Using principal eigenvectors of adjacency matrices with added diagonal weights to compose centrality measures and identify bowtie structures for a digraph

ABSTRACT

Principal eigenvectors of adjacency matrices are often adopted as measures of centrality for a graph or digraph. However, previous principal-eigenvector-like measures for a digraph usually consider only the strongly connected component whose adjacency submatrix has the largest eigenvalue. In this paper, for each and every strongly connected component in a digraph, we add weights to diagonal elements of its member nodes in the adjacency matrix such that the modified matrix will have the new unique largest eigenvalue and corresponding principal eigenvectors. Consequently, we use the new principal eigenvectors of the modified matrices, based on different strongly connected components, not only to compose centrality measures but also to identify bowtie structures for a digraph.

KEYWORDS:

• Principal eigenvector
• centrality
• bowtie structure

Acknowledgments

The author would like to thank the reviewers of previous research (Lu, 2017), whose insightful suggestions inspire the key idea of this paper, and the reviewers of this paper, whose valuable comments improve the quality of this paper.