Weighted TOPSIS (w-TOPSIS) centrality
Weighted TOPSIS (w-TOPSIS)
is a hybrid centrality measure that extends the classical TOPSIS method by incorporating attribute weights [2]. Let \(R\) be the normalized \(N \times m\) decision matrix, where each entry \(r_{ij}\) characterizes the normalized influence of node \(i\) with respect to centrality metric \(j\). Hu et al. [2] consider \(m=3\) metrics: degree, betweenness and closeness centralities.
The weights \(w_j\) for each centrality metric are derived based on the node spreading capability in the SIR model. Let \(F_i(t)\) denote the average number of infected and recovered nodes at time \(t\) if node \(i\) is initially infected, with spreading probability \(α=0.3\), recovery probability \(β=1\), and \(t=100\). The auxiliary variable \(v_{ij}\) aligns the normalized centrality with spreading influence:
\[
v_{ij} = \left| \frac{r_{ij}}{\sum_{l=1}^N r_{lj}} - \frac{F_i(t)}{\sum_{l=1}^N F_l(t)} \right|^{-1}.
\]
The weight of metric \(j\) is then defined as
\[
w_j = \frac{\sum_{i=1}^N v_{ij}}{\sum_{i=1}^N \sum_{j=1}^m v_{ij}}.
\]
The positive and negative ideal solutions are defined as
\[
A^{+} = \bigl[\max_i w_1 r_{i1}, \dots, \max_i w_m r_{im}\bigr], \quad
A^{-} = \bigl[\min_i w_1 r_{i1}, \dots, \min_i w_m r_{im}\bigr].
\]
The w-TOPSIS centrality of node \(i\) is computed as its relative closeness to the ideal solution:
\[
c_{\mathrm{w-TOPSIS}}(i) = \frac{S_i^{-}}{S_i^{-} + S_i^{+}},
\]
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 \(c_{\mathrm{w-TOPSIS}}(i)\) values are considered more influential, as they are simultaneously closer to the positive ideal and farther from the negative ideal solutions.