An extended family of measures for directed networks

Centrality indices such as β-centrality, Katz status, and Hubbell’s index are commonly generalized to directed networks by relating the in-centrality of nodes to the in-centrality of their in-neighbors and equivalently so for out-centrality. This paper proposes an extension of Bonacich’s β-centrality and related measures for directed networks where the in-centrality of a node depends on the out-centrality of their in-neighbors and their out-centrality on the in-centrality of their out-neighbors. The so defined indices extend hubs and authorities in the same way as β-centrality generalizes eigenvector centrality. Several technical results are presented including the extension of the range of permissible β parameters to negative values, similar to traditional β-centrality.