Influence capability (IC)
Influence capability
(IC) is a hybrid centrality measure that integrates node position information from the \(k\)-shell decomposition with the influence of the neighborhood [2]. The centrality of node \(i\) is defined as
\[
c_{\mathrm{IC}}(i) = w_1 \cdot IC_p(i) + (1 - w_1) \cdot IC_N(i),
\]
where \(IC_p(i)\) captures the influence of node \(i\) based on its positional attributes:
\[
IC_p(i) = d_r(i) + \sum_{j \in \mathcal{N}(i)} \frac{2}{π} \arctan \left( \bigl(Iter(j)\bigr)^{1/3} \right).
\]
Here, \(Iter(j)\) is the iteration at which node \(j\) is removed in the \(k\)-shell decomposition, and \(d_r(i)\) is the residual degree of node \(i\) after removing neighbors \(l\) with \(Iter(l) < Iter(j)\).
The neighborhood influence is measured by
\[
IC_N(i) = \sum_{j \in \mathcal{N}(i)} \sum_{l \in \mathcal{N}(j)} d_l,
\]
which aggregates the degrees of all neighbors of node \(i\) and of their neighbors, counting each node as many times as it appears in these neighbor sets. The parameter \(w_1 \in [0,1]\) balances the contributions of \(IC_p(i)\) and \(IC_N(i)\). Wang et al. [2] suggest setting
\[
w_1 = \frac{1 - E_1}{2 - E_1 - E_2},
\]
where \(E_1\) and \(E_2\) are the entropies of the \(IC_p(i)\) and \(IC_N(i)\) distributions, respectively.