Model comparison via simplicial complexes and persistent homology

TL;DR

This paper introduces a novel framework for comparing mathematical models using simplicial complexes and persistent homology, enabling systematic analysis of model structure and equivalence.

Contribution

It develops two new methods for model comparison based on topological representations, providing a rigorous way to assess structural similarities and differences.

Findings

Identified equivalence between positional-information and Turing-pattern models.

Provided a topological approach to model comparison.

Demonstrated the effectiveness of persistent homology in model analysis.

Abstract

In many scientific and technological contexts we have only a poor understanding of the structure and details of appropriate mathematical models. We often, therefore, need to compare different models. With available data we can use formal statistical model selection to compare and contrast the ability of different mathematical models to describe such data. There is, however, a lack of rigorous methods to compare different models \emph{a priori}. Here we develop and illustrate two such approaches that allow us to compare model structures in a systematic way {by representing models in terms of simplicial complexes}. Using well-developed concepts from simplicial algebraic topology, we define a distance between models based on their simplicial representations. Employing persistent homology with a flat filtration provides for alternative representations of the models as persistence intervals, which represent the structure of the models, from which we can also obtain the distances between models. We then expand on this measure of model distance to study the concept of model equivalence in order to determine the conceptual similarity of models. We apply our methodology for model comparison to demonstrate an equivalence between a positional-information model and a Turing-pattern model from developmental biology, constituting a novel observation for two classes of models that were previously regarded as unrelated.