Least Squares Ranking on Graphs
Anil N. Hirani, Kaushik Kalyanaraman, Seth Watts

TL;DR
This paper explores least squares ranking on graphs, connecting it to various mathematical and computational fields, and aims to clarify fundamental ideas while guiding future research and method development.
Contribution
It provides a comprehensive explanation of least squares ranking on graphs, highlighting its connections to multiple research areas and offering numerical insights for method improvement.
Findings
Connections to spectral graph theory and multilevel methods
Numerical experiments guide method selection
Highlights need for further development in algorithms
Abstract
Given a set of alternatives to be ranked, and some pairwise comparison data, ranking is a least squares computation on a graph. The vertices are the alternatives, and the edge values comprise the comparison data. The basic idea is very simple and old: come up with values on vertices such that their differences match the given edge data. Since an exact match will usually be impossible, one settles for matching in a least squares sense. This formulation was first described by Leake in 1976 for rankingfootball teams and appears as an example in Professor Gilbert Strang's classic linear algebra textbook. If one is willing to look into the residual a little further, then the problem really comes alive, as shown effectively by the remarkable recent paper of Jiang et al. With or without this twist, the humble least squares problem on graphs has far-reaching connections with many current areas…
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Taxonomy
TopicsTopological and Geometric Data Analysis · Graph theory and applications · Complex Network Analysis Techniques
