Weighted k-shell degree neighborhood (WKSDN)
The
weighted \textit{k
-shell degree neighborhood} (WKSDN) is a parameter-free hybrid centrality measure that integrates both the degree and \(k\)-shell index of nodes in a network [2]. Unlike the weighted \(k\)-shell degree (Wksd) measure proposed by Namtirtha et al. [3], Maji's formulation estimates the weight of each edge \((i,j)\) as
\[
w_{ij} = (k_s(i) + k_s(j)) + λ (d_i + d_j),
\]
where \(d_i\) and \(k_s(i)\) denote the degree and \(k\)-shell index of node \(i\), respectively. The parameter \(λ\) serves as a normalization factor and is defined as
\[
λ = \frac{\sum_{i=1}^{N} k_s(i)}{\sum_{i=1}^{N} d_i}.
\]
The centrality of node \(i\) is then calculated as the sum of the weights of all its incident edges:
\[
c_{\mathrm{WKSDN}}(i) = \sum_{j \in \mathcal{N}(i)} w_{ij},
\]
where \(\mathcal{N}(i)\) denotes the set of neighbors of node \(i\).