HYPERMATRIX ALGEBRA AND IRREDUCIBLE ARITY IN HIGHER-ORDER SYSTEMS: CONCEPTS AND PERSPECTIVES

Carlos Zapata-Carratalã, Maximilian Schich (), Taliesin Beynon () and Xerxes D. Arsiwalla ()
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Carlos Zapata-Carratalã: Wolfram Institute, Champaign 61820-7237, Illinois, USA
Maximilian Schich: ERA Chair for Cultural Data Analytics, Baltic Film, Media and Arts School, Tallinn University, Narva mnt 25, 10120, Tallinn, Estonia
Taliesin Beynon: Mathematics Department, University of Cape Town, Woolsack Drive, 7701, Western Cape, South Africa
Xerxes D. Arsiwalla: Wolfram Research, Champaign 61820-7237, Illinois, USA

Advances in Complex Systems (ACS), 2023, vol. 26, issue 06, 1-22

Abstract: Theoretical and computational frameworks of complexity science are dominated by binary structures. This binary bias, seen in the ubiquity of pair-wise networks and formal binary operations 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.

Keywords: Higher-order system; complexity; emergence; non-reductionism; hypergraph; network science; category theory; arity; multicomputation; nuclear physics; quantum entanglement; semiotics; mathematical cognition; protein interactions; genome topology; molecular computing; assembly theory; cultural analytics; interdisciplinary research (search for similar items in EconPapers)
Date: 2023
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DOI: 10.1142/S0219525923500078

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