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
\[
A^{+} = [\max_{i}{w_{1} r_{i1}}, \dots, \max_{i}{w_{m} r_{im}}], \quad
A^{-} = [\min_{i}{w_{1} r_{i1}}, \dots, \min_{i}{w_{m} r_{im}}],
\]
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.