Edge Percolated Component (EPC)
Edge Percolated Component
(EPC) quantifies the robustness of a node’s connectivity under random edge failures [2]. Specifically, it estimates the fraction of nodes that remain connected to node \(i\) when each edge in the graph \(G\) is independently removed with probability \(p\).
Let \(G^{(k)}\) denote the \(k\)-th realization of \(G\) after random edge removal. The EPC centrality of node \(i\) is then given by
\begin{equation*}
c_{\mathrm{EPC}}(i) = \frac{1}{N K} \sum_{k=1}^{K} \sum_{j=1}^{N} δ_{ij}^{(k)},
\end{equation*}
where \(N\) is the number of nodes, \(K\) is the total number of realizations and
\[
δ_{ij}^{(k)} =
\begin{cases}
1, & \text{if nodes \(i\) and \(j\) are connected in } G^{(k)},\\
0, & \text{otherwise.}
\end{cases}
\]
Intuitively, a higher EPC centrality indicates that node \(i\) remains connected to a larger fraction of the network under random edge failures, reflecting its structural resilience.