A graph interpretation of the least squares ranking method

TL;DR

This paper analyzes the least squares ranking method for generalized tournaments, revealing its interpretation through graph theory, its iterative computation process, and its relation to positional power measures, with implications for various comparison-based evaluations.

Contribution

It introduces a graph-based interpretation of the least squares ranking method and explores its connections and potential modifications, enhancing understanding of its computational and theoretical properties.

Findings

The rating vector can be obtained as a limit of an iterative process.

The method accounts for both direct results and the strength of comparisons.

Connections to positional power measures are established.

Abstract

The paper aims at analyzing the least squares ranking method for generalized tournaments with possible missing and multiple paired comparisons. The bilateral relationships may reflect the outcomes of a sport competition, product comparisons, or evaluation of political candidates and policies. It is shown that the rating vector can be obtained as a limit point of an iterative process based on the scores in almost all cases. The calculation is interpreted on an undirected graph with loops attached to some nodes, revealing that the procedure takes into account not only the given object's results but also the strength of objects compared with it. We explore the connection between this method and another procedure defined for ranking the nodes in a digraph, the positional power measure. The decomposition of the least squares solution offers a number of ways to modify the method.