## 5 A Computational Model of Network Leadership
### A Multiplex Model of Leadership Centrality  
In the previous sections, we justified our choice of preferring an eigenvector model in the style of PageRank to measure the centrality of network users. However, we are still faced with the problem of designing an appropriate model of users' interactions in the network; in other words, we need to formalize the notion of \"bonding\" in an ESN, such as TamTamy. Once more, we rely also on sociological literature to identify social behaviors that may indicate leadership. We illustrate our bonding model with the help of Figure 3. As explained in previous Section 4, co-workers play three roles: _author_, _commenter_, or _rater_. In agreement with Reference [15], the action of starting (authoring) a thread can be considered as an _empowerment_ action, since authors are willing to share knowledge and solicit the collaboration of colleagues. As also remarked in Reference [47], \"empowerment is to start a conversation\". In organizational conversations, when talking _with_ employees rather than _to_ them, the achieved interactivity makes the conversation open rather than directive [47] and enables participants to put their own ideas. Note that, in TamTamy, starting a thread is not a role obligation, except for content publishers who are a small minority. Furthermore, messages whose purpose is simply to broadcast information, rather than soliciting contributions, do not receive comments. Restricting to threads with at least one comment (referred to hereafter as _non-zero threads_) provides a reasonable support to the hypothesis that the author's motivation was indeed to involve other partners on specific aspects of their working life. Furthermore, as demonstrated in Section 5.4, on topic analysis, the large majority of non-zero threads actually includes discussions on a variety of technical and organizational issues. The interpretation in terms of leadership qualities of the other roles in Figure 3 is more straightforward. Commenters show their willingness and ability to collaborate. In fact, _collaborative leaders_[48] do not exert their role just deciding what to do: rather, they help and encourage the collaborative process work. Finally, raters express their trust (or distrust), thus acknowledging/questioning the competence of commenters and the authority of authors. As remarked in Reference [49], _trustworthy leaders_ are \"rewarded by employees who stretch, push their limits, and volunteer to go above and beyond\". Figure 3: Co-workers roles and actions in the enterprise social network. To represent the mutual reinforcement among an actor's _empowerment_, _collaboration_, and _trust centrality_, we use a three-way _multiplex network_. Multiplex networks [50] are a class of networks recently introduced to model systems in which the same set of nodes are connected via more than one type of link. They are a special case of multilayer networks, in which no links are allowed among different layers. In our multiplex network, the first layer, \(G_{1}(N,E_{1})\), represents the activity of authors; the second layer, \(G_{2}(N,E_{2})\), models the activity of commenters; and the third layer, \(G_{3}(N,E_{3})\), is that of raters (see the small artificial example of previous Figure 1, depicting a set of three threads and the corresponding multiplex network's layers). In the first layer, \(G_{1}(N,E_{1})\), for any thread \(\theta\) initiated by any author \(i\), we add an edge \(e_{1}(i,j)\) with \(w_{\theta}(i\to j)=1\), whenever \(j\) posts a comment in \(\theta\). This models the fact that author \(i\) has empowered user \(j\). The _cumulative empowering weight_ of an edge \(e_{1}(i,j)\) is computed as:  
\[w_{e}(i\to j)=\frac{\sum_{\forall\theta}w_{\theta}(i\to j)}{\sum_{\forall \theta\
eq k}w_{\theta}(k\to j)}. \tag{2}\]  
The adjacency matrix \(E\) (_empowerment matrix_) associated with \(G_{1}\) is left stochastic, with each column summing to 1.