The generalized subgraph centrality (GSC), also referred to as \(t\)-subgraph centrality, extends the classical subgraph centrality by allowing a flexible weighting of closed walks based on their length, capturing both local and global aspects of a node's influence [2].
The subgraph centrality of node \(i\), denoted \(c_s(i)\), is defined as
\[
c_s(i) = \sum_{k=0}^{\infty} \frac{(A^k)_{ii}}{k!} = [e^A]_{ii},
\]
where \(A\) is the adjacency matrix and \((A^k)_{ii}\) counts the number of closed walks of length \(k\) starting and ending at node \(i\). In the generalized form, the factorial weighting is rescaled by a parameter \(t\), allowing the emphasis on short or long walks to be adjusted:

  • Positive rescaling (\(t \geq 0\)): longer walks are increasingly penalized, yielding a more localized centrality measure \[ c_{GSC}(i) = \sum_{k=0}^{\infty} \frac{(A^k)_{ii}}{(t+k)!}. \]
  • Negative rescaling (\(t < 0\)): the emphasis shifts toward longer walks, capturing a node's global environment \[ c_{GSC}(i) = \sum_{k=0}^{|t|-1} (A^k)_{ii} + \sum_{k=0}^{\infty} \frac{(A^{|t|+k})_{ii}}{k!} = \sum_{k=0}^{|t|-1} (A^k)_{ii} + [A^{|t|} e^A]_{ii}. \]

By varying \(t\), one can tune the centrality to emphasize either local structure (large positive \(t\)) or global network connectivity (negative \(t\)). The generalized subgraph centrality has been applied to protein-protein interaction (PPI) networks, with \(t = 7\) shown to outperform subgraph centrality and other measures in identifying essential proteins in the yeast network.

References

[1] Shvydun, S. (2025). Zoo of Centralities: Encyclopedia of Node Metrics in Complex Networks. arXiv: 2511.05122 https://doi.org/10.48550/arXiv.2511.05122
[2] Estrada, E. (2010). Generalized walks-based centrality measures for complex biological networks. Journal of theoretical biology, 263(4), 556-565. doi: 10.1016/j.jtbi.2010.01.014.