# The topology of synergy: Linking topological and information-theoretic approaches to higher-order interactions in complex systems

## TL;DR

This paper compares topological and information-theoretic methods for studying complex systems and finds that they both detect similar higher-order interactions.

## Contribution

The paper provides the first direct comparison between topological data analysis and information-theoretic approaches to higher-order interactions.

## Key findings

- Higher-order synergistic information correlates with three-dimensional cavities in data structures like spheres and toroids.
- fMRI data shows strong correlations between synergistic information and the number and size of three-dimensional cavities.
- Dimensionality reduction techniques like PCA fail to preserve higher-order information and topological structures.

## Abstract

The study of irreducible higher-order interactions has become a core topic of study in complex systems, as they provide a formal scaffold around which to build a quantitative understanding of emergence and emergent properties. Two of the most well-developed frameworks, topological data analysis and multivariate information theory, aim to provide formal tools for identifying higher-order interactions in empirical data. Despite similar aims, however, these two approaches are built on markedly different mathematical foundations and have been developed largely in parallel - with limited interdisciplinary cross-talk between them. In this study, we present a head-to-head comparison of topological data analysis and information-theoretic approaches to describing higher-order interactions in multivariate data; with the goal of assessing the similarities, and differences, between how the frameworks define “higher-order structures.” We begin with toy examples with known topologies (spheres, toroids, planes, and knots), before turning to more complex, naturalistic data: fMRI signals collected from the human brain. We find that intrinsic, higher-order synergistic information is associated with three-dimensional cavities in an embedded point cloud: shapes such as spheres and hollow toroids are synergy-dominated, regardless of how the data is rotated. In fMRI data, we find strong correlations between synergistic information and both the number and size of three-dimensional cavities. Furthermore, we find that dimensionality reduction techniques such as PCA preferentially represent higher-order redundancies, and largely fail to preserve both higher-order information and topological structure, suggesting that common manifold-based approaches to studying high-dimensional data are systematically failing to identify important features of the data. These results point towards the possibility of developing a rich theory of higher-order interactions that spans topological and information-theoretic approaches while simultaneously highlighting the profound limitations of more conventional methods.

The problem of understanding when a set of interacting components of a complex systems produce behavior that is “greater than the sum of their parts" is foundational in many areas of modern science. Two different mathematical approaches have been developed to study higher-order interactions in data: one based on topology, and another based on information theory. These two frameworks are very different, and there has been little study of their overlap or the extent to which they are sensitive to the same “kind" of higher-order interactions. In this study, we compare both types of analyses directly. We find that there is indeed overlap: higher-order structures in the topological sense are correlated with irreducibly synergistic interactions in the information-theoretic sense. These results suggest that these two fields may share as-yet undiscovered mathematical connections, and deepen our understanding of emergent properties in complex systems.