θ-Centrality
The
\(θ\)-centrality
, also known as the improved method \(θ\), the KS-\(k\) method, the \(k\)-shell distance method, or the distance-to-network-core (DNC) method, is a \(k\)-shell decomposition-based approach designed to differentiate the spreading influence of nodes that share the same \(k\)-core value [2]. Denote by \(k_s(i)\) the \(k\)-shell index of node \(i\). The \(θ\)-centrality of node \(i\), denoted by \(c_{θ}(i)\), depends both on its \(k\)-core value and on its distance from the network core:
\begin{equation*}
c_{θ}(i) = \bigl(\max_{l} k_s(l) - k_s(i) + 1\bigr) \sum_{j \in J} d_{ij},
\end{equation*}
where \(d_{ij}\) is the shortest-path distance between nodes \(i\) and \(j\), and \(J\) denotes the set of nodes with the maximum \(k\)-shell index, i.e., the innermost core of the network:
\[
J = \{\, j \in \mathcal{N} \mid k_s(j) = \max_{l} k_s(l) \,\}.
\]
Nodes with lower \(θ\)-centrality values are considered more influential, as they are closer to the network core and occupy structurally central positions. If the network is disconnected, \(θ\)-centrality is undefined because shortest-path distances cannot be computed between all node pairs.