# Finding Top-\(k\) Nodes for Temporal Closeness in Large Temporal Graphs
## 5 Experimental Results  
In our experiments, we used 45 medium/large real-world temporal graphs taken from different application domains, that is, collaboration, communication, and transportation domains. For sake of brevity, we describe our results by referring to a sample of our dataset (see Table 1): the entire dataset and the entire set of experimental results are shown in the Appendix A (Table 1). Here, we are going to use the following temporal graphs.  
* TOPO. The nodes are autonomous systems and the temporal edges are connections between autonomous systems. The appearing time of an edge is the time-point of the corresponding connection [42; 43; 44].
* all, come, fant. Every node corresponds to an actor and two actors are connected by their collaboration in a movie, where the appearing time of an edge is the year of the movie. We use the whole temporal collaboration graph and the ones induced by the comedy and the fantasy genres [45].
* melb. Nodes are transport stops and temporal edges are connections traversed by a public vehicle: the edge appearing time is the arrival time (see [30; 31]).
* fbwa. The nodes of the graph are Facebook users, and each directed temporal edge links the user writing a post to the user whose wall the post is written on [43; 44; 46].
* linu. The communication graph of the Linux kernel mailing list. An edge \((u,v,t)\) means that user \(u\) sent an email to user \(v\) at time \(t\)[44].  
\begin{table}
\begin{tabular}{l r r r r r r r r} \hline \hline  & \multicolumn{3}{c}{**Undirected Graphs**} & \multicolumn{3}{c}{**Directed Graphs**} \\ \hline
**Name** & **Nodes** & **Edges** & **exact** & **apx-1024** & **Name** & **Nodes** & **Edges** & **exact** & **apx-1024** \\ \hline fant & 34,464 & 87,331 & 1815 & 33 & melb & 19,493 & 1,098,227 & 6258 & 380 \\ topo & 34,761 & 154,842 & 1649 & 47 & fbwa & 46,952 & 876,993 & 12,184 & 264 \\ come & 162,303 & 666,568 & 29,601 & 203 & linu & 63,400 & 1,096,400 & 19,313 & 317 \\ all & 527,535 & 3,152,994 & 484,906 & 941 & twit & 3,511,241 & 16,438,790 & * 97,553,304 & 28,449 \\ \hline \hline \end{tabular}
\end{table}
Table 1: A sample of our dataset. For each graph we report the number of nodes, the number of temporal edges, and the running times (in seconds) of exact (the cell marked with * is an estimation) and apx-1024 (average among 50 experiments). The running times of apx-\(h\), for any other value of \(h\), can be estimated as \(h\cdot t/1024\), where \(t\) is the running time of apx-1024.  
* twit. Tweets about the migrant crisis of 2015. A directed edge \((u,v,t)\) means that user \(u\) retweeted a tweet of user \(v\) at time \(t\)[47; 48].  
For all graphs (apart from twit), we have computed the exact values of the closeness centrality for all nodes, in order to evaluate the quality of our approximation algorithm and of our ranking algorithm. In the case of twit, we have executed only the approximation algorithm in order to deduce some properties of the graph. Our computing platform is a machine with Intel(R) Xeon(R) CPU E5-2620 v3 at 2.40 GHz, 24 virtual cores, 128 Gb RAM, running Ubuntu Linux version 4.4.0-22-generic. The code was written in Java, using Java 1.8, and it is available at [[https://github.com/piluc/TemporalCloseness](https://github.com/piluc/TemporalCloseness)]([https://github.com/piluc/TemporalCloseness](https://github.com/piluc/TemporalCloseness)). Hereafter, we will refer to the exact algorithm as exact, and to our approximation algorithm as apx-\(h\), where \(h\) denotes the sample size in the definition of the closeness estimator. For each graph in our dataset, we ran apx-\(h\) setting \(h\) equal to 32, 64, 128, 256, 512, 1024, and repeating each experiment 50 times.