Filtration-Domination in Bifiltered Graphs

TL;DR

This paper introduces algorithms to efficiently identify and remove filtration-dominated edges in bifiltered graphs, significantly reducing complexity and resource usage in topological data analysis.

Contribution

It presents two novel algorithms that detect filtration-dominated edges directly on bifiltered graphs, avoiding the costly extraction of clique complexes.

Findings

Over 90% of edges can be removed in most cases

Significant speedup in topological data analysis pipeline

Reduction in memory usage during computations

Abstract

Bifiltered graphs are a versatile tool for modelling relations between data points across multiple grades of a two-dimensional scale. They are especially popular in topological data analysis, where the homological properties of the induced clique complexes are studied. To reduce the large size of these clique complexes, we identify filtration-dominated edges of the graph, whose removal preserves the relevant topological properties. We give two algorithms to detect filtration-dominated edges in a bifiltered graph and analyze their complexity. These two algorithms work directly on the bifiltered graph, without first extracting the clique complexes, which are generally much bigger. We present extensive experimental evaluation which shows that in most cases, more than 90% of the edges can be removed. In turn, we demonstrate that this often leads to a substantial speedup, and reduction in the memory usage, of the computational pipeline of multiparameter topological data analysis.