The seformed graph laplacian and its applications to network centrality analysis

Journal article

We introduce and study a new network centrality measure based on the concept of nonbacktracking walks; that is, walks not containing subsequences of the form $uvu$ where $u$ and $v$ are any distinct connected vertices of the underlying graph. We argue that this feature can yield more meaningful rankings than traditional walk-based centrality measures. We show that the resulting Katz-style centrality measure may be computed via the so-called deformed graph Laplacian— a quadratic matrix polynomial that can be associated with any graph. By proving a range of new results about this matrix polynomial, we gain insights into the behavior of the algorithm with respect to its Katz-like parameter. The results also inform implementation issues. In particular we show that, in an appropriate limit, the new measure coincides with the nonbacktracking version of eigenvector centrality introduced by Martin, Zhang and Newman in 2014. Rigorous analysis on star and star-like networks illustrates the benefits of the new approach, and further results are given on real networks.

Authors: Grindrod, P; Highham, D; Noferini, V

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: Society for Industrial and Applied Mathematics
• Journal: SIAM Journal on Matrix Analysis and Applications
• Volume: 39
• Issue: 1
• Pages: 310–341
• Publication date: 2018-03-01
• Acceptance date: 2017-11-21
• DOI: 10.1137/17M1112297
• EISSN: 1095-7162
• ISSN: 0895-4798
• Keywords: nonbacktracking; complex network; matrix polynomial; deformed graph Laplacian; centrality index; generating function