A family of centrality measures for graph data based on subgraphs

We present the theoretical foundations and first experimental study of a new approach in centrality measures for graph data. The main principle is straightforward: the more relevant subgraphs around a vertex, the more central it is in the network. We formalize the notion of “relevant subgraphs” by choosing a family of subgraphs that, given a graph G and a vertex v, assigns a subset of connected subgraphs of G that contains v. Any of such families defines a measure of centrality by counting the number of subgraphs assigned to the vertex, i.e., a vertex will be more important for the network if it belongs to more subgraphs in the family. We show several examples of this approach. In particular, we propose the All-Subgraphs (All-Trees) centrality, a centrality measure that considers every subgraph (tree). We study fundamental properties over families of subgraphs that guarantee desirable properties over the centrality measure. Interestingly, All-Subgraphs and All-Trees satisfy all these properties, showing their robustness as centrality notions. To conclude the theoretical analysis, we study the computational complexity of counting certain families of subgraphs and show a linear time algorithm to compute the All-Subgraphs and All-Trees centrality for graphs with bounded treewidth. Finally, we implemented these algorithms and computed these measures over more than one hundred real-world networks. With this data, we present an empirical comparison between well-known centrality measures and those proposed in this work.