A Matrix Analysis of Functional Centrality Measures

We consider a broad class of walk-based, parameterized node centrality measures for network analysis. These measures are expressed in terms of functions of the adjacency matrix and generalize various well-known centrality indices, including Katz and subgraph centrality. We show that the parameter can be “tuned” to interpolate between degree and eigenvector centrality, which appear as limiting cases. Our analysis helps explain certain correlations often observed between the rankings obtained using different centrality measures, and provides some guidance for the tuning of parameters. We also highlight the roles played by the spectral gap of the adjacency matrix and by the number of triangles in the network. Our analysis covers both undirected and directed networks, including weighted ones. A brief discussion of PageRank is also given.