## 2 Materials and Methods
### Social Networks Analysis with PageRank  
Generally, PageRank, an objective technique for web page assessment, remains unaffected by readers' subjective judgments regarding web page importance [41]. We create five different kinds of interactions for the IT company. We also create six kinds of interactions for the international manufacturing company. A graph consists of nodes connected by edges with direction. A node refers to a member, and an edge between two nodes serves as the relation between them. Conventionally, to compute the importance of web pages by the PageRank algorithm, we need to analyze both the number and PageRank of other web pages that reference the web page. In our work, the importance of a particular member is quite similar to the importance of web pages due to leaders being always informed by all kinds of colleagues and usually communicating with different levels of colleagues. Therefore, we use the PageRank method to calculate the PageRank value to estimate the importance of staff. We analyze both the number and PageRank of people who initiate interactions with staff nodes. To perform this process, we use the five types of relationships defined previously. The information on the projects consists of two parts: *Vertices and *Arcs. Vertices contain information about the project team roles (PTRs) and their attributes associated. *Arcs comprise information about the links (source and target) between project team roles (PTRs) and the weight value (types of inter-colleague relationship) for each link. This value corresponds to the collaboration intensity (CI) which is used to measure the intensity of collaboration between PTR. It offers information about the degree of closeness to which PTR shares information. Its value ranges from 1 to 5, which represents the five relationships described for the projects A, B, C, and D. The value ranges from 1 to 6 represent the six relationships for projects E and F. We summarize CI into five and six categories based on their weight values. Then, we calculate the PageRank values for each of these categories, as shown in Table 1, which are included as new features in the model. In the PageRank algorithm mentioned in [PERSON] et al. [28], set \(V\) is the set of all nodes in the graph. It has \(n\) nodes \(P_{i}\in V\), where \(i\) refers to the label of the node \(n\). Also, there is a damping factor \(q\) for each \(P_{i}\) where \(q\) is a constant parameter. Equation (1) below shows the PageRank of the node \(P_{i}\):  
\[\text{PageRank}(P_{i})=\frac{1-q}{n}+q\sum_{P_{j}\in I(P_{i})}\frac{\text{PageRank }\left(P_{j}\right)}{O\left(P_{j}\right)} \tag{1}\] \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Projects** & **Intensity.1** & **Intensity.2** & **Intensity.3** & **Intensity.4** & **Intensity.5** & **Intensity.6** \\ \hline \multirow{2}{*}{A} & 0.00132 & 0.00203 & 0.00175 & 0.00224 & 0.00164 & NULL \\  & 0.00132 & 0.00160 & 0.00253 & 0.00209 & 0.00182 & NULL \\ \hline \multirow{2}{*}{B} & 0.00258 & 0.00190 & 0.00170 & 0.00175 & 0.00175 & NULL \\  & 0.00258 & 0.00190 & 0.00170 & 0.00175 & 0.00175 & NULL \\ \hline \multirow{2}{*}{C} & 0.01621 & 0.01209 & 0.01025 & 0.01150 & 0.01348 & NULL \\  & 0.01792 & 0.01908 & 0.001602 & 0.01415 & 0.01873 & NULL \\ \hline \multirow{2}{*}{D} & 0.01997 & 0.01429 & 0.00459 & 0.01509 & 0.01367 & NULL \\  & 0.01997 & 0.01429 & 0.00850 & 0.00696 & 0.00680 & NULL \\ \hline \multirow{2}{*}{E} & 0.01013 & 0.01452 & 0.00818 & 0.00559 & 0.00686 & 0.00686 \\  & 0.00802 & 0.01649 & 0.00961 & 0.00757 & 0.00814 & 0.00814 \\ \hline \multirow{2}{*}{F} & 0.01765 & 0.01005 & 0.00808 & 0.01336 & 0.00748 & 0.00748 \\  & 0.01765 & 0.01005 & 0.00649 & 0.01485 & 0.00762 & 0.00762 \\ \hline \hline \end{tabular}  
* Projects A, B, C, and D have five types of interactions, while projects E and F have six types of interactions.  
\end{table} 
Table 1: PageRank values estimated for each type of collaboration intensity in projects A, B, C, D, E, and F.  
which was originally proposed by [PERSON] et al. (2019) in 1998.