# Corner Centrality of Nodes in Multilayer Networks: A Case Study in the Network Analysis of Keywords
## 2 Mathematical Model for the Corner Centrality of Nodes in a Multilayer Network
### A Toy Example  
In this section, we will show the main steps of Algorithm 1 for the multilayer network MLN4 in Figure 1. If we want to obtain the corner centrality value for the nodes in layer \(L_{1}\), then, according to Algorithm 1\(L_{*}\) will be \(L_{1}\). Following the steps 1-9 in Algorithm 1, we obtained the relative importance of every node in \(L_{1}\), analyzed by every layer (see Table 1):  
To obtain the relative importance for layer \(L_{1}\) (see column \(I_{L_{1}}\) in Table 1) for every node in \(L_{1}\), we used the intralink information. Thus, using a fixed node in the L1 layer, we consider its neighbors in \(L_{1}\) and compute the differences of the initial value of importance between the node and its neighbors. In the same way, in layer \(L_{3}\) we obtained the relative importance by using Equation (1) (see column \(I_{L_{3}}\) in Table 1). However, for layer \(L_{2}\), we used the interlink information to establish the relative importance of the nodes in \(L_{1}\) for layer \(L_{2}\) by using Equation (2) (see column \(I_{L_{2}}\) in Table 1). In this case, the neighbors of the node in the \(L_{2}\) layer are considered. For example, the node \(b\) has a relative importance of 44 in the layer \(L_{1}\). This value was obtained by adding:  
\[I_{L_{1}}(b)=i_{L_{1}}(b)-i_{L_{1}}(a)+i_{L_{1}}(b)-i_{L_{1}}(e)+i_{L_{1}}(b)- i_{L_{1}}(f)=14+15+15\]  
Concerning layer \(L_{2}\) the relative importance of the node \(b\) was obtained using the interlinks between layers \(L1\) and \(L2\):  
\[I_{L_{2}}(b)=i_{L_{1}}(b)-i_{L_{2}}(n)=16-15=1\]  
Additionally, to obtain the relative importance for the node \(b\) in the layer \(L_{3}\), we used the intralinks in layer \(L_{3}\)  
\[I_{L_{3}}(b)=i_{L_{3}}(b)-i_{L_{3}}(a)=5\]  
Analyzing Table 1, we can see that node \(a\) has a high relative importance in layer \(L_{3}\) but a very low importance in layers \(L_{1}\) and \(L_{2}\). Moreover, though node \(a\) has a bigger initial importance than the importance of nodes \(c\) and \(d\) in layer \(L_{1}\), this relative importance stays low in regard to the importance of node \(b\). Additionally, we see in Table 1 that for all the layers, node \(b\) is essential. In steps 11-12, we normalized the relative importance by subtracting the mean and divided by the standard deviation. The result of the normalization is shown in Table 2: \begin{table}
\begin{tabular}{c c c c} \hline \hline \multicolumn{4}{c}{**Relative Importance for Every Node**} \\ \hline
**Node** & \(I_{L_{1}}\) & \(I_{L_{2}}\) & \(I_{L_{3}}\) \\ \hline a & \(-12\) & \(0\) & \(31\) \\ b & \(44\) & \(1\) & \(5\) \\ c & \(-1\) & \(0\) & \(-9\) \\ d & \(-1\) & \(0\) & \(-9\) \\ e & \(-15\) & \(0\) & \(-9\) \\ f & \(-15\) & \(0\) & \(-9\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Relative importance of every node in layer \(L1\) from MLN4 in Figure 1. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multicolumn{4}{c}{**Normalized Relative Importance for Every Node**} \\ \hline
**Node** & \(I_{L_{1}}\) & \(I_{L_{2}}\) & \(I_{L_{3}}\) \\ \hline a & \(-0.584\) & \(-0.447\) & \(2.098\) \\ b & \(2.142\) & \(2.236\) & \(0.338\) \\ c & \(-0.049\) & \(-0.447\) & \(-0.609\) \\ d & \(-0.049\) & \(-0.447\) & \(-0.609\) \\ e & \(-0.730\) & \(-0.447\) & \(-0.609\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Normalized relative importance of every node in layer \(L1\) from MLN4 in Figure 1.  
In steps 12-14, we calculated the corner centrality value for each node in layer \(L_{1}\). In step 13, we obtained the matrix \(M(u)\), which is \(3\times 3\) and symmetrical. We must recall that the relative global importance of a node (see Equation (9)) must attend to its relative importance as well as the relative importance of the nodes in its neighborhood within layer \(L_{1}\). For the node \(b\) in layer \(L_{1}\), it is defined as:  
\[M(b)=\left( \begin{array}{ccc}M_{11}(b)&M_{12}(b)&M_{13}(b)\\ M_{21}(b)&M_{22}(b)&M_{23}(b)\\ M_{31}(b)&M_{32}(b)&M_{33}(b)\end{array} \right)\]  
with  
\[