Analytic Hierarchy Process (AHP) centrality
The
Analytic Hierarchy Process (AHP) centrality
is a hybrid measure for identifying influential nodes by integrating multiple centrality metrics through the AHP framework [2]. Let \(R\) be the normalized \(N \times m\) decision matrix, where each entry \(r_{ij}\) represents the normalized influence of node \(i\) with respect to centrality metric \(j\). Bian et al. [2] consider \(m=3\) metrics: degree, betweenness and closeness centralities. The relative importance \(w_j\) of each metric is determined using a weighted TOPSIS approach (w-TOPSIS) [3].
For each centrality criterion \(j\), an \(N \times N\) pairwise comparison matrix \(B^{(j)}\) is constructed, where each entry \(b^{(j)}_{ik}\) quantifies the relative importance of node \(i\) compared to node \(k\) with respect to criterion \(j\) (e.g. \(b^{(j)}_{ik} = r_{ij} / r_{kj}\)). The AHP score of node \(i\) for criterion \(j\) is then computed as
\[
s_j(i) = \frac{1}{N} \sum_{k=1}^N \frac{b^{(j)}_{ik}}{\sum_{l=1}^N b^{(j)}_{lk}}.
\]
The AHP centrality of node \(i\) is obtained by aggregating the weighted scores across all criteria:
\[
c_{\mathrm{AHP}}(i) = \sum_{j=1}^m w_j s_j(i).
\]
Hence, AHP combines both the relative importance of each centrality metric and the comparative evaluation of nodes, providing a comprehensive assessment of node influence.