# Approximating the Temporal Neighbourhood Function of Large Temporal Graphs
## 5 Case Study: Comparison of 25 Public Transport Networks
One could suspect that results similar to the ones shown in the left column of Figure 6 could be obtained by simply considering the temporal density of the original transport networks. For each interval \(\mathbb{I}=[t_{\alpha},t_{\omega}]\), let \(m(\mathbb{I})\) denote the number of edges from \(u\) to \(v\) with both starting and arrival time in \(\mathbb{I}\). We then define the temporal density in the interval \(\mathbb{I}\) as \(\delta(\mathbb{I})=\frac{m(\mathbb{I})}{n}\), where \(n\) denotes the number of nodes of the network. In the right column of Figure 6, we show, for each city group, the _density diagrams_ of all cities included in the group. As it can be seen, in Groups 1, 2, 3 and 5, the first city is the same as in the reachability diagrams. In Group 4, instead, Paris seems to have the densest public transport network, even if it is not the first city in terms of reachability cone sizes. Moreover, while the reachability and the density diagrams of Groups 1 and 5 seem to be quite consistent, this is not the case for Groups 2 and 3. For instance, in Group 2, Rome is denser than Palermo, but Palermo outperforms Rome in terms of reachability, while, in Group 3, this phenomenon happens in the case of Prague and Winnipeg. In other words, the reachability diagram seems to provide some information that is not explicitly given by the density diagram. It is also worth observing how several cities present two peaks in their reachability and density diagrams, which we can assume correspond to the two rush hours of the day. This seems to  
Figure 5: The reachability diagram of the public transport network of the city of Adelaide, in which the interval between 6:00 a.m. and 9:00 p.m. has been split into 15 intervals of one hour each. indicate that these cities, in correspondence with these rush hours, increase the number of connections and provide a better service in terms of the number of pairs of connected nodes. Figure 6: The comparison of the reachability diagrams (**left**) and of the density diagrams (**right**) of the 25 public transport networks. In Figure 7, instead, we compare the reachability diagrams of the five groups obtained with time intervals of length 30 and 90 min, respectively (these diagrams should also be compared to the reachability diagrams shown in the left column of Figure 6, which are obtained with time intervals of length 60 min). It is worth observing that, in the second and in the third group, considering shorter time intervals makes the city with the better diagram perform even better compared to the others, and it basically nullifies the differences among the other cities. When larger time intervals are analyzed, it seems that the only effect we get is that the reachability diagrams are systematically shifted up. This is quite reasonable, since we can expect that, in a time interval of one hour and half, a significant percentage of all possible pairs of nodes are now connected. It is anyway worth noting that, in the case of the three cities in Group 4 and of the two cities in Group 5, even with time intervals of 90 min, this percentage is quite low (less than 15% in Group 4 and less than 3% in Group 5). This suggests that, in order to obtain a globally almost total connectivity, these cities require trips of quite great duration. Finally, in Figure 8, we show the reachability diagrams of three sample cities obtained with different values of \(k\) in atnf (the number of experiments is always 20). As it can be seen, the diagrams are almost identical, which might suggest that even \(k=32\) could be used in order to reduce the execution time. However, we decided to use \(k=64\) in the previously described experiments in order to further increase the approximation quality of the results. As we said at the beginning of this section, the main goal of this analysis has been to describe a case study in which, by using atnf, we can extremely reduce the total execution time of the experiments.