Hybrid degree and k-shell method
The
hybrid degree and
k
-shell method
is a variant of the local gravity model that ranks nodes in complex networks based on the degree and \(k\)-shell index of each node and its \(r\)-hop neighbors [2]. The centrality \(c_{x\text{-}ks}(i)\) of node \(i\) is defined as
\[
c_{x\text{-}ks}(i) = \sum_{j \in \mathcal{N}^{(\leq r)}(i)} \frac{k_s(i) d_j}{d_{ij}^2} = k_s(i) \sum_{j \in \mathcal{N}^{(\leq r)}(i)} \frac{d_j}{d_{ij}^2},
\]
where \(\mathcal{N}^{(\leq r)}(i)\) denotes the set of nodes within distance \(r\) from node \(i\) (excluding node \(i\)), \(k_s(i)\) and \(d_i\) are the \(k\)-shell index and degree of node \(i\), and \(d_{ij}\) is the shortest path distance between nodes \(i\) and \(j\). Maji et al. [2] consider a three-hop neighborhood, i.e., \(r = 3\).
The hybrid degree and \(k\)-shell method was evaluated using the susceptible-infected-recovered (SIR) model and metrics including spreadability, monotonicity, and Kendall’s tau, and it outperformed seven existing centrality measures in identifying influential seed nodes on real networks.