Percolation centrality
Percolation centrality
(PC) quantifies the importance of nodes in facilitating percolation through a network, such as in epidemic spreading or information diffusion [2]. In this model, each node \(i\) is assigned a percolation state \(x_i\) with \(0 \leq x_i \leq 1\), representing the extent to which the node is percolated.
The percolation centrality \(c_{PC}(i)\) of node \(i\) is defined as the fraction of ``percolated paths'', which are shortest paths whose source nodes are percolated, that pass through \(i\), i.e.,
\begin{equation*}
c_{PC}(i) = \frac{1}{N-2} \sum_{j \neq i \neq k} \frac{σ_{jk}(i)}{σ_{jk}} \cdot \frac{x_j}{\sum_{s=1}^{N} x_s - x_i},
\end{equation*}
where \(σ_{jk}\) denotes the total number of shortest paths from node \(j\) to node \(k\), and \(σ_{jk}(i)\) is the number of those paths that pass through node \(i\). Piraveenan et al. [2] demonstrated that percolation centrality reduces to standard
betweenness centrality
when all nodes have the same percolation state.