A Weighted Generalization of the Graham-Diaconis Inequality for Ranked List Similarity

TL;DR

This paper extends the Graham-Diaconis inequality to weighted ranked lists, establishing a theoretical link between weighted Spearman's footrule and Kendall's tau measures.

Contribution

It introduces weighted versions of the similarity measures and proves a generalized inequality relating them, broadening the original unweighted framework.

Findings

Weighted Spearman's footrule and Kendall's tau are defined.

A generalized inequality relating the weighted measures is proven.

The results extend the theoretical understanding of ranked list similarity.

Abstract

The Graham-Diaconis inequality shows the equivalence between two well-known methods of measuring the similarity of two given ranked lists of items: Spearman's footrule and Kendall's tau. The original inequality assumes unweighted items in input lists. In this paper, we first define versions of these methods for weighted items. We then prove a generalization of the inequality for the weighted versions.