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

2025-04-14

AI Summary

This work addresses the identification and characterization of irreducible higher-order interactions in complex systems, systematically comparing topological data analysis (TDA) and multivariate information theory (MIT) in modeling higher-order structural dependencies. Method: We integrate persistent homology, partial information decomposition (PID), fMRI-based modeling, and robustness evaluation of dimensionality reduction techniques. Contribution/Results: We establish, for the first time, a statistically significant positive correlation between three-dimensional topological voids (e.g., spherical structures) and intrinsic synergy—a key MIT measure of higher-order dependence. Furthermore, we demonstrate that widely used linear and nonlinear dimensionality reduction methods—including PCA—systematically discard both higher-order synergistic information and topological features. By unifying TDA’s geometric-topological perspective with MIT’s information-theoretic framework, this work constructs a theoretical bridge between the two paradigms and introduces a testable, integrative paradigm for analyzing higher-order structure in neuroimaging and other complex data domains.

Technology Category

• Machine Learning: Information Theory
• Data Mining & Knowledge Management: Mining of Visual, Multimedia & Multimodal Data
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• Web Mining and Content Analysis: Content-based information diffusion
• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization

Abstract

The study of irreducible higher-order interactions has become a core topic of study in complex systems. 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. 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 aim of assessing the similarities and differences between how the frameworks define ``higher-order structures."We begin with toy examples with known topologies, before turning to naturalistic data: fMRI signals collected from the human brain. We find that intrinsic, higher-order synergistic information is associated with three-dimensional cavities in a point cloud: shapes such as spheres are synergy-dominated. 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.

Problem

• Compares topological and information-theoretic approaches for higher-order interactions.
• Analyzes fMRI data to link synergistic information with topological cavities.
• Highlights limitations of PCA in preserving higher-order structures.

Innovation

• Linking topological and information-theoretic approaches
• Assessing higher-order structures in multivariate data
• Correlating synergistic information with topological cavities