s-shell index
The
s
-shell index
is an extension of the traditional \(k\)-shell decomposition, designed to identify influential spreaders in weighted networks [2]. Unlike the \(k\)-shell method, which removes nodes based on degree, the \(s\)-shell decomposition relies on node strength, incorporating asymmetric edge weights that reflect the potential of each link to facilitate spreading.
The strength \( s_i \) of node \( i \) is defined as the sum of the asymmetric weights of its outgoing links:
\[
s_i = \sum_{j \in \mathcal{N}(i)} w_{ij} = \sum_{j \in \mathcal{N}(i)} \left[ 1 + (d_i d_j^{out})^{a} \right],
\]
where \( w_{ij} \) quantifies the influence of edge \( (i,j) \) in the spreading process, \( d_i \) is the degree of node \( i \), \( d_j^{out}\) denotes the number of neighbors of node \( j \) not shared with \( i \), i.e.,
\[
d_j^{out} = \left| \{\, l \in \mathcal{N}(j) : l \notin \mathcal{N}(i) \cup \{i\} \,\} \right|,
\]
and \( a \) is a tunable parameter controlling the weight contribution (\( a = 0.5 \) in [2]).
The formulation of
s
-shell index reflects the idea that edges leading to new regions of the network contribute more to spreading than those confined within a node's local neighborhood. The resulting edge weights are asymmetric (\(w_{ij} \neq w_{ji}\)), capturing directional spreading potential. When \(a = 0\), the \(s\)-shell index reduces to the standard \(k\)-shell index.