## 4 Clique Percolation Community Detection
### Clique Percolation Community Detection Process  
The traditional static network framework limits the studies to certain properties of these networks. For instance, it allows identifications for the bottlenecks or the clusters of destinations without measuring the dynamic characteristic of the aviation system [32]. In contrast, the clique percolation method proposes an algorithm to detect the interaction patterns of cliques. Although [PERSON] et al. worried that the number and the size of \(k\)-clique may affect the quality of the detected communities [18], the nature of the airline network limits the cliques to three-/four-node communities in most cases. Subsequently, this study mainly examined the network dynamics of three-/four-clique communities in the system. Initially, the maximum edge weight is tested as the upper limits for \(I_{0}\), as was recommended by [PERSON] et al. [33]. For instance, the maximum edge weight of 119 is set as the upper limits for AA's codeshare network, in steps of 0.1. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multicolumn{6}{c}{**Operating Network**} & \multicolumn{6}{c}{**Codeshare**} \\  & & & & & & & **Network** & & \\ \hline
**Full-Service** & **IATA** & **Nodes** & **Edges** & **Edge-to-Node Ratio** & **Density** & **Nodes** & **Edges** & **Edge-to-Node Ratio** & **Density** \\ \hline American & AA & 204 & 1165 & 5.71 & 5.63\% & 583 & 3014 & 5.17 & 1.78\% \\ Airlines & DL & 234 & 1232 & 5.26 & 4.52\% & 611 & 3318 & 5.43 & 1.78\% \\ United Airlines & UA & 211 & 1155 & 5.47 & 5.21\% & 612 & 3327 & 5.44 & 1.78\% \\ China Southern & CZ & 222 & 1531 & 6.9 & 6.24\% & 349 & 2311 & 6.62 & 3.81\% \\ Airlines & LH & 181 & 533 & 2.94 & 3.27\% & 489 & 2037 & 4.17 & 1.71\% \\ China Eastern & MU & 237 & 1711 & 7.22 & 6.12\% & 403 & 2811 & 6.98 & 3.47\% \\ Airlines & BA & 208 & 453 & 2.18 & 2.10\% & 486 & 1893 & 3.9 & 1.61\% \\ Air China & CA & 195 & 976 & 5.01 & 5.16\% & 333 & 2247 & 6.75 & 4.06\% \\ \hline \multicolumn{6}{c}{**Operating Network**} & \multicolumn{6}{c}{**Edge-to-Node**} \\
**Low-Cost** & **IATA** & **Code** & **Nodes** & **Edges** & **Ratio** & **Density** & & & \\ \hline Southwest & WN & 101 & 1492 & 14.77 & 29.54\% & & & & \\ Airlines & FR & 220 & 3707 & 16.85 & 15.39\% & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Statistics results for selected 10 airlines (excluded subsidiaries). Although the airline codeshare networks seem to be sophisticated, UA's network, for example, can be divided into a maximum of ten communities. When the optimal \(I_{0}\) is identified by the emergence of the gigantic component, a small number of communities may lead to an unstable threshold. As a result, the maximal variance (\(\chi\)) is associated with the optimal \(I_{0}\) for three-/four-clique community identifications for AA, BA, CZ, DL, and MU. It also helps to detect the three-clique communities for CA, LH, UA, and WN. Take AA as an example, for \(k~{}=~{}3\), the maximal variance equals 4.25, which leads to an optimal intensity (\(I_{0}~{}=~{}14.2\)). Among 583 airports in AA's codeshare network, 270 airports are identified and classified into three three-clique communities, while 313 nodes are isolated, including 49 nodes found in three-clique communities and 264 nodes outside cliques. LAX and SEA are identified as the shared nodes, which interconnect the coexistence of structural subgraphs in the system. Similarly, optimal \(I_{0}\) (9.1) is identified at the point of maximal variance (\(\chi~{}=~{}2.69\)) for AA's four-clique communities. The increase in optimal \(k\) witnesses the rising number of isolated nodes (385). Three hundred and eighty-two of them are outside four-node cliques and sparsely connected to the network originally. Only 198 airports are identified in the three four-clique communities, while three airports (GRU, LAX, and MIA) are detected as shared nodes. When detecting four-clique communities for CA, LH, UA, and WN, the number of communities tends to be too small to establish a stable estimate of \(\chi\).