TOPSIS-RE centrality
TOPSIS-RE centrality is a hybrid method that combines relative entropy and the Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS) to evaluate node influence in a network [2]. Let \(R\) denote the normalized \(N \times m\) decision matrix, where each row corresponds to a node and each column represents a normalized centrality measure. Liu et al. [2] consider \(m=4\) centralities: degree, closeness, betweenness and IKSD. The positive ideal solution \(A^{+}\) and negative ideal solution \(A^{-}\) are defined as\[A^{+} = [\max_{i}{w_{1} r_{i1}}, \dots, \max_{i}{w_{m} r_{im}}], \quadA^{-} = [\min_{i}{w_{1} r_{i1}}, \dots, \min_{i}{w_{m} r_{im}}],\]where \(w_j\) is the weight assigned to centrality \(j\). Liu et al. [2] set\[[w_1, w_2, w_3, w_4] = [0.0625, 0.1875, 0.1875, 0.5625]\]using the Analytic Hierarchy Process (AHP). The centrality \(c_{TOPSIS-RE}(i)\) of node \(i\) is calculated as the relative closeness to the ideal solution:\begin{equation*}c_{TOPSIS-RE}(i) = \frac{S_i^{-}}{S_i^{-} + S_i^{+}},\end{equation*}where \(S_i^{+}\) and \(S_i^{-}\) measure the relative entropy between node \(i\) and the positive and negative ideal solutions, respectively:\[S_i^{+} = \sqrt{\sum_{j=1}^m \left( A_j^{+} \log \frac{A_j^{+}}{w_j r_{ij}} + (1-A_j^{+}) \log \frac{1-A_j^{+}}{1-w_j r_{ij}} \right)},\]\[S_i^{-} = \sqrt{\sum_{j=1}^m \left( A_j^{-} \log \frac{A_j^{-}}{w_j r_{ij}} + (1-A_j^{-}) \log \frac{1-A_j^{-}}{1-w_j r_{ij}} \right)}.\]Nodes with higher TOPSIS-RE scores are considered more influential, being closer to the positive ideal solution and farther from the negative ideal solution.