Meta-centrality is a hybrid centrality measure that integrates multiple centrality rankings using the Borda count from social choice theory [2]. Given \(n\) rankings of nodes derived from different centrality measures, the method aims to select the most informative rankings and aggregate them into a single meta-centrality score. For example, Madotto \& Liu [2] consider \(n=8\) measures for weighted networks: degree, strength, closeness, eigenvector, PageRank, \(k\)-shell, weighted \(k\)-shell, and expected force.
The Borda count aggregation-based meta-centrality proceeds in three steps:


  1. Slicing : identify subsets of rankings to be used in the aggregation. Form the set \(X\) as \[ X = H \cup L \cup HL, \] where \(H = \{h_i\}_{i=1,...,n}\), \(L = \{l_i\}_{i=1,...,n}\), and \(HL = \{h_i \cup l_i\}_{i=1,...,n}\) are defined based on the Spearman correlation \(m_{ij}\) between rankings \(i\) and \(j\): \begin{align*} h_i &= \{i\} \cup \{j \mid m_{ij} \geq t_b, i \leq j\},\\ l_i &= \{i\} \cup \{j \mid m_{ij} \leq t_s, i \leq j\}, \end{align*} with thresholds \(t_b = 0.8\) and \(t_s = 0.3\).

  2. Selection : choose the most informative subsets of rankings by selecting two sets \(T_1, T_2 \subset X\) with the highest entropy. The entropy of a set \(x_i \in X\) is given by \[ E(x_i) = \frac{1}{|x_i|}\sum_{j \in x_i} \frac{m_{ij}}{\sum_j m_{ij}} \log \frac{m_{ij}}{\sum_j m_{ij}}. \]

  3. Aggregation : compute the Borda count \(B(i)\) of node \(i\) using the selected subset \(T_1\) (or \(T_2\)): \[ B(i) = \sum_{j \in T_1} (N - τ_j(i)), \] where \(τ_j(i)\) is the position of node \(i\) in ranking \(j\).


References

[1] Shvydun, S. (2025). Zoo of Centralities: Encyclopedia of Node Metrics in Complex Networks. arXiv: 2511.05122 https://doi.org/10.48550/arXiv.2511.05122
[2] Madotto, A., & Liu, J. (2016). Super-spreader identification using meta-centrality. Scientific reports, 6(1), 38994. doi: 10.1038/srep38994.