## 5 A Computational Model of Network Leadership
### A Multiplex Model of Leadership Centrality
Edges \(e_{2}(i,j)\) in \(G_{2}(N,E_{2})\) are created whenever a user \(i\) posts a comment for the first time in a thread \(\theta\) authored by \(j\). Subsequent answers update the weight of \(e_{2}(i,j)\). Given the sequential thread structure, comments are considered as implicitly directed to the author, unless the message includes the name of another recipient in the thread (as in the example of Figure 2). Note that authors themselves can add comments during a discussion, and, in this case, they are treated as commenters. Any comment \(h\) in a thread \(\theta\) has a weight computed as follows:  
\[w_{h}(i\to j)=(\pi_{1}+\frac{\pi_{2}}{k}), \tag{3}\]  
where \(k\) is the sequential order of the comment in the thread (\(k=1\) if a user is the first who replied). This models the idea that the first commenter is more competent, or more willing, to collaborate, than others (alternatively, \(k\) can be set to represent the difference between the timestamp of the author's message and that of the commenter). In the formula, \(\pi_{1},\pi_{2}\) are parameters that we experimentally set to 0.5. The _cumulative collaboration weight_ of an edge \(e_{2}(i,j)\) sums all answers provided by user \(i\) to user \(j\) in any thread within a considered timespan \(W\):  
\[w_{c}(i\to j)=\frac{\sum_{\forall h,\theta}w_{h}(i\to j)}{\sum_{\forall h, \theta,k}w_{h}(k\to j)}. \tag{4}\]  
The _collaboration matrix_\(C\) (the adjacency matrix associated with \(G_{2}\)) is, as for \(E\), left stochastic. The third layer \(G_{3}(N,E_{3})\) models the rating activity of users. Edges \(e_{3}(i,j)\) are weighted with the trust \(w_{t}(i\to j)\) of rater \(i\) in user \(j\). We first introduce the quantity:  
\[trust(i\to j)=\pi_{3}+\pi_{4}\times\sum_{h\in m(j)}\delta_{h}(i\to j), \tag{5}\]  
where \(m(j)\) is the set of messages generated by user \(j\) (either as an author or as a commenter), \(h\) is a message in \(m(j)\), and \(\delta_{h}(i\to j)=+1\) if \(i\) likes \(h\), \(-1\) if \(i\) dislikes, and \(0\) if no opinion is expressed. As for Formula (9), parameters \(\pi_{3}\) and \(\pi_{4}\) are set to 0.5. According to Equation (7), \(0\leq trust<0.5\) indicates distrust. We then define:  
\[w_{t}(i\to j)=\frac{trust(i\to j)}{\sum_{k}trust(i\to k)}. \tag{6}\]  
The \(T\) matrix (the adjacency matrix associated with \(G_{3}\)), named the _credibility matrix_, is right stochastic. We further denote with \(\mathcal{M}\) the 3-way \(N\times N\times 3\) tensor of the multiplex network. The third dimension of the network represents the network leadership qualities discussed at the beginning of the section: _empowerment, collaboration_, and _credibility_. We now need to measure the _centrality_ of nodes in the multiplex network, to identify highly central agents. A simple assumption would be to compute, for every user, the empowerment, collaboration, and credibility ranks \(r^{e}\), \(r^{c}\), and \(r^{t}\) using monoplex PageRank [22], and then computing some heuristic function to combine these indicators, for example, using a regression, as in Reference [9]. However, a better assumption is to postulate a mutual reinforcement relation among layers, i.e., that the centrality of each node in one layer affects the centrality of the same node in any other layer. Therefore, to compute our measure of _network leadership centrality_, we use _Multiple PageRank (MPR)_, introduced in Reference [43]. First, we note that PageRank centrality depends on the in-degree of nodes, while, according to our formulation, _empowerment_ and _collaboration_ of a node depend on the weight of outgoing edges. Therefore, we need to invert the direction of edges in the corresponding graphs. We denote with \(E^{\prime}\), \(C^{\prime}\) (Where \({}^{\prime}\) denotes the transpose), and \(T\) the slices of an \(\mathcal{M}^{\prime}\) tensor; and, with \(w^{\prime}_{e}(j\to i)=w_{e}(i\to j)\), \(w^{\prime}_{c}(j\to i)=w_{c}(i\to j)\), and \(w_{t}(i\to j)\), the corresponding matrix cells.