# 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
Taking into account that \(\left({{I_{{L_{1}}}}(v)+{I_{{L_{2}}}}(v)}\right)^{2}={I_{{L_{1}}}}(v)^{2}+2{I_ {{L_{1}}}}(v){I_{{L_{2}}}}(v)+{I_{{L_{2}}}}(v)^{2}\), Equation (5) can be written in matrix form as:  
\[CC_{{L_{*}}}(u)=\sum\
olimits_{{v\in Neighbour_{{L_{*}}}(u)\cup u}}{\left({ \begin{array}{*{20}{c}}{1,}&{1}\\ \end{array} }\right)}\left({ \begin{array}{*{20}{c}}{{I_{{L_{1}}}}(v){I_{{L_{ 2}}}}(v)}\\ {{I_{{L_{1}}}}(v){I_{{L_{2}}}}(v)}&{{I_{{L_{2}}}}(v)^{2}}\\ \end{array} }\right)\left({ \begin{array}{*{20}{c}}{1}\\ {1}\\ \end{array} }\right) \tag{6}\]  
\(CC_{{L_{*}}}(u)\) can be rewritten as:  
\[CC_{{L_{*}}}(u)=\left({ \begin{array}{*{20}{c}}{1,}&{1}\\ \end{array} }\right)M(u)\left({ \begin{array}{*{20}{c}}{1}\\ {1}\\ \end{array} }\right) \tag{7}\]  
where \(M\) is the structure tensor, and it is defined as:  
\[M(u)=\sum\
olimits_{{v\in Neighbour_{{L_{*}}}(u)\cup u}}\left({ \begin{array}[ ]{*{20}{c}}{{I_{{L_{1}}}}(v)^{2}}&{{I_{{L_{1}}}}(v){I_{{L_{2}}}}(v)}\\ {{I_{{L_{1}}}}(v){I_{{L_{2}}}}(v)}&{{I_{{L_{2}}}}(v)^{2}}\\ \end{array} }\right) \tag{8}\]  
Note that the matrix \(M\) is derived from the differentials of the initial value of nodes' importance in the target layer with respect to the neighboring nodes' initial importance in the reference layers. For a number of layers \(N\) bigger than two, the matrix \(M\) is defined as:  
\[M(u)=\sum\
olimits_{{v\in Neighbour_{{L_{*}}}(u)\cup u}}{\left({ \begin{array} []{*{20}{c}}{{I_{{L_{1}}}}(v){I_{{L_{2}}}}(v)}&{\ldots}&{{I_{{L_{1}}}}(v){I_ {{L_{N}}}}(v)}\\ \vdots&{\ddots}\\ {{I_{{L_{N}}}}(v){I_{{L_{1}}}}(v)}&{{I_{{L_{N}}}}(v){I_{{L_{2}}}}(v)}&{\ldots }&{{I_{{L_{N}}}}(v)^{2}}\\ \end{array} }\right)} \tag{9}\]  
Equation (8) can be rewritten as:  
\[M(u)=\left({ \begin{array}{*{20}{c}}{{\sum\
olimits_{{v\in Neighbour_{{L_{*} }}(u)\cup u}}{{I_{{L_{1}}}}(v)^{2}}}}&{{\sum\
olimits_{{v\in Neighbour_{{L_{*} }}(u)\cup u}}{{I_{{L_{1}}}}(v){I_{{L_{2}}}}(v)}}}\\ {{\sum\
olimits_{{v\in Neighbour_{{L_{*}}}(u)\cup u}}{{I_{{L_{1}}}}(v){I_{{L_{ 2}}}}(v)}}}&{{\sum\
olimits_{{v\in Neighbour_{{L_{*}}}(u)\cup u}}{{I_{{L_{ 2}}}}(v)^{2}}}}\\ \end{array} }\right) \tag{10}\]  
To summarize the distribution of the relative importance of a node across different layers, we obtain the eigenvalues \(\lambda_{1}\) and \(\lambda_{2}\) of the matrix \(M\). In the case that \(\lambda_{2}\ll\lambda_{1}\), the node has a high relative importance in layer one, but the relative importance in layer 2 is low. The opposite situation is achieved when \(\lambda_{1}\ll\lambda_{2}\). In this paper, we want to obtain the nodes with highest relative importance across all layers,so for this we need \(\lambda_{1}\approx\lambda_{2}\) (the two eigenvalues are large and similar in magnitude). Therefore, to obtain the minimum between \(\lambda_{1}\) and \(\lambda_{2}\), we use  
\[\lambda_{min}\approx\frac{\lambda_{1}\lambda_{2}}{\lambda_{1}+\lambda_{2}}= \frac{\mathit{det}(M)}{\mathit{trace}(M)} \tag{11}\]  
where \(\mathit{det}\) and \(\mathit{trace}\) are the determinant and trace operators of the matrix \(M\). For example, if \(\lambda_{1}\gg\lambda_{2}\), then \(\frac{\lambda_{1}\lambda_{2}}{\lambda_{1}+\lambda_{2}}=\frac{\lambda_{2}}{ \frac{\lambda_{2}}{\lambda_{1}}+1}\approx\lambda_{2}\). To summarize, our method generates a score for each node in layer \(L_{*}\) as:  
\[S_{L_{*}}(u)=\frac{\mathit{det}(M(u))}{\mathit{trace}(M(u))+\epsilon} \tag{12}\]  
where \(\epsilon\) is a small constant to avoid divisions by zero. The score \(S_{L_{*}}\) defines the grade of the corner centrality of a node. Algorithm 1 shows the steps to obtain the ranking proposed in this paper. The algorithm can be used for multilayer networks complying with the following conditions:  
1. The multilayer network can contain layers with the same set of nodes. In this case, the layers do not have interlinks between them. 2. The multilayer network can contain layers with different nodes. In this case, the layers can be connected using interlinks between them. 3. The multilayer network can contain interlinks and intralinks between layers.