A hierarchical walk-based measure of centrality based on reachability between strongly connected components in a digraph

ABSTRACT

For measuring the centrality in a digraph, Bonacich and Lloyd summarized a vector, from the power series of an attenuated adjacency matrix, as the alpha centrality. However, scores of alpha centrality are usually dominated by nodes in the strongly connected component, which owns the largest eigenvalue of the adjacency matrix. In this paper, based on reachability between strongly connected components, we consider not only the largest eigenvalue but also the other smaller ones to attenuate the adjacency matrix hierarchically; and obtain a measure from the power series of the hierarchically attenuated adjacency matrix. Consequently, we propose the hierarchical alpha centrality, which can yield higher scores for nodes at higher hierarchies of reachability in a digraph.

KEYWORDS:

• Adjacency matrix
• attenuation
• centrality

Acknowledgments

The author would like to thank the editors and reviewers for helping the academic career of the author going. Special thanks for the insightful suggestions and valuable references, including the numbers of Strahler (1952) and main path analysis (Hummon & Doreian, 1989), to improve the paper, which was supported in part by the Ministry of Science and Technology, Taiwan, R.O.C., under Grant MOST 108-2410-H-182-017.

Notes

¹ In symmetry, there is also the lower block-triangular form, a transpose of the upper one.