# Finding Top-\(k\) Nodes for Temporal Closeness in Large Temporal Graphs
## Appendix A Further Experiments  
In the following tables we will show our experimental test-bed and the results we obtained for all the graphs. In Table 11, we report the full list of our graphs, with their number of nodes and edges. Moreover, consistently with respect to Table 1, we also report the running time of exact and the average running time of apx-1024 among 50 experiments. Recall that the running times of apx-\(h\), for any other value of \(h\), can be obtained as \(h\cdot t/1024\), where \(t\) is the running time of apx-1024. In Table 11, we report the average RE (together with the coefficient of variation) achieved by apx-256, apx-512, apx-1024, for the top-100 nodes, according to the exact ranking, and for the remaining nodes. The results largely confirm what we have shown in Figure 5, namely that the RE for top-nodes is almost always very small if compared to the RE of all the other nodes. The graphs in the upper part are undirected. In Table 11, consistently with respect to Figure 6, we show the average [PERSON]'s \(\tau\) for all the graphs, comparing the ranking found by apx-\(h\) for \(h=32,64,128,256,512,1024\) with the exact ranking (except for the twitter graph, where we refer to the ranking computed by apx-1024). Finally, in Table 11, similarly to Table 2, we report the maximum position of the top-\(k\) nodes (for the exact ranking) in the approximate ranking computed by apx-\(h\) (over 50 experiments), with \(k=1,5,10,20,100\) and \(h=256,512,1024\). As we have estimated, obtaining the exact ranking and closeness of twitter requires more than three years. For this reason, we were not able to provide these results for this graph, so that it has been excluded from Tables 11 and 12. \begin{table}
\begin{tabular}{l r r r r} \hline \hline \multicolumn{5}{c}{**Undirected Graphs**} \\ \hline
**Name** & **Nodes** & **Edges** & **exact** & **apx-1024** \\ \hline music & 16,417 & 36,217 & 346 & 21 \\ musical & 21,102 & 66,853 & 971 & 33 \\ mystery & 34,787 & 87,086 & 1863 & 33 \\ scifi & 24,551 & 54,578 & 916 & 30 \\ war & 19,690 & 51,980 & 617 & 27 \\ western & 11,344 & 58,230 & 382 & 29 \\ \hline \multicolumn{5}{c}{**Directed Graphs**} \\ \hline
**Name** & **Nodes** & **Edges** & **exact** & **apx-1024** \\ \hline election & 7119 & 103,675 & 201 & 34 \\ facebook & 46,952 & 876,993 & 12,184 & 264 \\ twitter * & 3,511,241 & 16,438,790 & 97,553,304 & 28,449 \\ linux & 6340 & 1,096,440 & 19,313 & 317 \\ adelaide & 7548 & 404,300 & 889 & 143 \\ belfast & 1917 & 122,693 & 62 & 33 \\ berlin & 4601 & 1,048,218 & 1358 & 352 \\ bordeaux & 3435 & 236,595 & 231 & 68 \\ brisbane & 9645 & 392,805 & 1051 & 110 \\ canberra & 2764 & 124,305 & 95 & 35 \\ detroit & 5683 & 214,863 & 350 & 63 \\ dublin & 4571 & 407,240 & 527 & 117 \\ grenoble & 1547 & 114,492 & 46 & 30 \\ helsinki & 6986 & 686,457 & 1342 & 196 \\ kuopio & 549 & 32,122 & 5 & 8 \\ lisbon & 7073 & 526,179 & 1019 & 167 \\ luxembourg & 1367 & 186,752 & 70 & 52 \\ melbourne & 19,493 & 1,098,227 & 6258 & 380 \\ nantes & 2353 & 196,421 & 126 & 55 \\ palermo & 2176 & 226,215 & 142 & 66 \\ paris & 11,950 & 1,823,872 & 6149 & 550 \\ prague & 5147 & 670,423 & 947 & 190 \\ rennes & 1407 & 109,075 & 42 & 30 \\ rome & 7869 & 1,051,211 & 2451 & 364 \\ sydney & 24,063 & 1,265,135 & 8635 & 411 \\ toulouse & 3329 & 224,516 & 204 & 63 \\ turku & 1850 & 133,512 & 69 & 38 \\ venice & 1874 & 118,519 & 59 & 32 \\ winnipeg & 5079 & 333,882 & 492 & 99 \\ \hline \hline \end{tabular}
\end{table}