# 统计代写|网络分析代写Network Analysis代考|Topological characteristics of networks

## 统计代写|网络分析代写Network Analysis代考|Topological characteristics of networks

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.

Average path length $(\mathrm{L})$ is one of the most robust assessmentmeasures for network topology study. It quantifies how complex real-world networks are “wired” and evolving. Moreover, the average path length is a measure of network size, and it indicates the rate of (quick) transfer of information throughout the network. Average path length of a network $\mathcal{G}(\mathcal{V}, \mathcal{E})$, is the mean distance of all possible shortest path $\left(d_{i, j}\right)$ followed between any two nodes, $v_i$ and $v_j$. Mathematically, an average path length for a directed graph can be expressed as
$$L=\frac{1}{|\mathcal{V}|(|\mathcal{V}|-1)} \sum_{i=1}^{|\mathcal{V}|} \sum_{j=1}^{|\mathcal{V}|} d_{i, j}$$
where, $d_{i, j}$ is the shortest path between any two nodes, $i$ and $j$, and $|\mathcal{V}|(|\mathcal{V}|-1)$ is the total number of expected edges. In the case of an undirected graph, where $e(i, j)=e(j, i) \forall e(i, j) \in \mathcal{E}$, the average path length for an undirected graph can be represent as
$$\mathrm{L}=\frac{2}{|\mathcal{V}|(|\mathcal{V}|-1)} \sum_{i=1}^{|\mathcal{V}|} \sum_{j=1}^{|\mathcal{V}|} d_{i, j}$$
Most of the real-world networks have a small average path length, where every node is connected through the shortest path to every other nodes. With the change in the number of nodes in a network, the average path length is also affected, but that change is not drastic.

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

Clustering coefficient $(\mathrm{cc})$ is a measure of affinity (likelihood), to which nodes in a network tends to create tightly connected group with each others. The tendency of likelihood of adjacent nodes in a network is higher in comparison to the nonadjacent nodes. There exists several alternatives [20], [10], [30] for defining clustering coefficient. Clemente et al. [5] generalized clustering coefficient measure for weighted and directed networks. Latapy et al. [19] and Opsahl [24] defined a new clustering coefficient measure for bipartite graph. However, among all, Watts and Strogatz [33] definitions of clustering coefficient is widely accepted. Furthermore, they introduced the concept of local and global, or network average clustering coefficient in their proposed approach. Local clustering coefficient $\left(\mathrm{CC}{v_i}\right)$ is the ratio of total number of edges that are present among the neighbors of a node $v_i$ to the total number of possible edges that could exist among the neighbors of $v_i$. Thus $\mathrm{CC}{v_i}$ for a directed graph is given as
$$C C_{v_i}=\frac{\sum_{j=1}^{\left|N_{v_i}\right|} \lambda\left(v_i, v_j\right)}{\left|N_{v_i}\right|\left(\left|N_{v_i}\right|-1\right)},$$
where, $\lambda\left(v_i, v_j\right)= \begin{cases}1, & \text { if }\left(v_i, v_j\right) \text { is connected, } \forall v_j \in N_{v_i}, i \neq j \ 0, & \text { otherwise. }\end{cases}$

$N_{v_i}=\left{v_k \mid e(i, k) \in \mathcal{E} \vee e(k, i) \in \mathcal{E}\right}$ is the set of adjacent nodes of $v_i$ in $\mathcal{V}$, and $\left|N_{v_i}\right|\left(\left|N_{v_i}\right|-1\right)$ is the total number of expected edges.
In the case of an undirected graph, the total number of expected edges will be $\frac{\left|N_{v_i}\right|\left(\left|N_{v_i}\right|-1\right)}{2}$, since $e(i, j)=e(j, i)$. Thus $\mathrm{CC}{v_i}$ for an undirected graph can be represent as $$C C{v_i}=\frac{2 \times \sum_{j=1}^{\left|N_{v_i}\right|} \lambda\left(v_i, v_j\right)}{\left|N_{v_i}\right|\left(\left|N_{v_i}\right|-1\right)} .$$
The average (global) clustering coefficient [33] is the mean of $K$ local clustering coefficient. Therefore the global clustering coefficient for a graph $\mathcal{G}$ can be defined as
$$\overline{C C}=\frac{\sum_{i=1}^{\mathcal{V}} \mathrm{CC}_{v_i}}{\mathcal{V}}$$
where the range of $\overline{C C}$ values lies within $0 \leq \overline{C C} \leq 1$.

# 网络分析代考

## 统计代写|网络分析代写Network Analysis代考|Topological characteristics of networks

$$L=\frac{1}{|\mathcal{V}|(|\mathcal{V}|-1)} \sum_{i=1}^{|\mathcal{V}|} \sum_{j=1}^{|\mathcal{V}|} d_{i, j}$$

$$\mathrm{L}=\frac{2}{|\mathcal{V}|(|\mathcal{V}|-1)} \sum_{i=1}^{|\mathcal{V}|} \sum_{j=1}^{|\mathcal{V}|} d_{i, j}$$

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

$$C C_{v_i}=\frac{\sum_{j=1}^{\left|N_{r_i}\right|} \lambda\left(v_i, v_j\right)}{\left|N_{v_i}\right|\left(\left|N_{v_i}\right|-1\right)},$$

$\lambda\left(v_i, v_j\right)=\left{1, \quad\right.$ if $\left(v_i, v_j\right)$ is connected, $\forall v_j \in N_{v_i}, i \neq j 0, \quad$ otherwise.

$$C C v_i=\frac{2 \times \sum_{j=1}^{\left|N_{v_i}\right|} \lambda\left(v_i, v_j\right)}{\left|N_{v_i}\right|\left(\left|N_{v_i}\right|-1\right)} .$$

$$\overline{C C}=\frac{\sum_{i=1}^{\mathcal{V}} \mathrm{CC}_{v_i}}{\mathcal{V}}$$其中的范围 $\overline{C C}$ 价值在于 $0 \leq \overline{C C} \leq 1$.

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