## 4 Detailed Design
### Social Tie Classifier System
#### 4.1.1 Nodes' Acquaintances Mechanism  
To discover the node's acquaintances, FSF introduces a metric called Node's Acquaintanceship Metric (NAM). The NAM metric extends the metric similarity proposed by [PERSON] et al. [38]. The main goal of this metric is to take advantage of people' daily routines to discover their acquaintances. It is largely known that people's pattern mobility can be useful to determine when a pair of nodes will meet each other in the future [39; 40]. NAM tries to take advantage of this by looking at the similarity of the contacts between a pair of nodes. To the best understanding of the reader, consider the example shown in Figure 2. In this figure, we can see the history of encounters in four different weeks between a pair of nodes (id 13 and 22) participating in a real experiment entitled Sassy [11]. According to their self-reported friendship, these nodes are not friends, despite meeting each other frequently. It is worth mentioning that the history of contacts among them every week has similar values for some characteristics, like the duration, frequency (five days a week), and the time between contacts, among others. Based on this history of encounters, it is reasonable to suppose that there is a good chance that they will meet each other again along the following week. In the real world, it is easy to realize that this behaviour commonly occurs in people' daily routines. For example, this behaviour can be applied to colleagues at work or at university, and to people sharing the bus or the metro every day, among others. Based on these assumptions, the NAM metric takes into account that a pair of nodes \(n_{x}\) and \(n_{y}\) will be acquaintances of each other if, and only if, the history of contacts among them at times \(T_{i}\) and \(T_{i+1}\) are similar. To do this, we use the history of encounters among \(n_{x}\) and \(n_{y}\). To measure the similarity between the contacts of a pair of nodes, we have used the following characteristics: number of contacts (NC), contact duration (CD), time between contacts (TBC), and the most common time of contacts (MCTC).These specific characteristics can be estimated as follows:  
\[NC\ (x,y)=n, \tag{1}\]  
where \(n\) is the number of encounters among the nodes (\(x\),\(y\)) at every period of time \(T\),  
\[CD\ (xy)=\frac{\sum_{i=1}^{n}(t_{end(i)}-t_{start(i)})}{n}, \tag{2}\]  
where \(t_{start(i)}\) and \(t_{end(i)}\) are the start time and the end time of the contact \(i\) between nodes \(x\) and \(y\),  
\[TBC\ (xy)=\frac{\sum_{i=1}^{n}(t_{start(i+1)}-t_{end(i)})}{n}, \tag{3}\]  
where \(t_{start(i+1)}\) is the next contact start time, and \(t_{end(i)}\) is the end time of the previous contact. To decrease the impact of data variability, for both the contact duration and the time between contacts, NAM takes the standard deviation into account. The last characteristic used by NAM is the most common time of encounters. To calculate it, NAM first stores the time when a specific contact takes place. Then, it uses the bimodal mode procedure to determine the most common hour for the contacts between \(n_{x}\) and \(n_{y}\). In order to select this type of mode, we have performed some experiments in the mobility traces used in our work. From the results, we have noted that the biggest number of modes in each collected sample is two. However, most of the time the samples used has just one mode. In the cases in which the samples have two modes, the FSF algorithm takes into account the lowest one. If there is no mode in the sample, our algorithm does not take into account this characteristic when calculating the metric NAM. Lastly, to determine how similar the contacts are between \(n_{x}\) and \(n_{y}\), NAM uses a metric called cosine similarity. Let \(u_{i}\) and \(v_{i}\) be the vectors containing the history of contacts at times \(T_{i}\) and \(T_{i+1}\), respectively.