Beyond Binary: Hypermatrix Algebra and Irreducible Arity in Higher-Order Systems

TL;DR

This paper introduces a hypermatrix algebra framework to model irreducible higher-order interactions, surpassing binary limitations in networks, with applications across sciences and insights into emergence phenomena.

Contribution

It develops a novel hypergraph and hypermatrix-based formalism to capture complex polyadic interactions beyond traditional binary models.

Findings

Irreducible higher-order interactions are widespread in natural and social systems.

The framework extends current higher-order network techniques with hypermatrix algebra.

Emergence may relate to variations in interaction arity over space and time.

Abstract

Theoretical and computational frameworks of modern science are dominated by binary structures. This binary bias, seen in the ubiquity of pair-wise networks and formal operations of two arguments in mathematical models, limits our capacity to faithfully capture irreducible polyadic interactions in higher-order systems. A paradigmatic example of a higher-order interaction is the Borromean link of three interlocking rings. In this paper we propose a mathematical framework via hypergraphs and hypermatrix algebras that allows to formalize such forms of higher-order bonding and connectivity in a parsimonious way. Our framework builds on and extends current techniques in higher-order networks -- still mostly rooted in binary structures such as adjacency matrices -- and incorporates recent developments in higher-arity structures to articulate the compositional behavior of adjacency hypermatrices. Irreducible higher-order interactions turn out to be a widespread occurrence across natural sciences and socio-cultural knowledge representation. We demonstrate this by reviewing recent results in computer science, physics, chemistry, biology, ecology, social science, and cultural analysis through the conceptual lens of irreducible higher-order interactions. We further speculate that the general phenomenon of emergence in complex systems may be characterized by spatio-temporal discrepancies of interaction arity.