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:

References

1. Abrams, P. A., Higher-order interactions stabilize dynamics in competitive network models, Am. Nat. 121 (1983) 887–891. Crossref, Web of Science, Google Scholar
2. Ainsworth, M. D. S., Attachments and other affectional bonds across the life cycle, in Attachment Across the Life Cycle (Routledge, 2006), pp. 41–59. Google Scholar
3. Ampatzoglou, I. and Pavlovic, N., Rigorous derivation of a ternary boltzmann equation for a classical system of particles, Commun. Math. Phys. 387 (2021) 793–863. Crossref, Web of Science, Google Scholar
4. Andrews, G. R., Foundations of Multithreaded, Parallel, and Distributed Programming, Vol. 11 (Addison-Wesley Reading, 2000). Google Scholar
5. Arsiwalla, X. D. and Gorard, J., Pregeometric spaces from wolfram model rewriting systems as homotopy types, preprint (2021), arXiv:2111.03460. Google Scholar
6. Arsiwalla, X. D., Gorard, J. and Elshatlawy, H., Homotopies in multiway (nondeterministic) rewriting systems as n-fold categories, Complex Syst. 31(3) (2022) 261–277. Crossref, Web of Science, Google Scholar
7. Arsiwalla, X. D., Sole, R., Moulin-Frier, C., Herreros, I., Sanchez-Fibla, M. and Verschure, P., The morphospace of consciousness, NeuroSci 4(2) (2023) 79–102. Crossref, Web of Science, Google Scholar
8. Arsiwalla, X. D. and Verschure, P., Measuring the complexity of consciousness, Front. Neurosci. 12 (2018) 424. Crossref, Web of Science, Google Scholar
9. Arsiwalla, X. D. and Verschure, P. F., Integrated information for large complex networks, in The 2013 Int. Joint Conf. Neural Networks (IJCNN) (IEEE, 2013), pp. 1–7. Crossref, Google Scholar
10. Arsiwalla, X. D. and Verschure, P. F., The global dynamical complexity of the human brain network, Appl. Netw. Sci. 1 (2016) 1–13. Crossref, Google Scholar
11. Arsiwalla, X. D. and Verschure, P. F., High integrated information in complex networks near criticality, in Int. Conf. Artificial Neural Networks (Springer, 2016), pp. 184–191. Crossref, Google Scholar
12. Attiya, H. and Welch, J., Distributed Computing: Fundamentals, Simulations, and Advanced Topics, Vol. 19 (John Wiley & Sons, 2004). Crossref, Google Scholar
13. Baas, N., New structures in complex systems, Eur. Phys. J.: Spec. Top. 178 (2009) 25–44. Crossref, Web of Science, Google Scholar
14. Baas, N. A., On structure and organization: An organizing principle, Int. J. Gen. Syst. 42 (2013) 170–196. Crossref, Web of Science, Google Scholar
15. Baas, N. A., Higher order architecture of collections of objects, Int. J. Gen. Syst. 44 (2015) 55–75. Crossref, Web of Science, Google Scholar
16. Baas, N. A., On the mathematics of higher structures, Int. J. Gen. Syst. 48 (2019) 603–624. Crossref, Web of Science, Google Scholar
17. Baas, N. A., On the philosophy of higher structures, Int. J. Gen. Syst. 48 (2019) 463–475. Crossref, Web of Science, Google Scholar
18. Baas, N. A., Fedorov, D., Jensen, A., Riisager, K., Volosniev, A. and Zinner, N., Higher-order Brunnian structures and possible physical realizations, Phys. At. Nucl. 77 (2014) 336–343. Crossref, Web of Science, Google Scholar
19. Baez, J. C. and Dolan, J., Higher-dimensional algebra III. n-categories and the algebra of opetopes, Adv. Math. 135 (1998) 145–206. Crossref, Web of Science, Google Scholar
20. Baez, J. C. and Huerta, J., An invitation to higher gauge theory, Gen. Relativ. Gravit. 43 (2011) 2335–2392. Crossref, Web of Science, Google Scholar
21. Balzarini, R. N., Dharma, C., Kohut, T., Holmes, B. M., Campbell, L., Lehmiller, J. J. and Harman, J. J., Demographic comparison of American individuals in polyamorous and monogamous relationships, J. Sex Res. 56 (2019) 681–694. Crossref, Web of Science, Google Scholar
22. Battiston, F., Amico, E., Barrat, A., Bianconi, G., de Arruda, G. F., Franceschiello, B., Iacopini, I., Kéfi, S., Latora, V., Moreno, Y., Murray, M. M., Peixoto, T. P., Vaccarino, F. and Petri, G., The physics of higher-order interactions in complex systems, Nat. Phys. 17 (2021) 1093–1098. Crossref, Web of Science, Google Scholar
23. Battiston, F., Cencetti, G., Iacopini, I., Latora, V., Lucas, M., Patania, A., Young, J.-G. and Petri, G., Networks beyond pairwise interactions: Structure and dynamics, Phys. Rep. 874 (2020) 1–92. Crossref, Web of Science, Google Scholar
24. Battiston, F. and Petri, G., Higher-Order Systems (Springer Nature, 2022). Crossref, Google Scholar
25. Benini, M., Schenkel, A. and Woike, L., Operads for algebraic quantum field theory, Commun. Contemp. Math. 23 (2021) 2050007. Link, Web of Science, Google Scholar
26. Benson, A. R., Abebe, R., Schaub, M. T., Jadbabaie, A. and Kleinberg, J., Simplicial closure and higher-order link prediction, Proc. Natl. Acad. Sci. USA 115 (2018) E11221–E11230. Crossref, Web of Science, Google Scholar
27. Benson, A. R., Gleich, D. F. and Leskovec, J., Higher-order organization of complex networks, Science 353 (2016) 163–166. Crossref, Web of Science, Google Scholar
28. Bertoni, M., Kiefer, F., Biasini, M., Bordoli, L. and Schwede, T., Modeling protein quaternary structure of homo-and hetero-oligomers beyond binary interactions by homology, Sci. Rep. 7 (2017) 1–15. Crossref, Web of Science, Google Scholar
29. Bhalla, U. S., Molecular computation in neurons: A modeling perspective, Curr. Opin. Neurobiol. 25 (2014) 31–37. Crossref, Web of Science, Google Scholar
30. Bhasin, V. S. and Mazumdar, I., Few Body Dynamics, Efimov Effect and Halo Nuclei (Springer, 2021). Crossref, Google Scholar
31. Biamonte, J. and Bergholm, V., Tensor networks in a nutshell, preprint (2017), arXiv:1708.00006. Google Scholar
32. Bianconi, G. et al., Complex systems in the spotlight: Next steps after the 2021 nobel prize in physics, J. Phys.: Complex. 4 (2023) 010201. Crossref, Google Scholar
33. Billick, I. and Case, T. J., Higher order interactions in ecological communities: What are they and how can they be detected? Ecology 75 (1994) 1529–1543. Crossref, Web of Science, Google Scholar
34. Bon, M., Vernizzi, G., Orland, H. and Zee, A., Topological classification of RNA structures, J. Mol. Biol. 379 (2008) 900–911. Crossref, Web of Science, Google Scholar
35. Boynton, R. M., Human color perception, in Science of Vision (Springer, 1990), pp. 211–253. Crossref, Google Scholar
36. Bratseth, J., Pettersen, H. and Solbakken, L., A tensor formalism for computer science, in 2021 Int. Symp. Electrical, Electronics and Information Engineering (ACM, 2021), pp. 295–300. Crossref, Google Scholar
37. Burr, D. C., Turi, M. and Anobile, G., Subitizing but not estimation of numerosity requires attentional resources, J. Vis. 10 (2010) 20. Crossref, Web of Science, Google Scholar
38. Carlsson, R., N-ary algebras, Nagoya Math. J. 78 (1980) 45–56. Crossref, Web of Science, Google Scholar
39. Carlucci, L., Ciani, G. and Proserpio, D. M., Borromean links and other non-conventional links in ‘polycatenated’ coordination polymers: Re-examination of some puzzling networks, CrystEngComm 5 (2003) 269–279. Crossref, Web of Science, Google Scholar
40. Cayley, A., The Collected Mathematical Papers of Arthur Cayley, Vol. 7 (Cambridge University Press, 1894). Google Scholar
41. Chen, J., Rauch, C. A., White, J. H., Englund, P. T. and Cozzarelli, N. R., The topology of the kinetoplast dna network, Cell 80 (1995) 61–69. Crossref, Web of Science, Google Scholar
42. Cheng, E., Weak n-categories: Opetopic and multitopic foundations, J. Pure Appl. Algebra 186 (2004) 109–137. Crossref, Web of Science, Google Scholar
43. Chichak, K. S., Cantrill, S. J., Pease, A. R., Chiu, S.-H., Cave, G. W., Atwood, J. L. and Stoddart, J. F., Molecular borromean rings, Science 304 (2004) 1308–1312. Crossref, Web of Science, Google Scholar
44. Clements, D. H., Sarama, J. and MacDonald, B. L., Subitizing: The neglected quantifier, in Constructing Number (Springer, 2019), pp. 13–45. Crossref, Google Scholar
45. Coecke, B. and Duncan, R., Interacting quantum observables: Categorical algebra and diagrammatics, New J. Phys. 13 (2011) 043016. Crossref, Web of Science, Google Scholar
46. Coecke, B. and Kissinger, A., Picturing quantum processes, in Int. Conf. Theory and Application of Diagrams (Springer, 2018), pp. 28–31. Crossref, Google Scholar
47. Cohen, E., Fifty years of kinetic theory, Physica A 194 (1993) 229–257. Crossref, Web of Science, Google Scholar
48. Cornelius, T. and Glockle, W., Efimov states for three 4He atoms? J. Chem. Phys. 85 (1986) 3906–3912. Crossref, Web of Science, Google Scholar
49. Corry, L., Modern Algebra and the Rise of Mathematical Structures (Springer Science & Business Media, 2003). Google Scholar
50. Courtney, O. T. and Bianconi, G., Generalized network structures: The configuration model and the canonical ensemble of simplicial complexes, Phys. Rev. E 93 (2016) 062311. Crossref, Web of Science, Google Scholar
51. David, A., Islam, S., Tankhilevich, E. and Sternberg, M. J., The alphafold database of protein structures: A biologist’s guide, J. Mol. Biol. 434 (2022) 167336. Crossref, Web of Science, Google Scholar
52. Davvaz, B., Dudek, W. A. and Vougiouklis, T., A generalization of n-ary algebraic systems, Commun. Algebra 37 (2009) 1248–1263. Crossref, Web of Science, Google Scholar
53. de Azcárraga, J. A. and Izquierdo, J. M., n-ary algebras: A review with applications, J. Phys. A: Math. Theor. 43 (2010) 293001. Crossref, Web of Science, Google Scholar
54. del Mármol, J., Yedlin, M. and Ruta, V., The structural basis of odorant recognition in insect olfactory receptors, Nature 597 (2021) 126–131. Crossref, Web of Science, Google Scholar
55. Dobnikar, J., Brunner, M., von Grünberg, H.-H. and Bechinger, C., Three-body interactions in colloidal systems, Phys. Rev. E 69 (2004) 031402. Crossref, Web of Science, Google Scholar
56. Dobnikar, J., Chen, Y., Rzehak, R. and von Grünberg, H.-H., Many-body interactions in colloidal suspensions, J. Phys. Condens. Matter 15 (2002) S263. Crossref, Web of Science, Google Scholar
57. Dudek, W. A., Remarks to glazek’s results on n-ary groups, preprint (2007), arXiv:0704.2749. Google Scholar
58. Dunbar, R. I., The social brain: Mind, language, and society in evolutionary perspective, Annual Review of Anthropology 32 (2003) 163–181. Crossref, Web of Science, Google Scholar
59. Efimov, V., Energy levels arising from resonant two-body forces in a three-body system, Phys. Lett. B 33 (1970) 563–564. Crossref, Web of Science, Google Scholar
60. Eliassi-Rad, T., Latora, V., Rosvall, M. and Scholtes, I., Higher-order graph models: From theoretical foundations to machine learning (Dagstuhl seminar 21352), Dagstuhl Rep. 11 (2021) 139–178. Google Scholar
61. Estrada, E. and Knight, P. A., A First Course in Network Theory (Oxford University Press, USA, 2015). Google Scholar
62. Fatemi, B., Taslakian, P., Vazquez, D. and Poole, D., Knowledge hypergraphs: Prediction beyond binary relations, preprint (2019), arXiv:1906.00137. Google Scholar
63. Fedorov, D. V., Jensen, A. S. and Riisager, K., Efimov states in halo nuclei, Phys. Rev. Lett. 73 (1994) 2817–2820. Crossref, Web of Science, Google Scholar
64. Filippov, V. T., n-lie algebras, Sib. Math. J. 26 (1985) 879–891. Crossref, Web of Science, Google Scholar
65. Flack, J. C., Coarse-graining as a downward causation mechanism, Philos. Trans. Roy. Soc. A 375 (2017) 20160338. Crossref, Web of Science, Google Scholar
66. Fraisse, R., Theory of Relations (Elsevier, 2000). Google Scholar
67. Gil-Ramírez, G., Leigh, D. A. and Stephens, A. J., Catenanes: Fifty years of molecular links, Angew. Chem. Int. Ed. 54 (2015) 6110–6150. Crossref, Web of Science, Google Scholar
68. Girard-Buttoz, C., Surbeck, M., Samuni, L., Boesch, C., Fruth, B., Crockford, C., Hohmann, G. and Wittig, R. M., Variable use of polyadic grooming and its effect on access to social partners in wild chimpanzees and bonobos, Anim. Behav. 168 (2020) 211–224. Crossref, Web of Science, Google Scholar
69. Gnang, E. K., A combinatorial approach to the algebra of hypermatrices, preprint (2014), arXiv:1403.3134. Google Scholar
70. Gnang, E. K., A hypermatrix analog of the general linear group, Notices Amer. Math. Soc. 68 (2021) 1693–1699. Crossref, Google Scholar
71. Gnang, E. K. and Filmus, Y., On the Bhattacharya-Mesner rank of third order hypermatrices, Linear Algebra Appl. 588 (2020) 391–418. Crossref, Web of Science, Google Scholar
72. Gorard, J., Namuduri, M. and Arsiwalla, X. D., Zx-calculus and extended hypergraph rewriting systems I: A multiway approach to categorical quantum information theory, preprint (2020), arXiv:2010.02752. Google Scholar
73. Gorard, J., Namuduri, M. and Arsiwalla, X. D., Fast automated reasoning over string diagrams using multiway causal structure, preprint (2021), arXiv:2105.04057. Google Scholar
74. Gorard, J., Namuduri, M. and Arsiwalla, X. D., Zx-calculus and extended wolfram model systems II: Fast diagrammatic reasoning with an application to quantum circuit simplification, preprint (2021), arXiv:2103.15820. Google Scholar
75. Gunaratne, S. A., Paper, printing and the printing press: A horizontally integrative macrohistory analysis, Gazette 63 (2001) 459–479. Crossref, Google Scholar
76. Hammer, H.-W. and Platter, L., Efimov states in nuclear and particle physics, Annu. Rev. Nucl. Part. Sci. 60 (2010) 207–236. Crossref, Web of Science, Google Scholar
77. Hogan, A. et al., Knowledge graphs, ACM Comput. Surv. 54 (2021) 1–37. Crossref, Web of Science, Google Scholar
78. Högberg, A., Gärdenfors, P. and Larsson, L., Knowing, learning and teaching–how homo became docens, Camb. Archaeol. J. 25 (2015) 847–858. Crossref, Web of Science, Google Scholar
79. Hollings, C. and Lawson, M. V., Wagner’s Theory of Generalised Heaps, Vol. 21 (Springer, 2017). Crossref, Google Scholar
80. Homans, G. C., Hare, A. P. and Polley, R. B., The Human Group (Routledge, 2017). Crossref, Google Scholar
81. Huang, F. W. and Reidys, C. M., Shapes of topological RNA structures, Math. Biosci. 270 (2015) 57–65. Crossref, Web of Science, Google Scholar
82. Iacopini, I., Karsai, M. and Barrat, A., The temporal dynamics of group interactions in higher-order social networks, preprint (2023), arXiv:2306.09967. Google Scholar
83. Kay, E. R. and Leigh, D. A., Rise of the molecular machines, Angew. Chem. Int. Ed. 54 (2015) 10080–10088. Crossref, Web of Science, Google Scholar
84. Kempes, C. P. and Krakauer, D. C., The multiple paths to multiple life, J. Mol. Evol. 89 (2021) 415–426. Crossref, Web of Science, Google Scholar
85. Kerner, R., The cubic chessboard, Class. Quantum Grav. 14 (1997) A203. Crossref, Web of Science, Google Scholar
86. Kerner, R., Ternary and non-associative structures, Int. J. Geom. Methods Mod. Phys. 5 (2008) 1265–1294. Link, Web of Science, Google Scholar
87. King, T. D., Human color perception, cognition, and culture: Why red is always red, in Color Imaging X: Processing, Hardcopy, and Applications, Vol. 5667 (International Society for Optics and Photonics, 2005), pp. 234–242. Crossref, Google Scholar
88. Klamt, S., Haus, U.-U. and Theis, F., Hypergraphs and cellular networks, PLoS Comput. Biol. 5 (2009) e1000385. Crossref, Web of Science, Google Scholar
89. Koch, C., Massimini, M., Boly, M. and Tononi, G., Neural correlates of consciousness: Progress and problems, Nat. Rev. Neurosci. 17 (2016) 307–321. Crossref, Web of Science, Google Scholar
90. Kohler, T. A., Bird, D. and Wolpert, D. H., Social scale and collective computation: Does information processing limit rate of growth in scale? J. Soc. Comput. 3 (2022) 1–17. Crossref, Google Scholar
91. Kolar, Z., Heap-ternary algebraic structure, Math. Commun. 5 (2000) 87–95. Google Scholar
92. Krakauer, D., Bertschinger, N., Olbrich, E., Flack, J. C. and Ay, N., The information theory of individuality, Theory Biosci. 139 (2020) 209–223. Crossref, Web of Science, Google Scholar
93. Krieg, M., Fläschner, G., Alsteens, D., Gaub, B. M., Roos, W. H., Wuite, G. J., Gaub, H. E., Gerber, C., Dufrêne, Y. F. and Müller, D. J., Atomic force microscopy-based mechanobiology, Nat. Rev. Phys. 1 (2019) 41–57. Crossref, Google Scholar
94. Leinster, T., Higher Operads, Higher Categories, Vol. 298 (Cambridge University Press, 2004). Crossref, Google Scholar
95. Letten, A. D. and Stouffer, D. B., The mechanistic basis for higher-order interactions and non-additivity in competitive communities, Ecol. Lett. 22 (2019) 423–436. Crossref, Web of Science, Google Scholar
96. Levine, J. M., Bascompte, J., Adler, P. B. and Allesina, S., Beyond pairwise mechanisms of species coexistence in complex communities, Nature 546 (2017) 56–64. Crossref, Web of Science, Google Scholar
97. Lurie, J., Higher Topos Theory (Princeton University Press, 2009). Crossref, Google Scholar
98. Main, K. H., Provan, J. I., Haynes, P. J., Wells, G., Hartley, J. A. and Pyne, A. L., Atomic force microscopy–a tool for structural and translational dna research, APL Bioeng. 5 (2021) 031504. Crossref, Web of Science, Google Scholar
99. Markl, M., Shnider, S. and Stasheff, J. D., Operads in Algebra, Topology and Physics, Vol. 96 (American Mathematical Society, 2002). Google Scholar
100. Márquez, L. M., Redman, R. S., Rodriguez, R. J. and Roossinck, M. J., A virus in a fungus in a plant: Three-way symbiosis required for thermal tolerance, Science 315 (2007) 513–515. Crossref, Web of Science, Google Scholar
101. Marshall, S. M., Moore, D. G., Murray, A. R., Walker, S. I. and Cronin, L., Formalising the pathways to life using assembly spaces, Entropy 24 (2022) 884. Crossref, Web of Science, Google Scholar
102. Massouros, C. and Massouros, G., An overview of the foundations of the hypergroup theory, Mathematics 9 (2021) 1014. Crossref, Web of Science, Google Scholar
103. Mayfield, M. M. and Stouffer, D. B., Higher-order interactions capture unexplained complexity in diverse communities, Nat. Ecol. Evol. 1 (2017) 1–7. Crossref, Web of Science, Google Scholar
104. Mediano, P. A., Rosas, F. E., Farah, J. C., Shanahan, M., Bor, D. and Barrett, A. B., Integrated information as a common signature of dynamical and information-processing complexity, Chaos 32 (2022) 013115. Crossref, Web of Science, Google Scholar
105. Mesner, D. M. and Bhattacharya, P., Association schemes on triples and a ternary algebra, J. Combin. Theory Ser. A 55 (1990) 204–234. Crossref, Web of Science, Google Scholar
106. Mesner, D. M. and Bhattacharya, P., A ternary algebra arising from association schemes on triples, J. Algebra 164 (1994) 595–613. Crossref, Web of Science, Google Scholar
107. Meyer, C. D., Forgan, R. S., Chichak, K. S., Peters, A. J., Tangchaivang, N., Cave, G. W. V., Khan, S. I., Cantrill, S. J. and Stoddart, J. F., The dynamic chemistry of molecular borromean rings and solomon knots, Chemistry 16 (2010) 12570–12581. Crossref, Web of Science, Google Scholar
108. Michieletto, D., Marenduzzo, D. and Orlandini, E., Is the kinetoplast DNA a percolating network of linked rings at its critical point? Phys. Biol. 12 (2015) 036001. Crossref, Web of Science, Google Scholar
109. Moskovich, D. and Carmi, A. Y., Tangle machines, Proc. R. Soc. A 471 (2015) 20150111. Crossref, Google Scholar
110. Naidon, P. and Endo, S., Efimov physics: A review, Rep. Prog. Phys. 80 (2017) 056001. Crossref, Web of Science, Google Scholar
111. Nambu, Y., Generalized hamiltonian dynamics, Phys. Rev. D 7 (1973) 2405–2414. Crossref, Web of Science, Google Scholar
112. Nambu, Y. and Han, M.-Y., Three triplets, paraquarks, and colored quarks, Phys. Rev. D 10 (1974) 674. Crossref, Web of Science, Google Scholar
113. Nayak, C., Simon, S. H., Stern, A., Freedman, M. and Sarma, S. D., Non-abelian anyons and topological quantum computation, Rev. Mod. Phys. 80 (2008) 1083. Crossref, Web of Science, Google Scholar
114. Neuhäuser, L., Lambiotte, R. and Schaub, M. T., Consensus dynamics on temporal hypergraphs, Phys. Rev. E 104 (2021) 064305. Crossref, Web of Science, Google Scholar
115. Neuhäuser, L., Mellor, A. and Lambiotte, R., Multibody interactions and nonlinear consensus dynamics on networked systems, Phys. Rev. E 101 (2020) 032310. Crossref, Web of Science, Google Scholar
116. Newman, M. E., The structure of scientific collaboration networks, Proc. Natl. Acad. Sci. USA 98 (2001) 404–409. Crossref, Web of Science, Google Scholar
117. Nickel, M., Murphy, K., Tresp, V. and Gabrilovich, E., A review of relational machine learning for knowledge graphs, Proc. IEEE 104 (2015) 11–33. Crossref, Web of Science, Google Scholar
118. Niemyska, W., Rubach, P., Gren, B. A., Nguyen, M. L., Garstka, W., da Silva, F. B., Rawdon, E. J. and Sulkowska, J. I., Alphaknot: Server to analyze entanglement in structures predicted by alphafold methods, Nucleic Acids Res. 50 (2022) W44–W50. Crossref, Web of Science, Google Scholar
119. Nooren, I. M. and Thornton, J. M., Diversity of protein–protein interactions, EMBO J. 22 (2003) 3486–3492. Crossref, Web of Science, Google Scholar
120. Núñez, R. and Fias, W., Ancestral mental number lines: What is the evidence? Cogn. Sci. 41 (2017) 2262–2266. Crossref, Web of Science, Google Scholar
121. Núñez, R. E., Is there really an evolved capacity for number? Trends Cogn. Sci. 21 (2017) 409–424. Crossref, Web of Science, Google Scholar
122. Oizumi, M., Albantakis, L. and Tononi, G., From the phenomenology to the mechanisms of consciousness: Integrated information theory 3.0, PLoS Comput. Biol. 10 (2014) e1003588. Crossref, Web of Science, Google Scholar
123. Ornes, S., How big data carried graph theory into new dimensions, Quanta Magazine (2021). Google Scholar
124. Ortiz-Muñoz, A., Medina-Abarca, H. F. and Fontana, W., Combinatorial protein–protein interactions on a polymerizing scaffold, Proc. Natl. Acad. Sci. USA 117 (2020) 2930–2937. Crossref, Web of Science, Google Scholar
125. Overmann, K. A., The material difference in human cognition, Adapt. Behav. 29 (2021) 123–135. Crossref, Web of Science, Google Scholar
126. Overmann, K. A., Early writing: A cognitive archaeological perspective on literacy and numeracy, Visible Lang. 56 (2022) 8–44. Crossref, Google Scholar
127. Pan, M. and Su, C.-Y., Coordination assembly of borromean structures, CrystEngComm 16 (2014) 7847–7859. Crossref, Web of Science, Google Scholar
128. Patania, A., Petri, G. and Vaccarino, F., The shape of collaborations, EPJ Data Sci. 6 (2017) 1–16. Crossref, Web of Science, Google Scholar
129. Peirce, C. S., Description of a Notation for the Logic of Relatives, Resulting from an Amplification of the Conceptions of Boole’s Calculus of Logic (Welch, Bigelow, 1870). Google Scholar
130. Peirce, C. S., On the algebra of logic, Am. J. Math. 3 (1880) 15–57. Crossref, Google Scholar
131. Pellissier Tanon, T., Vrandečić, D., Schaffert, S., Steiner, T. and Pintscher, L., From freebase to wikidata: The great migration, in Proc. 25th Int. Conf. World Wide Web (ACM, 2016), pp. 1419–1428. Crossref, Google Scholar
132. Phizicky, E. M. and Fields, S., Protein-protein interactions: Methods for detection and analysis, Microbiol. Rev. 59 (1995) 94–123. Crossref, Google Scholar
133. Ravasz, E., Somera, A. L., Mongru, D. A., Oltvai, Z. N. and Barabási, A.-L., Hierarchical organization of modularity in metabolic networks, Science 297 (2002) 1551–1555. Crossref, Web of Science, Google Scholar
134. Roopin, M., Thornhill, D. J., Santos, S. R. and Chadwick, N. E., Ammonia flux, physiological parameters, and symbiodinium diversity in the anemonefish symbiosis on red sea coral reefs, Symbiosis 53 (2011). Crossref, Web of Science, Google Scholar
135. Rosas, F. E., Mediano, P. A., Luppi, A. I., Varley, T. F., Lizier, J. T., Stramaglia, S., Jensen, H. J. and Marinazzo, D., Disentangling high-order mechanisms and high-order behaviours in complex systems, Nat. Phys. 18 (2022) 476–477. Crossref, Web of Science, Google Scholar
136. Rybolowicz, B., Topics in the theory of trusses, Ph.D. thesis, Swansea University (2021). Google Scholar
137. Rybołowicz, B. and Zapata-Carratalá, C., Biunit pairs in semiheaps and associated semigroups, preprint (2022), arXiv:2209.07203. Google Scholar
138. Scagliarini, T., Marinazzo, D., Guo, Y., Stramaglia, S. and Rosas, F. E., Quantifying high-order interdependencies on individual patterns via the local o-information: Theory and applications to music analysis, preprint (2021), arXiv:2108.11625. Google Scholar
139. Schich, M., Revealing matrices, in Beautiful Visualization: Looking at Data Through the Eyes of Experts (O’Reilly Media, 2010), pp. 227–254. Google Scholar
140. Schill, G., Catenanes, Rotaxanes, and Knots, Vol. 22 (Elsevier, 2017). Google Scholar
141. Schoenly, K., Beaver, R. and Heumier, T., On the trophic relations of insects: A food-web approach, Am. Nat. 137 (1991) 597–638. Crossref, Web of Science, Google Scholar
142. Schweitzer, F., Agents, Networks, Evolution (World Scientific, 2022). Link, Google Scholar
143. Sienko, T., Adamatzky, A. and Rambidi, N., Molecular Computing (MIT Press, 2003). Crossref, Google Scholar
144. Simpson, C., Homotopy Theory of Higher Categories: From Segal Categories to n-Categories and Beyond, Vol. 19 (Cambridge University Press, 2011). Crossref, Google Scholar
145. Solé, R. and Bascompte, J., Self-Organization in Complex Ecosystems. (MPB-42) (Princeton University Press, 2012). Google Scholar
146. Song, Y. and Buchwald, P., Tnf superfamily protein-protein interactions: Feasibility of small-molecule modulation, Curr. Drug Targets 16 (2015) 393–408. Crossref, Web of Science, Google Scholar
147. Stoddart, J. F., Dawning of the age of molecular nanotopology, Nano Lett. 20 (2020) 5597–5600. Crossref, Web of Science, Google Scholar
148. Tarski, A., On the calculus of relations, J. Symb. Log. 6 (1941) 73–89. Crossref, Google Scholar
149. Taylor, W. R. and Lin, K., Protein knots: A tangled problem, Nature 421 (2003) 25–25. Crossref, Web of Science, Google Scholar
150. Tononi, G., An information integration theory of consciousness, BMC Neurosci. 5 (2004) 1–22. Crossref, Web of Science, Google Scholar
151. Tononi, G., Boly, M., Massimini, M. and Koch, C., Integrated information theory: From consciousness to its physical substrate, Nat. Rev. Neurosci. 17 (2016) 450–461. Crossref, Web of Science, Google Scholar
152. Tononi, G. and Edelman, G. M., Consciousness and complexity, Science 282 (1998) 1846–1851. Crossref, Web of Science, Google Scholar
153. Torfing, J., Governance networks, in The Oxford Handbook of Governance (Oxford University Press, 2012), pp. 99–113. Google Scholar
154. Twyman, M. and Contractor, N., Team assembly, in Strategies for Team Science Success (Springer, 2019), pp. 217–240. Crossref, Google Scholar
155. Valverde, S., Vidiella, B., Montanez, R., Fraile, A., Sacristán, S. and García-Arenal, F., Coexistence of nestedness and modularity in host–pathogen infection networks, Nat. Ecol. Evol. 4 (2020) 568–577. Crossref, Web of Science, Google Scholar
156. Wagner, A. and Fell, D. A., The small world inside large metabolic networks, Proc. R. Soc. B 268 (2001) 1803–1810. Crossref, Web of Science, Google Scholar
157. Wikipedia, T. F. E., Arity (2022), https://en.wikipedia.org/w/index.php?title= Arity&oldid=1062674633. Google Scholar
158. Wolfram, S., A New Kind of Science (Wolfram Media, USA, 2002). Google Scholar
159. Wolfram, S., Multicomputation as a fourth general paradigm for theoretical science, Wolfram Media (2021). Google Scholar
160. Wootton, J. T., Putting the pieces together: Testing the independence of interactions among organisms, Ecology 75 (1994) 1544–1551. Crossref, Web of Science, Google Scholar
161. You, Y., Bibo, J. and Pollmann, F., Higher-order entanglement and many-body invariants for higher-order topological phases, Phys. Rev. Res. 2 (2020) 033192. Crossref, Google Scholar
162. Yu, S., Yang, H., Nakahara, H., Santos, G. S., Nikolic, D. and Plenz, D., Higher-order interactions characterized in cortical activity, J. Neurosci. 31 (2011) 17514–17526. Crossref, Web of Science, Google Scholar
163. Zapata-Carratala, C., Higher-arity science project (2023), https://www.arity.science/. Google Scholar
164. Zapata-Carratala, C. and Arsiwalla, X. D., An invitation to higher arity science, preprint (2022), arXiv:2201.09738. Google Scholar
165. Zapata-Carratala, C., Arsiwalla, X. D. and Beynon, T., Heaps of fish: Arrays, generalized associativity and heapoids, preprint (2022), arXiv:2205.05456. Google Scholar
166. Zhou, D., Huang, J. and Schölkopf, B., Learning with hypergraphs: Clustering, classification, and embedding, Adv. Neural Inf. Process. Syst. 19 (2006) 1601–1608. Google Scholar