HodgeRank is the limit of Perron Rank

TL;DR

This paper establishes that HodgeRank can be viewed as the limit of Perron Rank through a mathematical analysis of eigenvectors and Hadamard powers, revealing a new connection between two pairwise ranking methods.

Contribution

It demonstrates that HodgeRank is the limit of Perron Rank, providing a novel mathematical link between these two pairwise ranking techniques.

Findings

HodgeRank is the limit of Perron Rank as k approaches 0.

The principal eigenvector of the k-th Hadamard power converges to the row geometric mean.

The convergence has geometric significance in pairwise comparison ranking.

Abstract

We study the map which takes an elementwise positive matrix to the k-th root of the principal eigenvector of its k-th Hadamard power. We show that as $k$ tends to 0 one recovers the row geometric mean vector and discuss the geometric significance of this convergence. In the context of pairwise comparison ranking, our result states that HodgeRank is the limit of Perron Rank, thereby providing a novel mathematical link between two important pairwise ranking methods.