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arXiv · 2301.07494

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

Carlos Zapata-Carratalá·Maximilian Schich·Taliesin Beynon·Xerxes D. Arsiwalla
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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.

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Carlos Zapata-Carratalá, Maximilian Schich, Taliesin Beynon, Xerxes D. Arsiwalla. 2023-01-05. Beyond Binary: Hypermatrix Algebra and Irreducible Arity in Higher-Order Systems. https://arxiv.org/abs/2301.07494

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