TY - JOUR
T1 - On the graph Laplacian and the rankability of data
AU - Cameron, Thomas R.
AU - Langville, Amy N.
AU - Smith, Heather C.
N1 - Funding Information:
The authors wish to acknowledge Paul Anderson, Kathryn Behling, and Tim Chartier for many stimulating conversations that helped motivate some of the topics in this article. We are particularly grateful to Tim Chartier for providing the college football data used in this article. In addition, we wish to acknowledge an anonymous referee whose thoughtful comments greatly improved this article.
Publisher Copyright:
© 2019
Copyright:
Copyright 2019 Elsevier B.V., All rights reserved.
PY - 2020/3/1
Y1 - 2020/3/1
N2 - Recently, Anderson et al. (2019) proposed the concept of rankability, which refers to a dataset's inherent ability to produce a meaningful ranking of its items. In the same paper, they proposed a rankability measure that is based on an integer program for computing the minimum number of edge changes made to a directed graph in order to obtain a complete dominance graph, i.e., an acyclic tournament graph. In this article, we prove a spectral-degree characterization of complete dominance graphs and apply this characterization to produce a new measure of rankability that is cost-effective and more widely applicable. We support the details of our algorithm with several results regarding the conditioning of the Laplacian spectrum of complete dominance graphs and the Hausdorff distance between their Laplacian spectrum and that of an arbitrary directed graph with weights between zero and one. Finally, we analyze the rankability of datasets from the world of chess and college football.
AB - Recently, Anderson et al. (2019) proposed the concept of rankability, which refers to a dataset's inherent ability to produce a meaningful ranking of its items. In the same paper, they proposed a rankability measure that is based on an integer program for computing the minimum number of edge changes made to a directed graph in order to obtain a complete dominance graph, i.e., an acyclic tournament graph. In this article, we prove a spectral-degree characterization of complete dominance graphs and apply this characterization to produce a new measure of rankability that is cost-effective and more widely applicable. We support the details of our algorithm with several results regarding the conditioning of the Laplacian spectrum of complete dominance graphs and the Hausdorff distance between their Laplacian spectrum and that of an arbitrary directed graph with weights between zero and one. Finally, we analyze the rankability of datasets from the world of chess and college football.
UR - http://www.scopus.com/inward/record.url?scp=85075715689&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=85075715689&partnerID=8YFLogxK
U2 - 10.1016/j.laa.2019.11.026
DO - 10.1016/j.laa.2019.11.026
M3 - Article
AN - SCOPUS:85075715689
VL - 588
SP - 81
EP - 100
JO - Linear Algebra and Its Applications
JF - Linear Algebra and Its Applications
SN - 0024-3795
ER -