All-subgraph centrality
All-subgraph centrality (ASC) quantifies the importance of a node based on its participation in all connected subgraphs of a network [2]. For any subset of nodes \(S \subseteq \mathcal{N}\), let \(G_S\) denote the subgraph induced by \(S\). The set of all connected subgraphs that contain node \(i\) is \[\mathcal{A}(i, G) = \{\, S \subseteq \mathcal{N} \mid i \in S \text{ and } G_S \text{ is connected}\}.\]The all-subgraph centrality of node \(i\) is\[C_{\mathrm{ASC}}(i, G) = \log \bigl| \mathcal{A}(i, G) \bigr|,\]i.e., the logarithm of the number of connected subgraphs that include \(i\). Intuitively, all-subgraph centrality counts only connected subgraphs, since there is no justification for increasing the centrality of a node by including subgraphs in which it is not directly connected to other nodes. Nodes gain higher centrality if they participate in a larger number of connected subgraphs, reflecting their involvement in more diverse structural arrangements. The exact computation of the number of connected subgraphs \(|\mathcal{A}(i, G)|\) containing node \(i\) is generally intractable for large networks, since the number of connected subgraphs grows exponentially with graph size. In practice, \(|\mathcal{A}(i, G)|\) is estimated using approximation methods, such as restricting subgraphs to a fixed radius around \(i\) or using sampling techniques.