# Approximating the Temporal Neighbourhood Function of Large Temporal Graphs
## 4 Experimental Results
### Dataset  
In order to perform our experiments, we used the following temporal graphs (other temporal networks will be considered in Section 5, in which we will perform a case study based on the directed public transport networks of 25 cities):  
* college: Private messages sent on an online social network at the University of California, Irvine. An edge \((u,v,t)\) means that user \(u\) sent a private message to user \(v\) at time \(t\)[26; 27].
* enron: Emails between Enron employees 1999 and 2003. An edge \((u,v,t)\) means that user \(u\) sent an email to user \(v\) at time \(t\).
* facebook-wallpost: A small subset of posts to another user's wall on Facebook. The nodes of the network are Facebook users, and each directed temporal edge represents one post, linking the users writing a post to the user whose wall the post is written on [28; 29; 30].
* imdb: 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 will consider both the whole temporal collaboration graph and the graphs induced by the following genres: adventure,horror, thriller, crime, romance, action, comedy, drama (for each genre, we consider only the edges corresponding to movies classified in that genre) [31].
* ROLLERNET: Opportunistic sighting of Bluetooth devices by groups of rollerbladers carrying Intel iMotes during a roller tour. The 62 iMotes performed neighborhood scans every 15 s [32; 33].
* TOPOLOGY: The nodes are autonomous systems and the edges are connections between autonomous systems. The appearing time of an edge is the time-point of the corresponding connection [29; 30; 34].
* twitter: Tweets about the migrant crisis of 2015 [35; 36]. A directed edge \((u,v,t)\) means that user \(u\) retweeted a tweet of user \(v\) at time \(t\).
* Wiki-election: The network of users from the English Wikipedia that voted for and against each other in admin elections. Nodes represent individual users, and edges represent votes. Each edge is annotated with the date of the vote [29; 30; 37].  
The dimensions of the above temporal graphs are summarized in Table 2, where we report for each graph its number of nodes and its number of edges, and whether the graph is directed or not. For a given network \(G\), let \(t_{\alpha}\) and \(t_{\omega}\) denote the minimum and maximum appearing time, respectively, of the temporal edges included in \(G\). We have then considered different intervals \(\mathbb{I}=[t_{\alpha},t_{\beta}]\), for increasing values of \(t_{\beta}\) with \(t_{\beta}\in\mathcal{T}(G)\), where \(\mathcal{T}(G)\) is the time horizon of \(G\) (that is, the union of all the appearing times of its temporal edges). Both etnf and atnf compute (or approximate) the number of pairs of nodes \(u\) and \(v\) such that, starting from \(u\) not before time \(t_{\alpha}\), we can reach \(v\) within time \(t_{\beta}\). As a result, we get an exact cumulative frequency distribution, running etnf, and its approximation, running atnf. Section 4.1 is devoted to measure the quality of this approximation, while Section 4.2 shows the running times.