Shell clustering coefficient (SCC)
The
shell clustering coefficient
(SCC) quantifies node influence by considering the hierarchical similarity between a node and its neighbors [2]. Zareie et al. introduce the
shell vector
of node \(i\) as
\[
sv(i) = \left(|N_{ks}^{(1)}(i)|,...,|N_{ks}^{(f)}(i)|\right),
\]
where \(|N_{ks}^{(k)}(i)|\) is the number of neighbors of node \(i\) belonging to hierarchy \(k\) with respect to the \(k\)-shell centrality, and \(f\) is the maximum hierarchy in the network.
The
shell clustering coefficient
\(SCC(i)\) of node \(i\) is defined as
\begin{equation*}
SCC(i) = \sum_{j \in \mathcal{N}(i)} \left[ 2 - \mathrm{corr}[sv(i), sv(j)] + \left(\frac{2 d_j}{\max_l d_l} + 1 \right) \right],
\end{equation*}
where \(\mathrm{corr}[sv(i), sv(j)]\) is the Pearson correlation between the shell vectors of nodes \(i\) and \(j\), and \(d_j\) is the degree of neighbor \(j\).
Intuitively, a high correlation between node \(i\) and its neighbors, combined with neighbors of low degree, negatively affects the spreading ability of node \(i\).