Simplicial complexes and complex systems

TL;DR

This paper introduces topological data analysis using simplicial complexes as a novel approach to studying complex systems, emphasizing its ability to capture multi-body interactions and data shape features.

Contribution

It provides an overview of topological data analysis methods, discusses their relevance to complex systems, and highlights future challenges in the field.

Findings

Topological methods can identify holes and cavities in data.

These methods are robust under data deformations.

Empirical studies show relevance to complex systems.

Abstract

We provide a short introduction to the field of topological data analysis and discuss its possible relevance for the study of complex systems. Topological data analysis provides a set of tools to characterise the shape of data, in terms of the presence of holes or cavities between the points. The methods, based on notion of simplicial complexes, generalise standard network tools by naturally allowing for many-body interactions and providing results robust under continuous deformations of the data. We present strengths and weaknesses of current methods, as well as a range of empirical studies relevant to the field of complex systems, before identifying future methodological challenges to help understand the emergence of collective phenomena.