# Approximating the Temporal Neighbourhood Function of Large Temporal Graphs
## 5 Case Study: Comparison of 25 Public Transport Networks
As can be seen, in the case of the first and the last group, the two cities included in the group behave quite similarly, even if the city of Sydney seems to be more efficient in correspondence with the two peaks of its diagram at 8:00 a.m. and at 5:00 p.m. (which might be considered as the rush hours of the day). In the other three groups, instead, there is clearly a city more efficient than the others of the group (that is, Luxembourg in Group 2, Berlin in Group 3, and Adelaide in Group 4). In each of these groups, the existence of a second more efficient city (that is, Palermo, Winnipeg, and Paris) is also evident. While, in Group 2, two cities (that is, Rome and Rennes) fight for the third position, in Group 3, all of the other cities are basically equivalent. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**City** & **Stops** & **Edges** & **Day** & **R** & **G** \\ \hline Adelaide & 7548 & \(404,300\) & \(12/12/16\) & 40 & 4 \\ Berlin & 4601 & \(1,048,218\) & \(04/25/16\) & 30 & 3 \\ Brisbane & 9645 & \(392,805\) & \(12/12/16\) & 40 & 4 \\ Detroit & 5683 & \(214,863\) & \(12/12/16\) & 30 & 3 \\ Grenoble & 1547 & \(114,492\) & \(11/14/16\) & 20 & 2 \\ Kuopio & 549 & \(32,122\) & \(12/12/16\) & 10 & 1 \\ Luxembourg & 1367 & \(186,752\) & \(11/28/16\) & 20 & 2 \\ Nantes & 2353 & \(196,421\) & \(12/12/16\) & 20 & 2 \\ Paris & \(11,950\) & \(1,823,872\) & \(12/12/16\) & 35 & 4 \\ Rennes & 1407 & \(109,075\) & \(12/19/16\) & 20 & 2 \\ Sydney & \(24,063\) & \(1,265,135\) & \(12/19/16\) & 50 & 5 \\ Turku & 1850 & \(133,512\) & \(12/12/16\) & 10 & 1 \\ Winnipeg & 5079 & \(333,882\) & \(12/12/16\) & 30 & 3 \\ Belfast & 1917 & \(122,693\) & \(09/05/16\) & 30 & 3 \\ Bordeaux & 3435 & \(236,595\) & \(12/12/16\) & 30 & 3 \\ Canberra & 2764 & \(124,305\) & \(01/09/17\) & 30 & 3 \\ Dublin & 4571 & \(407,240\) & \(12/12/16\) & 20 & 2 \\ Helsinki & 6986 & \(686,457\) & \(12/12/16\) & 30 & 3 \\ Lisbon & 7073 & \(526,179\) & \(11/21/16\) & 30 & 3 \\ Melbourne & \(19,493\) & \(1,098,227\) & \(12/12/16\) & 50 & 5 \\ Palermo & 2176 & \(226,215\) & \(09/22/14\) & 20 & 2 \\ Prague & 5147 & \(670,423\) & \(12/12/16\) & 30 & 3 \\ Rome & 7869 & \(1,051,211\) & \(11/06/17\) & 20 & 2 \\ Toulouse & 3329 & \(224,516\) & \(12/12/16\) & 20 & 2 \\ Venice & 1874 & \(118,519\) & \(12/12/16\) & 20 & 2 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: The 25 cities included in the public transport network dataset [14]. The Stops column indicates the number of nodes, the Edges column the number of temporal edges, the Day column the day in which the data were collected, the R column the city’s radius (in km), and the G column the group the city belongs to. Note that we are not evaluating the whole quality of a public transport service, which of course depends on many other factors, such as robustness, safety, reliability, and customer service; instead, we are only comparing different public transport networks in terms of their temporal neighborhood functions during a day. In particular, the reachability diagram of a public transport network should only be interpreted as an indicator of the percentage of the pairs of nodes that are connected in each interval, with respect to the total number of pairs of nodes. It is also worth emphasizing the fact that, in order to use these diagrams for comparing two public transport networks of two cities, we must implicitly assume that the nodes of a network are uniformly spread in the circle of radius \(R\) corresponding to the network itself. It clearly does not make sense to compare a transportation network in which nodes are concentrated in a small sector of the circle with one in which nodes are uniformly spread, since this would turn out into a greater efficiency of the first network with respect to the second one. Unfortunately, we are not able to verify that nodes are uniformly spread in the circle, and we can only assume that this requirement is satisfied.