Node importance contribution correlation matrix (NICCM) method
The
node importance contribution correlation matrix
(NICCM) method extends node centrality analysis by considering that a node contributes unevenly to the importance of both adjacent and non-adjacent nodes within a limited radius [2]. This measure incorporates the influence of a node on others based on their shortest-path distance and the relative change in their centrality upon node removal.
The NICCM centrality of node \(i\) is defined as
\[
c_{\mathrm{NICCM}}(i) = c_{h}(i) \cdot \left( \sum_{j \in \mathcal{N}^{(\leq r)}(i)} \frac{c_{h}(j)\,δ_{ij}}{d_{ij}} \right),
\]
where \(c_{h}(i)\) is the harmonic centrality of node \(i\), \(d_{ij}\) is the shortest-path distance between nodes \(i\) and \(j\), and \( \mathcal{N}^{(\leq r)}(i)\) denotes the set of nodes within radius \(r\) from node \(i\). The contribution probability \(δ_{ij}\) quantifies the extent to which node \(i\) affects node \(j\) and is given by
\[
δ_{ij} = \frac{\Delta I_j}{\sum_{k \in \mathcal{N}^{(\leq r)}(i)} \Delta I_k},
\]
where \(\Delta I_j\) represents the change in the harmonic centrality of node \(j\) after the removal of node \(i\) from the network \(G\). Following Hu et al. [2], the radius parameter is typically set to \(r = 2\).
Nodes with high NICCM values exert strong influence on the importance of both nearby and moderately distant nodes, reflecting their broader structural impact on network connectivity.