## 5 Tie Strength Applied to MANETs
### Social Spheres  
A _social sphere_ represents the social contacts and topics of interest of a user when using social networks. In a previous research work [10], a model was built exclusively taking into account the interaction data gathered from public APIs (with users' permission) in Facebook and Twitter. Interesting interaction evidences or tie signs (private messages, retweets, mentions, etc.) were firstly identified, as it is summarized in Table 1 (Facebook evidences) and Table 2 (Twitter evidences). Then, they were analyzed to assess the strength of the social ties between users. For this to be possible, tie signs were tagged as direct or indirect interactions and as public or private interactions. In order to contextualize these tie signs, different social contexts were also taken into account, such as personal, familiar, professional, etc. Therefore, the tie signs between two users are weighted according to the following parameters: the different categories of the tie sign or evidence (private/public, direct/indirect) and the different context in which the tie sign occurred. There is a web service, coined as _mySocialSphere_, that can be accessed throughout a public API to obtain the public social sphere of any user. Thus, given a user \(u\), this web service analyzes the strength of his/her tie signs with other users and provides a tie strength index (a value between 0 and 1, being 1 the highest tie strength possible) that assesses the number and quality of the interactions of \(u\) with any other user \(v\). It is needed to remark that the tie strength index is inherently asymmetric, since the tie strength \begin{table}
\begin{tabular}{l c c c} \hline \hline
**Tie Signs (\(S_{u|k}(v)\))** & **Direct** & **Indirect** & **Public** & **Private** \\ \hline Wall-posts in friend’s Wall & x & & & x \\ Private messages exchanged & x & & & x \\ Comments in friend’s objects & x & & x & \\ Comments in the same objects & & x & x & \\ Likes in friend’s objects & x & & x & \\ Likes in the same objects & & x & x & \\ Being tagged in the same photos or videos & & x & & x \\ Belonging to the same private group & & x & & x \\ Belonging to the same public group & & x & x & \\ Attending to the same private event & & x & & x \\ Attending to the same public event & & x & x & \\ Being subscribed to the same user & & x & & \\ Being subscribed by the same user & & x & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Tie signs classification on Facebook [10]. perceived for \(u\) about his/her relationship with \(v\) may be different from the tie strength perceived for \(v\) about his/her relationship with \(u\). A simple and extreme example is the following one: user \(u\) often chats with \(v\) but also with other users, whereas \(v\) only chats with \(u\), the tie from \(v\)'s perspective will be stronger than from \(u\)'s one. Although all the mathematical details to compute the tie strength between two users are explained in [10], we consider that it is adequate for the purpose of this paper to mention here the following most relevant aspects. The tie strength of the relationship between user \(u\) and user \(v\), from the user \(u\)'s perspective, is denoted as \(T_{u}(v)\) and it is obtained as follows:  
\[T_{u}(v)=\sum_{k=1}^{N}\alpha_{k}\cdot f(|S_{u|k}(v)|),\ \ \ \text{where}\ \ \ \sum_{k=1}^{N}\alpha_{k}=1 \tag{5}\]  
where \(S_{u|k}(v)\) refers to the number of interactions of type \(k\) that \(u\) has with \(v\); \(N\) is the number of different tie signs analyzed (shown in Tables 1 and 2); and \(k\) refers to the type of interaction. \(\alpha_{k}\) denotes the weights of an interaction that belongs to type \(k\) (i.e., the influence of each type of interaction in the calculation of the tie strength). For instance, private interactions have a higher weight than public interactions; consequently, the \(\alpha_{k}\) is higher for private interactions than for public ones. Finally, function \(f\) is the following normalisation function:  
\[f(x)=