Carlos Zapata-Carratala, Maximilian Schich, Taliesin Beynon, Xerxes D. Arsiwalla
{"title":"Hypermatrix Algebra and Irreducible Arity in Higher-Order Systems: Concepts and Perspectives","authors":"Carlos Zapata-Carratala, Maximilian Schich, Taliesin Beynon, Xerxes D. Arsiwalla","doi":"10.1142/s0219525923500078","DOIUrl":null,"url":null,"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.","PeriodicalId":50871,"journal":{"name":"Advances in Complex Systems","volume":null,"pages":null},"PeriodicalIF":0.7000,"publicationDate":"2023-11-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Advances in Complex Systems","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1142/s0219525923500078","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"MATHEMATICS, INTERDISCIPLINARY APPLICATIONS","Score":null,"Total":0}
引用次数: 1
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.
期刊介绍:
Advances in Complex Systems aims to provide a unique medium of communication for multidisciplinary approaches, either empirical or theoretical, to the study of complex systems. The latter are seen as systems comprised of multiple interacting components, or agents. Nonlinear feedback processes, stochastic influences, specific conditions for the supply of energy, matter, or information may lead to the emergence of new system qualities on the macroscopic scale that cannot be reduced to the dynamics of the agents. Quantitative approaches to the dynamics of complex systems have to consider a broad range of concepts, from analytical tools, statistical methods and computer simulations to distributed problem solving, learning and adaptation. This is an interdisciplinary enterprise.