Extended hybrid degree and MDD method
The
extended hybrid degree and MDD method
ranks nodes in complex networks by considering both the mixed degree decomposition (MDD) index and the degree of each node and its neighbors [2]. It extends the hybrid degree and MDD centrality (\(c_{x\text{-}MDD}\)) by defining the centrality of node \(i\) as the sum of the \(c_{x\text{-}MDD}\) scores of its immediate neighbors:
\[
c_{\text{ex-MDD}}(i) = \sum_{j \in \mathcal{N}(i)} c_{x\text{-}MDD}(j)
= \sum_{j \in \mathcal{N}(i)} \left( MDD(j) \sum_{k \in \mathcal{N}^{(\leq r)}(j)} \frac{d_k}{d_{jk}^2} \right),
\]
where \(\mathcal{N}(i)\) denotes the set of neighbors of node \(i\), \(\mathcal{N}^{(\leq r)}(j)\) is the set of nodes within distance \(r\) from node \(j\) (excluding \(j\) itself), \(MDD(j)\) and \(d_j\) are the MDD index and degree of node \(j\), and \(d_{jk}\) is the shortest path distance between nodes \(j\) and \(k\). Maji et al. [2] consider a three-hop neighborhood, i.e., \(r = 3\).
The extended hybrid degree and MDD method has been evaluated using the susceptible-infected-recovered (SIR) model and metrics such as spreadability, monotonicity, and Kendall’s tau, demonstrating superior performance compared to several existing centrality measures in identifying influential seed nodes on real networks.