A matrix analysis of different centrality measures

Node centrality measures including degree, eigenvector, Katz and subgraph centralities are analyzed for both undirected and directed networks. We show how parameter-dependent measures, such as Katz and subgraph centrality, can be “tuned” to interpolate between degree and eigenvector centrality, which appear as limiting cases of the other measures. We interpret our finding in terms of the local and global influence of a given node in the graph as measured by graph walks of different length through that node. Our analysis gives some guidance for the choice of parameters in Katz and subgraph centrality, and provides an explanation for the observed correlations between different centrality measures and for the stability exhibited by the ranking vectors for certain parameter ranges. The important role played by the spectral gap of the adjacency matrix is also highlighted.