhou and Mondragon in the context of the Internet topology [36], refers to the tendency of high-degree nodes (i.e., the hubs) in the network, to be very well-connected to other hub nodes. The name “rich-club” arises from the metaphor that the nodes with a large number of links, i.e., the hubs are “rich”, and they tend to be tightly and wellinterconnected between themselves, forming subgraphs called “club”. The rich-club coefficient is nothing but the measure of connectedness density within the club. A network with a rich club organization is shown in Fig. $4.4$ for better understanding.

The nodes in a network can be categorized by a ranking scheme [36] or by their degree [8]. The rank $r$ of a node represents the corresponding position of the node in the list of descending order of node degrees, i.e., the most highly-connected node is ranked as $r=1$, the second best-connected node is $r=2$, and so on. The density of connections between the $r$ richest nodes is evaluated by the rich-club coefficient [36],
$$\Phi(r)=\frac{2 E(r)}{r(r-1)},$$
where $E(r)$ is the total number of links between $r$ hub nodes and $r(r-1) / 2$ is the maximum possible number of links among these nodes. Similarly, the rich-club coefficient [8] in terms of node de-gree can be represented as follows:
$$\Phi(k)=\frac{2 E_k}{N_k\left(N_k-1\right)},$$
where $E_k$ is the number of links present between the nodes of degree greater than or equal to $k$, and $N_k$ is the number of such nodes. Therefore, $\Phi(k)$ measures the fraction of actual links connecting those nodes and the maximum number of possible links. This measure explicitly reflects how densely connected are the nodes within a network.

The behavior of the rich-club coefficient is proportional to the value of $k$. It means, a rich-club coefficient increasing with the degree $k$ indicates that there exists a rich-club of nodes, which are densely interconnected than the nodes with smaller degrees. Contrarily, a decrease in the value of $\Phi(k)$ indicates the presence of many loosely connected and relatively independent subgroups. It is known as rich-club phenomenon.

## 统计代写|网络分析代写Network Analysis代考|Rich club coefficient

$$\Phi(r)=\frac{2 E(r)}{r(r-1)},$$

$$\Phi(k)=\frac{2 E_k}{N_k\left(N_k-1\right)},$$

Rich-club 系数的行为与 $k$. 这意味着，一个rich-club系数随着度数的增加而增加 $k$ 表示存在丰富的节点倶乐 部，与度数较小的节点相比，这些节点之间的互连更密集。反之，价值下降 $\Phi(k)$ 表示存在许多松散连接且 相对独立的子群。它被称为富人倶乐部现象。

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