On the Graph Laplacian and the Rankability of Data

TL;DR

This paper introduces a new, efficient spectral-based measure of data rankability using graph Laplacian properties, improving upon previous methods and demonstrating applications in sports and game datasets.

Contribution

It provides a spectral-degree characterization of complete dominance graphs and develops a cost-effective, broadly applicable rankability measure.

Findings

New spectral measure of rankability based on Laplacian spectrum

Analysis of the conditioning of Laplacian spectra in dominance graphs

Application of the measure to chess and college football datasets

Abstract

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 a 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.