Hyperharmonic analysis for the study of high-order information-theoretic signals

TL;DR

This paper introduces a novel method combining harmonic analysis and topology to efficiently represent complex high-order information-theoretic signals, overcoming exponential growth challenges.

Contribution

It presents a new workflow using the Laplace-de Rham operator for hyperharmonic analysis of high-order signals in complex systems.

Findings

Efficient representation of high-order signals achieved

Structural properties encoded via Laplace-de Rham operator

Applicable to diverse complex system scenarios

Abstract

Network representations often cannot fully account for the structural richness of complex systems spanning multiple levels of organisation. Recently proposed high-order information-theoretic signals are well-suited to capture synergistic phenomena that transcend pairwise interactions; however, the exponential-growth of their cardinality severely hinders their applicability. In this work, we combine methods from harmonic analysis and combinatorial topology to construct efficient representations of high-order information-theoretic signals. The core of our method is the diagonalisation of a discrete version of the Laplace-de Rham operator, that geometrically encodes structural properties of the system. We capitalise on these ideas by developing a complete workflow for the construction of hyperharmonic representations of high-order signals, which is applicable to a wide range of scenarios.