Dowker Complexes and filtrations on self-relations

TL;DR

This paper explores the topological properties of Dowker complexes derived from relations and self-relations, establishing their homotopy equivalences under conjugacy and shift equivalence, and introduces a new filtration based on these complexes.

Contribution

It demonstrates homotopy invariance of Dowker complexes under conjugacy and shift equivalence, and proposes a novel filtration method for analyzing relations using these complexes.

Findings

Dowker complexes are homotopically equivalent for conjugate relations.

Shift equivalent relations have Dowker complexes that become homotopically equivalent at some power.

A new filtration based on Dowker complexes and relation powers is introduced.

Abstract

Given a relation on $ X \times Y $, we can construct two abstract simplicial complexes called Dowker complexes. The geometric realizations of these simplicial complexes are homotopically equivalent. We show that if two relations are conjugate, then they have homotopically equivalent Dowker complexes. From a self-relation on $ X $, this is a directed graph, and we use the Dowker complexes to study their properties. We show that if two relations are shift equivalent, then, at some power of the relation, their Dowker complexes are homotopically equivalent. Finally, we define a new filtration based on Dowker complexes with different powers of a relation.