p-means centrality
The
p
-means centrality
is a distance-based measure designed to identify the most influential nodes in a network by generalizing several classical centrality metrics [2]. The centrality is parameterized by \(p\), which governs the aggregation of distances from node \(i\) to all other nodes in the network. For a node \(i \in \mathcal{N}\), the
p
-means centrality is then defined as
\[
c_{p\text{-means}}(i) =
\begin{cases}
\displaystyle \left( \frac{\sum_{j \neq i} d_{ji}^p}{N-1} \right)^{-\frac{1}{p}}, & \text{if } p \neq 0,\\[2mm]
\displaystyle \left( \prod_{j \neq i} d_{ji} \right)^{-\frac{1}{N-1}}, & \text{if } p = 0,
\end{cases}
\]
where \(d_{ji}\) is the shortest-path distance from node \(j\) to node \(i\), and \(N\) is the total number of nodes.
By varying \(p\), p-means centrality interpolates between several classical measures:
- \(p = 1\) corresponds to closeness centrality;
- \(p = -1\) corresponds to harmonic centrality;
- \(p \to \infty\) approaches eccentricity centrality;
- \(p \to -\infty\) yields the same node ranking as degree centrality.
The performance of p -means centrality has been evaluated using the susceptible-infected-recovered (SIR) model. Additionally, Andrade and Rêgo [2] investigated the values of \(p\) for which p-means centrality satisfies the size, density, and score monotonicity axioms.