Super mediator degree (SMD)
Super mediator degree
(SMD) is a vitality-based centrality measure that quantifies the importance of a node by evaluating the reduction in global influence spread caused by its removal [2]. The influence of each node is approximated using repeated bond percolation simulations. In each simulation \(m = 1, \dots, M\), a percolated graph \(G_m\) is generated by independently retaining each edge of the original graph \(G\) with probability \(μ\). A node is considered a super-mediator if its removal substantially decreases the average influence degree under the underlying diffusion model.
The SMD score of node \(i\) is defined as
\begin{equation}
c_{\mathrm{SMD}}(i) =
\frac{1}{M} \sum_{j \in \mathcal{N}} \sum_{m=1}^{M} |R(j, G_m)| \, κ(j)
- \frac{1}{M} \sum_{j \in \mathcal{N} \setminus \{ i \}} \sum_{m=1}^{M} |R(j, G_m \setminus \{i\})| \, κ(j),
\end{equation}
where \(R(j, G_m)\) denotes the set of nodes reachable from \(j\) in the percolated graph \(G_m\), and \(κ(j)\) is the probability that node \(j\) becomes an initial active node. This formulation captures the expected decrease in overall influence due to the removal of node \(i\), averaged across \(M\) independent percolation instances.
For general graphs, Saito et al. [2] set the initial activation probability uniformly as \(κ(j) = 1/N\) and define the edge retention probability as \(μ = r / \overline{d}\), where \(r \in \{0.25, 0.5, 1\}\) and \(\overline{d}\) is the average out-degree of \(G\). Nodes with high SMD scores tend to appear frequently in longer diffusion paths but less often in short ones, reflecting their role as critical bridges that facilitate sustained, deep information propagation across the network.