## 2 Complex Network Model and Related Indexes
### Statistical Indexes of Complex Network Characteristics  
Small-world characteristics and scale-free characteristics are the basis for distinguishing general networks from complex networks. The moderate, clustering coefficient, network efficiency, centrality, path length, etc., are the metrics they commonly use.  
The degree \(k_{i}\) is an index that can measure the importance of node \(i\) in a complex network. It represents the number of edges connected with point \(i\), as shown in Table 1 Formula (1).  
The clustering coefficient \(C_{i}\) of node \(i\) is a proportion, which is the number of adjacent nodes' connected edges in the maximum possible number of connected edges. It can reflect the connectivity efficiency of the local network around node \(i\), as shown in Table 1 Formula (2). The mean value of all node clustering coefficients of all node sets V in the network is the global clustering coefficient \(C\), as shown in Table 1 Formula (3).  
There are \(m\) paths between two nodes \(i\) and \(j\) in a network, \(d_{ij}\) is the minimum of all path lengths, then \(d_{ij}\) is called the shortest path length between nodes \(i\) and \(j\), as shown in Table 1 Formula (4). Network efficiency is a measure of connectivity efficiency between nodes, the reciprocal of the path length between nodes \(i\) and \(j\) is the node pair efficiency,  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Indexes** & \multicolumn{2}{c}{**Formulas**} \\ \hline Degree & \(k_{i}=\sum\limits_{j}a_{ij}\) & (1) \\ Node clustering coefficient & \(C_{i}=E_{i}/C^{2}{}_{k_{i}}\) & (2) \\ Global clustering coefficient & \(C=\sum\limits_{i=1}^{N}C_{i}/N\) & (3) \\ Average path length & \(D=\sum\limits_{i\
eq j}d_{ij}/N(N-1)\) & (4) \\ Network efficiency & \(E(G)=\sum\limits_{i\
eq j\in G}(1/d_{ij})/N(N-1)\) & (5) \\ Degree centrality & \(C_{i}^{D}=k_{i}/(N-1)\) & (6) \\ Near-centrality & \(C_{i}^{C}=(N-1)/\sum\limits_{i\
eq j}d_{ij}\) & (7) \\ Intermediate centrality & \(C_{i}^{B}=\sum\limits_{s\
eq i\
eq v}\sigma_{st}(i)/\sigma_{st}\) & (8) \\ Connectivity & \(r=e^{G}/e^{G}{}_{\max}\) & (9) \\ Network diameter & \(L=\max{d_{ij}}\) & (10) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Formulas for calculating static indexes.  
Figure 1: Representations of some subway networks. (**a**) Original network. (**b**) Space L network. (**c**) Space P network. (**d**) Space C network.  
and in a network, the average of all node pairs efficiency is the network efficiency, as shown in Table 1 Formula (5).  
The more prominent the \"central position\" of a node is, the more important its position in a network is. Degree centrality \(C_{i}^{D}\), near-centrality \(C_{i}^{C}\) and intermediate centrality \(C_{i}^{B}\) are three important indexes of node centrality. Degree centrality is the number of nodes connected to it in a network. Near-centrality is the average shortest distance between a node and all other nodes in a network. Intermediate centrality is the number of shortest paths between nodes passing through that node, as shown in Table 1 Formulas (6) to (8).  
The connectivity is an index to measure the degree of network development, and the ratio of the actual number of edges to the maximum possible number of edges, as shown in Table 1 Formula (9). \(d_{ij}\) is the shortest path length between two points, the network diameter \(L\) is the maximum value in \(d_{ij}\), as shown in Table 1 Formula (10). The relevant parameters described above are shown in Table 2.