# Finding Top-\(k\) Nodes for Temporal Closeness in Large Temporal Graphs
## 6 Conclusions  
We proposed a sampling-based approximation algorithm for the temporal closeness centrality measure, and we experimentally showed that this algorithm can be extremely efficient in computing the top-\(k\) nodes in real-world temporal graphs. An interesting open question is to understand why, in the case of few graphs, our method is not as efficient as in the case of all the others: some preliminary experimental results suggest that this might happen because all nodes have basically the same temporal closeness which is very small. In order to attack this problem, we believe that it would be interesting to study the performances of our algorithm on random temporal graphs. Notice, however, that there is yet no consensus on how to generate random temporal graphs or which features to select to make the random selection on, see for instance [50] and references within. Moreover, an interesting future research line is to explore the extension and application of our approach (by still referring to [32]) to the case in which temporal edges have a traveling time. Finally, it would be worth exploring the possibility of applying to the temporal closeness the approach of [10; 51] for static graphs, which basically consists of executing breadth-first searches starting from all the nodes of the graph and in \"cutting\" this visits as soon as it can be deduced (by using some appropriate bounds on the static closeness value) that the source of the visit is not among the top-\(k\). \begin{table}
\begin{tabular}{l c c c c c c c c c c c c c c c} \hline \hline  & \multicolumn{3}{c}{\(k=1\)} & \multicolumn{3}{c}{\(k=5\)} & \multicolumn{3}{c}{\(k=10\)} & \multicolumn{3}{c}{\(k=20\)} & \multicolumn{3}{c}{\(k=100\)} \\ \cline{2-13}
**Name** & \multicolumn{3}{c}{\(h\)} & & \multicolumn{3}{c}{\(h\)} & & \multicolumn{3}{c}{\(h\)} & & \multicolumn{3}{c}{\(h\)} & & \multicolumn{3}{c}{\(h\)} \\ \cline{2-13}  & **256** & **512** & **1024** & **256** & **512** & **1024** & **256** & **512** & **1024** & **256** & **512** & **1024** & **256** & **512** & **1024** \\ \hline Fant & 1072 & 382 & 47 & 2117 & 954 & 396 & 2859 & 1262 & 396 & 2957 & 1262 & 396 & 3409 & 1550 & 647 \\ topo & 15 & 4 & 5 & 65 & 22 & 11 & 65 & 33 & 19 & 110 & 74 & 36 & 281 & 167 & 137 \\ come & 5 & 2 & 1 & 33 & 17 & 17 & 37 & 23 & 20 & 74 & 37 & 36 & 340 & 190 & 173 \\ all & 6 & 8 & 3 & 25 & 19 & 11 & 30 & 32 & 18 & 71 & 67 & 35 & 278 & 232 & 166 \\ \hline fbwa & 56 & 42 & 30 & 95 & 93 & 51 & 95 & 93 & 51 & 163 & 104 & 94 & 544 & 504 & 474 \\ linu & 5 & 4 & 3 & 79 & 53 & 19 & 143 & 78 & 37 & 175 & 103 & 49 & 279 & 198 & 192 \\ melb & 124 & 58 & 15 & 256 & 116 & 51 & 256 & 116 & 66 & 306 & 185 & 125 & 1467 & 1157 & 706 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Maximum position of the top-\(k\) nodes (for the exact ranking) in the approximate ranking computed by apx-\(h\) (over 50 experiments) in the case of the temporal graphs included in our sample dataset (excluding twit for which the exact ranking could not be computed). Figure 7: Box-and-whisker plots of the maximum position of top 20 nodes in the approximate ranking as a function of the sample size in the case of the temporal graphs come, fbwa, and melb. **Author Contributions:** Conceptualization, P.C., C.M. and A.M.; methodology, P.C., C.M. and A.M.; software, P.C., C.M. and A.M.; validation, P.C., C.M. and A.M.; formal analysis, P.C., C.M. and A.M.; investigation, P.C., C.M. and A.M.; resources, P.C., C.M. and A.M.; data curation, P.C., C.M. and A.M.; writing-original draft preparation, P.C., C.M. and A.M.; writing-review and editing, P.C., C.M. and A.M. All authors have read and agreed to the published version of the manuscript. **Funding:** C.M. was partly funded by the ANR (French National Agency of Research) by the Limass project (under grant ANR-19-CE23-0010), and by the ANR FiT LabCom.