TOPSIS centrality
TOPSIS centrality
is a hybrid centrality measure that ranks nodes based on their similarity to an ideal solution using the TOPSIS (Technique for Order Preference by Similarity to Ideal Solution) approach [2]. Let \(R\) denote the normalized \(N \times m\) decision matrix, where each entry characterizes the normalized influence of a node with respect to \(m\) centrality metrics. Du et al. [2] consider \(m=3\) (degree, closeness and betweenness/eigenvector centralities).
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 of the \(j\)-th centrality metric (e.g., \(w_j = 1/m\) for equal weighting).
The TOPSIS centrality of node \(i\) is then given by its relative closeness to the ideal solution:
\[
c_{\mathrm{TOPSIS}}(i) = \frac{S_i^{-}}{S_i^{-} + S_i^{+}},
\]
where \(S_i^{+}\) and \(S_i^{-}\) are the Euclidean distances from node \(i\) to the positive and negative ideal solutions, respectively:
\[
S_i^{+} = \sqrt{\sum_{j=1}^m (A_j^{+} - w_j r_{ij})^2}, \quad
S_i^{-} = \sqrt{\sum_{j=1}^m (A_j^{-} - w_j r_{ij})^2}.
\]
Nodes with higher TOPSIS centrality scores are simultaneously closer to the positive ideal solution and farther from the negative ideal solution, reflecting high overall importance across the selected centrality metrics.