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

Modularity measures the strength and quality of partition of a network into communities or clusters $[14,22]$. The modularity is the difference between the fraction of edges falling within a community and the expected fraction of such edges in an equivalent network, with edges placed at random. Therefore we can define the modularity measure for partitioning a network into $c$ communities as follows:
$$Q=\sum_{i=1}^c\left(e_{i i}-a_i^2\right)$$
where $e_{i i}$ is the fraction of edges in module $i$, and $a_i$ is the fraction of edges that have at least one vertice in community $i$. Mathematically, $a_i$ is defined as follows:
$$a_i=\frac{k_i}{2 m}=\sum_j e_{i j}$$
where $k_i$ is the degree of the node, and $m=\frac{1}{2} \sum_i k_i$ is the total number of links in the network. On the other hand, $e_{i j}$ is the fraction of edges with one end vertex in community $i$ and the other in community $j$, and can be represented as
$$e_{i j}=\sum_{u v} \frac{A_{u v}}{2 m},$$
where $A_{u v}$ represent the adjacency matrix for all $u \in c_i$ and $v \in c_j$. $A_{u v}=0$ means that there is no link, and $A_{u v}=1$ means that there is an edge between the two nodes $u$ and $v$.

The value of $Q$ is bounded between $-1$ and 1. In general, a large value of $Q$ corresponds to better results.

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

There are two models of a random network. Accordingly, the following are two definitions of random networks commonly used:

1. Erdors-Rényi (ER) Mode1 [9]: The ER model represents an abstract representation of a random network in which a specified probability describes the existence of an edge between each couple of nodes. Formally, a random graph, $\mathcal{G}(\mathcal{V}, P)$ is a graph with $\mathcal{V}$ nodes, where each possible edge has probability $P$ of existence. Consequently, the number of edges in such a graph is a random variable.
2. Gilbert Model [13]: In this model, each pair of $\mathcal{V}$ nodes is connected with probability $P$. Each edge is introduced to a network of nodes $\mathcal{V}$ with equal probability $P^{\mathcal{E}}(1-P)\left(\begin{array}{c}(\mathcal{V} \mid \ 2\end{array}\right)-\mathcal{E}$, where $\mathcal{E}$ denotes the number of edges in the network.

In a network, any two nodes are likely to show a high clustering coefficient if they have a common third node. However, in the case of Erdős-Rényi [9] and Gilbert [13] random model, the probability of the presence of an edge between any two nodes is independent. Therefore Erdős-Rényi random network has a low clustering coefficient. The average path length of a random network grows logarithmically with the size of the network, i.e., $\langle L\rangle \sim \ln (|\mathcal{V}|)$. In a random network [13], the probability of a node that has $k$ links follows the binomial distribution and can be defined as
$$P_k=\left(\begin{array}{c} |\mathcal{V}|-1 \ k \end{array}\right) P^k(1-P)^{|\mathcal{V}|-1-k},$$
where $\mathcal{V}$ is the set of nodes connected with probability $P$ of a random network $\mathcal{G}=(\mathcal{V}, P)$. $\left({ }^{|\mathcal{V}|-1}\right)$ represents how many different ways we could pick $k$ links from $\mathcal{V}-1$ potential links a node can have. $P_k$ and $(1-P)^{\mathcal{V}-1-k}$ represent the probability of links present $(k)$ and missing $(\mathcal{V}-1-k)$. The shape of this distribution depends on network size $(\mathcal{V})$ and the probability $(P)$. The degree distribution of the random graph follows the Poisson distribution. The average degree $(\langle k\rangle)$ specifies the connectivity within random networks, as shown in Fig. 4.5. When $\langle k\rangle<1$, there exists a large number of small subgraphs. If $\langle k\rangle \gg 1$, it indicates the presence of both large and small subgraphs in the network. The transition phase occurs when $\langle k\rangle=1$.

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

$$Q=\sum_{i=1}^c\left(e_{i i}-a_i^2\right)$$

$$a_i=\frac{k_i}{2 m}=\sum_j e_{i j}$$

$$e_{i j}=\sum_{u v} \frac{A_{u v}}{2 m},$$

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

1. Erdors-Rényi (ER) Mode1 [9]：ER 模型表示随机网络的抽象表示，其中指定的概率描述每对节点之间存 在一条边。形式上，一个随机图， $\mathcal{G}(\mathcal{V}, P)$ 是一个图 $\mathcal{V}$ 节点，每个可能的边都有概率 $P$ 的存在。因此， 这种图中的边数是一个随机变量。 $P^{\mathcal{E}}(1-P)((\mathcal{V} \mid 2)-\mathcal{E}$ ， 在哪里 $\mathcal{E}$ 表示网络中的边数。
在一个网络中，任何两个节点如果有一个共同的第三个节点，它们都可能表现出较高的聚类系数。然而， 在 Erdôs-Rényi [9] 和 Gilbert [13] 随机模型的情况下，任何两个节点之间存在边的概率是独立的。因此 Erdôs-Rényi 随机网络具有较低的聚类系数。随机网络的平均路径长度随网络大小呈对数增长，即 $\langle L\rangle \sim \ln (|\mathcal{V}|)$. 在随机网络 [13] 中，节点具有 $k$ 链接蒖循二项分布，可以定义为
$$P_k=(|\mathcal{V}|-1 k) P^k(1-P)^{|\mathcal{V}|-1-k},$$
在哪里 $\mathcal{V}$ 是与概率相连的节点集 $P$ 随机网络 $\mathcal{G}=(\mathcal{V}, P)$. ( $\left.{ }^{|\mathcal{V}|-1}\right)$ 表示我们可以选择多少种不同的方式 $k$ 来自 的链接 $\mathcal{V}-1$ 个个节点可以拥有的潜在链接。 $P_k$ 和 $(1-P)^{\mathcal{V}-1-k}$ 表示存在链接的概率 $(k)$ 和失踪 $(\mathcal{V}-1-k)$. 这种分布的形状取决于网络大小 $(\mathcal{V})$ 和概率 $(P)$. 随机图的度分布遵循泊松分布。平均学历 $(\langle k\rangle)$ 指定随机网络内的连通性，如图 $4.5$ 所示。什么时候 $\langle k\rangle\langle 1$ ，存在大量的小子图。如果 $\langle k\rangle \gg 1$ ， 它表示网络中同时存在大子图和小子图。过渡阶段发生在 $\langle k\rangle=1$.

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