Integral k-shell centrality
The
integral \(k\)-shell
(IKS) centrality is an extension of the \(k\)-shell centrality that incorporates both 2-step neighborhood information and the historical \(k\)-shell values of nodes [2]. The historical \(k\)-shell of a node \(i\) is defined as the sum of its \(k\)-shell values across previous iterations of the decomposition.
The integral \(k\)-shell value \(c_{IKS}(i)\) of node \(i\) is given by
\begin{equation*}
c_{IKS}(i) = Q_2(i) + \sum_{j=1}^{m_i} k_s^{(j)},
\end{equation*}
where \(Q_2(i)\) denotes the number of nodes within 2 hops of node \(i\), \(m_i\) is the iteration at which node \(i\) is removed during the \(k\)-shell decomposition, and \(k_s^{(j)}\) is the historical \(k\)-shell index assigned to the nodes removed at iteration \(j\).
Nodes with high IKS values are considered influential both because of their local 2-step connectivity and their persistence in the core layers of the network during the \(k\)-shell decomposition.