# Reliability Analysis of an Air Traffic Network: From Network Structure to Transport Function
## 4 Empirical Analysis
### Calculation of the CORN, TTRN and CARN  
(1) Calculation of the CORN  
We calculated the CORN of the US air traffic network based on the method mentioned in Section 3.2. There are 2924 direct routes, 6610 routes with the shortest distance of 2, and 366 routes with the shortest distance of 3. The direct rate of the network (the ratio of the number of direct routes to the number of routes) is 29.5%. More than 96% of the routes can connect any two airports by no more than one transfer. Therefore, the accessibility of the air traffic network under direct requirements is lower, while the accessibility of the air traffic network under fewer than one transit requirement is higher. In the calculation of the connectivity efficiency of any two airports, the connectivity rate of a single airport can be obtained, which reflects the hub function of the airport and the connectivity capability with other airports, and also reflects the importance of the airport in the air traffic network to a certain extent. The top 10 airports in terms of high connectivity are shown in Table 4, most of which are hub airports. From Table 3, we find that the airports with the highest connectivity rate are the airports with the highest domestic passenger volume ranking in the United States, which reflects the positive relationship between the connectivity rate and hub airport circulation capacity. To some extent, the connectivity rate reflects the capacity of the airport and the level of openness to the outside world. Then, we calculate the value of the CORN as follows:  
\[R_{con}=\delta_{con}E=(1-\frac{\sigma_{con}}{\mu_{con}})\times\mu_{con}=\mu_{ con}-\sigma_{con}=E-\sigma_{con}\approx 0.6415-0.2342=0.4073\]  
where the value of 0.2342 is the standard deviation of connectivity efficiency \(\varepsilon_{ij}\); the value of 0.6415 is the mean of \(\varepsilon_{ij}\), which can be found in the Equations (2) and Equations (3). \begin{table}
\begin{tabular}{c c c} \hline \hline
**Ranking** & **Airports** & **Connectivity** \\ \hline
1 & ATL & 92.00\% \\
2 & ORD & 92.00\% \\
3 & DEN & 90.00\% \\
4 & DFW & 84.50\% \\
5 & IAH & 84.50\% \\
6 & MSP & 84.50\% \\
7 & DTW & 83.50\% \\
8 & EWR & 83.50\% \\
9 & LAS & 83.30\% \\
10 & PHX & 82.50\% \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Top 10 airports for connectivity. The CORN is 0.4073, indicating that the US air traffic network has low connectivity reliability. This is consistent with the hub radiation structure of the US airline network. The connectivity reliability of the US air traffic network can be improved by increasing the number of direct routes. (2) Calculation of the TTRN  
The TTRN of the US air traffic network is determined based on the method in Section 3.2. Taking January 2016 as an example, there are 2921 routes, 93.46% of which are routes with travel-time reliability above 0.9 and 72.71% of which are routes with travel-time reliability above 0.95. Then, we calculate the value of the TTRN as follows:  
\[R_{tra}=\delta_{tra}\frac{1}{l}\sum_{l\in L}V_{l}=(1-\frac{\sigma_{tra}}{\mu_{ tra}})\times\mu_{tra}=\mu_{tra}-\sigma_{tra}\approx 0.9602-0.0422=0.9180\]  
where the value of 0.0422 is the standard deviation of travel-time reliability \(V\); the value of 0.9602 is the mean of \(V\). The top 10 airports in terms of high travel-time reliability are shown in Table 5.  
Comparing Tables 3 and 4, the travel-time reliability of some airports is not as high as their connectivity, which shows the disadvantage of airports with high traffic volume in terms of travel-time reliability. Then, we obtain the TTRN of every month from September 2015 to February 2016, and the results are shown in Table 6.  
As seen from Table 5, the average TTRN is very high, and the monthly variation of the TTRN is very small, which reflects the stable travel-time reliability of the US air traffic network. The travel-time reliability of November is the highest and the travel-time reliability of September is the lowest. This may because there are many thundershowers in September and it is often sunny in November.