# 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  
We consider that a multilayer network \(\mathcal{G}\) is composed of a set of layers \(\{G_{1},G_{2},\ldots,G_{M}\}\). Every layer \(G_{i}=(X_{i},E_{i})\) is a directed simple graph with \(X_{i}=\left\{e_{i1},e_{i2},\ldots,e_{iN_{i}}\right\}\) being the set of nodes, and with \(E_{i}=\left\{\left(e_{il},e_{i\mathbb{R}}\right)\right\}\) being the set of edges such that \(l\in\{0,1,\ldots,N_{i}\}\) and \(k\in\{0,1,\ldots,N_{j}\}\) with \(N_{i}\) and \(N_{j}\) being the number of nodes in the graphs \(G_{i}\) and \(G_{j}\), respectively. Different examples of multilayer networks are shown in Figure 1.  
Additionally, every node \(u\) in a layer \(L\) has an initial importance value \(i_{L}(u)\). For example, in Figure 1, in the multilayer network \(MLN1\), the \(a\) node in layer one has an importance of 2, while in layer two it has a value of 1. This type of situation can be seen in a team project, when a researcher has a more relevant contribution in one task while in other tasks, the importance of that researcher is not as high. A node should be perceived with more relative importance in a layer if its importance is higher than the importance of the nodes in its neighborhood in that layer. In this way, we can define the relative importance of a node \(u\) in the layer \(L\) as:  
\[I_{L}(u)=\sum_{\forall v\in Neighbour_{L}(u)}i_{L}(u)-i_{L}(v) \tag{1}\]  
Equation (1) uses the intralinks defined in layer \(L\). On the other hand, we can also define the relative importance in other layers, separate from the layer of the node, by using the interlinks as:  
\[I_{L}(u)=\sum_{\forall v\in Neighbour_{L}(u)}i_{L*}(u)-i_{L}(v) \tag{2}\]  
with \(L\) being different from the node's layer \(L_{*}\). In this case, Equation (2) uses the interlinks defined between the \(L\) and \(L_{*}\) layers. \(L_{*}\) is defined as the _target layer_ while \(L\) is the _reference layer_. For example, by using Equation (1) for the node \(b\), in the multilayer network MLN3 of Figure 1, we obtained the relative importance \(I_{L_{1}}(b)\), and by using Equation (2), we obtained the relative importance in the layer \(L_{2}\), \(I_{L_{2}}(b)\). The relative importance of a layer is normalized across the nodes to obtain a mean and standard deviation value of 0 and 1, respectively. Finally, when we analyze the multilayer network, the relative global importance of a node will be higher if high relative importance is met in every layer for that node. In this case, we can define the relative global importance of a node as:  
\[I(u)=\left(\sum_{i=0}^{M}I_{L_{i}}(u)\right)^{2} \tag{3}\]In Figure 1 in MLN1, the importance for node \(a\) would be: \(I(a)=\left({{I_{{L_{1}}}}(a)+{I_{{L_{2}}}}(a)}\right)^{2}\). To simplify, we are going to suppose that we have a multilayer network with only two layers: \({L_{1}}\) and \({L_{2}}\). Then, Equation (3) would be:  
\[I(u)=\left({{I_{{L_{1}}}}(u)+{I_{{L_{2}}}}(u)}\right)^{2} \tag{4}\]  
Additionally, if a node has a high global relative importance, this value must be transmitted across the nodes in its neighborhood, in the layer to which it belongs. In this sense, let \({L_{*}}\) be the target layer to which the node \(u\) belongs; then, the \(u\) node would have a high global relative importance across all the layers, and also, in layer \({L_{*}}\), the nodes that surround it would also have a high global relative importance:  
\[CC_{{L_{*}}}(u)=\sum\
olimits_{{v\in Neighbour_{{L_{*}}}(u)\cup u}}{I(v)=\sum \
olimits_{{v\in Neighbour_{{L_{*}}}(u)\cup u}}{\left({{I_{{L_{1}}}}(v)+{I_{ {L_{2}}}}(v)}\right)^{2}}} \tag{5}\]  
In Equation (5), the summation that iterates over the neighbors of \(u\) (including \(u\)) in the target layer adds to the relative importance of its neighbors in the target layer.