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The physics of higher-order interactions in complex systems

Nature Physics volume 17, pages 1093–1098 (2021)Cite this article

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Abstract

Complex networks have become the main paradigm for modelling the dynamics of interacting systems. However, networks are intrinsically limited to describing pairwise interactions, whereas real-world systems are often characterized by higher-order interactions involving groups of three or more units. Higher-order structures, such as hypergraphs and simplicial complexes, are therefore a better tool to map the real organization of many social, biological and man-made systems. Here, we highlight recent evidence of collective behaviours induced by higher-order interactions, and we outline three key challenges for the physics of higher-order systems.

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Fig. 1: Pairwise and higher-order representations.
Fig. 2: Higher-order interactions lead to explosive phenomena.
Fig. 3: Higher-order systems are fully dynamical.
Fig. 4: Inference of higher-order systems is still an open and challenging problem.

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References

  1. Barabási, A.-L. The network takeover. Nat. Phys. 8, 14–16 (2011).

    Article  Google Scholar 

  2. Boccaletti, S., Latora, V., Moreno, Y., Chavez, M. & Hwang, D.-U. Complex networks: structure and dynamics. Phys. Rep. 424, 175–308 (2006).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  3. Barrat, A., Barthelemy, M. & Vespignani, A. Dynamical Processes on Complex Networks (Cambridge Univ. Press, 2008).

  4. Vespignani, A. Modelling dynamical processes in complex socio-technical systems. Nat. Phys. 8, 32–39 (2012).

    Article  Google Scholar 

  5. Pastor-Satorras, R. & Vespignani, A. Epidemic spreading in scale-free networks. Phys. Rev. Lett. 86, 3200–3203 (2001).

    Article  ADS  Google Scholar 

  6. Boguná, M., Pastor-Satorras, R. & Vespignani, A. Absence of epidemic threshold in scale-free networks with degree correlations. Phys. Rev. Lett. 90, 028701 (2003).

    Article  ADS  MATH  Google Scholar 

  7. Barahona, M. & Pecora, L. M. Synchronization in small-world systems. Phys. Rev. Lett. 89, 054101 (2002).

    Article  ADS  Google Scholar 

  8. Del Vicario, M. et al. The spreading of misinformation online. Proc. Natl Acad. Sci. USA 113, 554–559 (2016).

    Article  ADS  Google Scholar 

  9. Battiston, F. et al. Networks beyond pairwise interactions: structure and dynamics. Phys. Rep. 874, 1–92 (2020).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  10. Schneidman, E., Berry, M. J.II, Segev, R. & Bialek, W. Weak pairwise correlations imply strongly correlated network states in a neural population. Nature 440, 1007–1012 (2006).

    Article  ADS  Google Scholar 

  11. Schneidman, E. et al. Network information and connected correlations. Phys. Rev. Lett. 91, 238701 (2003).

    Article  ADS  Google Scholar 

  12. Yu, S. et al. Higher-order interactions characterized in cortical activity. J. Neurosci. 31, 17514–17526 (2011).

    Article  Google Scholar 

  13. Giusti, C., Pastalkova, E., Curto, C. & Itskov, V. Clique topology reveals intrinsic geometric structure in neural correlations. Proc. Natl Acad. Sci. USA 112, 13455–13460 (2015).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  14. Gardner, R. J. et al. Toroidal topology of population activity in grid cells. Preprint at bioRxiv https://doi.org/10.1101/2021.02.25.432776 (2021).

  15. Ganmor, E., Segev, R. & Schneidman, E. Sparse low-order interaction network underlies a highly correlated and learnable neural population code. Proc. Natl Acad. Sci. USA 108, 9679–9684 (2011).

    Article  ADS  Google Scholar 

  16. Merchan, L. & Nemenman, I. On the sufficiency of pairwise interactions in maximum entropy models of networks. J. Stat. Phys. 162, 1294–1308 (2016).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  17. Mayfield, M. M. & Stouffer, D. B. Higher-order interactions capture unexplained complexity in diverse communities. Nat. Ecol. Evol. 1, 0062 (2017).

    Article  Google Scholar 

  18. Cervantes-Loreto, A., Ayers, C., Dobbs, E., Brosi, B. & Stouffer, D. The context dependency of pollinator interference: how environmental conditions and species abundances impact floral visitation. Ecol. Lett. 24, 1443–1454 (2021).

    Article  Google Scholar 

  19. Bairey, E., Kelsic, E. D. & Kishony, R. High-order species interactions shape ecosystem diversity. Nat. Commun. 7, 12285 (2017).

    Article  ADS  Google Scholar 

  20. Grilli, J., Barabás, G., Michalska-Smith, M. J. & Allesina, S. Higher-order interactions stabilize dynamics in competitive network models. Nature 548, 210–213 (2017).

    Article  ADS  Google Scholar 

  21. Ritz, A., Tegge, A. N., Kim, H., Poirel, C. L. & Murali, T. Signaling hypergraphs. Trends Biotechnol. 32, 356–362 (2014).

    Article  Google Scholar 

  22. Centola, D., Becker, J., Brackbill, D. & Baronchelli, A. Experimental evidence for tipping points in social convention. Science 360, 1116–1119 (2018).

    Article  ADS  Google Scholar 

  23. Milojevi´c, S. Principles of scientific research team formation and evolution. Proc. Natl Acad. Sci. USA 111, 3984–3989 (2014).

    Article  ADS  Google Scholar 

  24. Povh, B., Rith, K., Scholz, C. & Zetsche, F. in Particles and Nuclei 281–303 (Springer, 1995).

  25. Duck, I. Three-alpha-particle resonances via the Fadeev equation. Nucl. Phys. 84, 586–594 (1966).

    Article  Google Scholar 

  26. Kim, H.-Y., Sofo, J. O., Velegol, D., Cole, M. W. & Lucas, A. A. van der Waals forces between nanoclusters: importance of many-body effects. J. Chem. Phys. 124, 074504 (2006).

    Article  ADS  Google Scholar 

  27. Zeiher, J. et al. Coherent many-body spin dynamics in a long-range interacting Ising chain. Phys. Rev. X 7, 041063 (2017).

    Google Scholar 

  28. Schaub, M. T., Benson, A. R., Horn, P., Lippner, G. & Jadbabaie, A. Random walks on simplicial complexes and the normalized Hodge Laplacian. SIAM Rev. 62, 353–391 (2020).

    Article  MathSciNet  MATH  Google Scholar 

  29. Carletti, T., Battiston, F., Cencetti, G. & Fanelli, D. Random walks on hypergraphs. Phys. Rev. E 101, 022308 (2020).

    Article  ADS  MathSciNet  Google Scholar 

  30. Millán, A. P., Torres, J. J. & Bianconi, G. Synchronization in network geometries with finite spectral dimension. Phys. Rev. E 99, 022307 (2019).

    Article  ADS  Google Scholar 

  31. Skardal, P. S. & Arenas, A. Abrupt desynchronization and extensive multistability in globally coupled oscillator simplexes. Phys. Rev. Lett. 122, 248301 (2019).

    Article  ADS  Google Scholar 

  32. Iacopini, I., Petri, G., Barrat, A. & Latora, V. Simplicial models of social contagion. Nat. Commun. 10, 2485 (2019).

    Article  ADS  Google Scholar 

  33. de Arruda, G. F., Petri, G. & Moreno, Y. Social contagion models on hypergraphs. Phys. Rev. Res. 2, 023032 (2020).

    Article  Google Scholar 

  34. Neuhäuser, L., Mellor, A. & Lambiotte, R. Multibody interactions and nonlinear consensus dynamics on networked systems. Phys. Rev. E 101, 032310 (2020).

    Article  ADS  MathSciNet  Google Scholar 

  35. Alvarez-Rodriguez, U. et al. Evolutionary dynamics of higher-order interactions in social networks. Nat. Hum. Behav. 5, 586–595 (2021).

    Article  Google Scholar 

  36. Achlioptas, D., D’Souza, R. M. & Spencer, J. Explosive percolation in random networks. Science 323, 1453–1455 (2009).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  37. da Costa, R. A., Dorogovtsev, S. N., Goltsev, A. V. & Mendes, J. F. F. Explosive percolation transition is actually continuous. Phys. Rev. Lett. 105, 255701 (2010).

    Article  ADS  Google Scholar 

  38. Riordan, O. & Warnke, L. Explosive percolation is continuous. Science 333, 322–324 (2011).

    Article  ADS  Google Scholar 

  39. D’Souza, R. M., Gómez-Gardeñes, J., Nagler, J. & Arenas, A. Explosive phenomena in complex networks. Adv. Phys. 68, 123–223 (2019).

    Article  ADS  Google Scholar 

  40. Gómez-Gardenes, J., Gómez, S., Arenas, A. & Moreno, Y. Explosive synchronization transitions in scale-free networks. Phys. Rev. Lett. 106, 128701 (2011).

    Article  ADS  Google Scholar 

  41. Matamalas, J. T., Gómez, S. & Arenas, A. Abrupt phase transition of epidemic spreading in simplicial complexes. Phys. Rev. Res. 2, 012049 (2020).

    Article  Google Scholar 

  42. Chowdhary, S., Kumar, A., Cencetti, G., Iacopini, I. & Battiston, F. Simplicial contagion in temporal higher-order networks. J. Phys. Complex. 2, 035019 (2021).

    Article  ADS  Google Scholar 

  43. St-Onge, G., Sun, H., Allard, A., Hébert-Dufresne, L. & Bianconi, G. Universal nonlinear infection kernel from heterogeneous exposure on higher-order networks. Preprint at https://arxiv.org/abs/2101.07229 (2021).

  44. Landry, N. W. & Restrepo, J. G. The effect of heterogeneity on hypergraph contagion models. Chaos 30, 103117 (2020).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  45. St-Onge, G. et al. Influential groups for seeding and sustaining hypergraph contagions. Preprint at https://arxiv.org/pdf/2105.07092.pdf (2021).

  46. Sun, H. & Bianconi, G. Higher-order percolation processes on multiplex hypergraphs. Preprint at https://arxiv.org/pdf/2104.05457.pdf (2021).

  47. Kuramoto, Y. Self-entrainment of a population of coupled non-linear oscillators. In Proc. International Symposium on Mathematical Problems in Theoretical Physics 420–422 (Springer, 1975).

  48. Skardal, P. S. & Arenas, A. Higher order interactions in complex networks of phase oscillators promote abrupt synchronization switching. Commun. Phys. 3, 218 (2020).

    Article  Google Scholar 

  49. Lucas, M., Cencetti, G. & Battiston, F. Multiorder Laplacian for synchronization in higher-order networks. Phys. Rev. Res. 2, 033410 (2020).

    Article  Google Scholar 

  50. Gambuzza, L. et al. Stability of synchronization in simplicial complexes. Nat. Commun. 12, 1255 (2021).

    Article  ADS  Google Scholar 

  51. Zhang, Y., Latora, V. & Motter, A. E. Unified treatment of synchronization patterns in generalized networks with higher-order, multilayer, and temporal interactions. Commun. Phys. 4, 195 (2021).

    Article  Google Scholar 

  52. de Arruda, G. F., Tizzani, M. & Moreno, Y. Phase transitions and stability of dynamical processes on hypergraphs. Commun. Phys. 4, 24 (2021).

    Article  Google Scholar 

  53. St-Onge, G., Thibeault, V., Allard, A., Dubé, L. J. & Hébert-Dufresne, L. Master equation analysis of mesoscopic localization in contagion dynamics on higher-order networks. Phys. Rev. E 103, 032301 (2021).

    Article  ADS  MathSciNet  Google Scholar 

  54. Kuehn, C. & Bick, C. A universal route to explosive phenomena. Sci. Adv. 7, eabe3824 (2021).

    Article  ADS  Google Scholar 

  55. Cisneros-Velarde, P. & Bullo, F. Multi-group SIS epidemics with simplicial and higher-order interactions. Preprint at https://arxiv.org/pdf/2005.11404.pdf (2020).

  56. Porter, M. & Gleeson, J. Dynamical Systems on Networks: A Tutorial. Frontiers in Applied Dynamical Systems: Reviews and Tutorials (Springer, 2016).

  57. Millán, A. P., Torres, J. J. & Bianconi, G. Explosive higher-order Kuramoto dynamics on simplicial complexes. Phys. Rev. Lett. 124, 218301 (2020).

    Article  ADS  MathSciNet  Google Scholar 

  58. Ghorbanchian, R., Restrepo, J. G., Torres, J. J. & Bianconi, G. Higher-order simplicial synchronization of coupled topological signals. Commun. Phys. 4, 120 (2021).

    Article  Google Scholar 

  59. Schaub, M. T. & Segarra, S. Flow smoothing and denoising: graph signal processing in the edge-space. In Proc. 2018 IEEE Global Conference on Signal and Information Processing (GlobalSIP) 735–739 (IEEE, 2018).

  60. Barbarossa, S. & Sardellitti, S. Topological signal processing over simplicial complexes. IEEE Trans. Signal Process. 68, 2992–3007 (2020).

    Article  ADS  Google Scholar 

  61. Holme, P. & Saramäki, J. Temporal networks. Phys. Rep. 519, 97–125 (2012).

    Article  ADS  Google Scholar 

  62. Gross, T. & Blasius, B. Adaptive coevolutionary networks: a review. J. R. Soc. Interface 5, 259–271 (2008).

    Article  Google Scholar 

  63. Young, J.-G., Petri, G. & Peixoto, T. P. Hypergraph reconstruction from network data. Commun. Phys. 4, 135 (2021).

    Article  Google Scholar 

  64. Saebi, M., Ciampaglia, G. L., Kaplan, L. M. & Chawla, N. V. HONEM: learning embedding for higher order networks. Big Data 8, 255–269 (2020).

    Article  Google Scholar 

  65. Newman, M. E. J. Network structure from rich but noisy data. Nat. Phys. 14, 542–545 (2018).

    Article  Google Scholar 

  66. Peixoto, T. P. Reconstructing networks with unknown and heterogeneous errors. Phys. Rev. X 8, 041011 (2018).

    Google Scholar 

  67. Musciotto, F., Battiston, F. & Mantegna, R. N. Detecting informative higher-order interactions in statistically validated hypergraphs. Commun. Phys. https://doi.org/10.1038/s42005-021-00710-4 (2021).

  68. Bohland, J. W. et al. A proposal for a coordinated effort for the determination of brainwide neuroanatomical connectivity in model organisms at a mesoscopic scale. PLoS Comput. Biol. 5, e1000334 (2009).

    Article  Google Scholar 

  69. Bassett, D. S. & Sporns, O. Network neuroscience. Nat. Neurosci. 20, 353–364 (2017).

    Article  Google Scholar 

  70. Glomb, K. et al. Computational models in electroencephalography. Brain Topogr. https://doi.org/10.1007/s10548-021-00828-2 (2021).

  71. Skudlarski, P. et al. Measuring brain connectivity: diffusion tensor imaging validates resting state temporal correlations. NeuroImage 43, 554–561 (2008).

    Article  Google Scholar 

  72. Tass, P. et al. Detection of n:m phase locking from noisy data: application to magnetoencephalography. Phys. Rev. Lett. 81, 3291–3294 (1998).

    Article  ADS  Google Scholar 

  73. Kralemann, B., Cimponeriu, L., Rosenblum, M., Pikovsky, A. & Mrowka, R. Phase dynamics of coupled oscillators reconstructed from data. Phys. Rev. E 77, 066205 (2008).

    Article  ADS  MathSciNet  Google Scholar 

  74. Kralemann, B., Pikovsky, A. & Rosenblum, M. Reconstructing phase dynamics of oscillator networks. Chaos 21, 025104 (2011).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  75. Kralemann, B., Pikovsky, A. & Rosenblum, M. Detecting triplet locking by triplet synchronization indices. Phys. Rev. E 87, 052904 (2013).

    Article  ADS  Google Scholar 

  76. Kralemann, B., Pikovsky, A. & Rosenblum, M. Reconstructing effective phase connectivity of oscillator networks from observations. N. J. Phys. 16, 085013 (2014).

    Article  Google Scholar 

  77. Pikovsky, A. Reconstruction of a random phase dynamics network from observations. Phys. Lett. A 382, 147–152 (2018).

    Article  ADS  MATH  Google Scholar 

  78. Barnett, L., Barrett, A. B. & Seth, A. K. Granger causality and transfer entropy are equivalent for Gaussian variables. Phys. Rev. Lett. 103, 238701 (2009).

    Article  ADS  Google Scholar 

  79. Schreiber, T. Measuring information transfer. Phys. Rev. Lett. 85, 461–464 (2000).

    Article  ADS  Google Scholar 

  80. Rosas, F. E., Mediano, P. A., Gastpar, M. & Jensen, H. J. Quantifying high-order interdependencies via multivariate extensions of the mutual information. Phys. Rev. E 100, 032305 (2019).

    Article  ADS  Google Scholar 

  81. Pearl, J. Causality (Cambridge Univ. Press, 2009).

  82. Peixoto, T. P. Network reconstruction and community detection from dynamics. Phys. Rev. Lett. 123, 128301 (2019).

    Article  ADS  Google Scholar 

  83. Bianconi, G., Darst, R. K., Iacovacci, J. & Fortunato, S. Triadic closure as a basic generating mechanism of communities in complex networks. Phys. Rev. E 90, 042806 (2014).

    Article  ADS  Google Scholar 

  84. Courtney, O. T. & Bianconi, G. Generalized network structures: the configuration model and the canonical ensemble of simplicial complexes. Phys. Rev. E 93, 062311 (2016).

    Article  ADS  Google Scholar 

  85. Chodrow, P. S. Configuration models of random hypergraphs. J. Complex Netw. 8, cnaa018 (2020).

    Article  MathSciNet  MATH  Google Scholar 

  86. Chodrow, P. S., Veldt, N. & Benson, A. R. Generative hypergraph clustering: from blockmodels to modularity. Sci. Adv. 7, eabh1303 (2021).

    Article  ADS  Google Scholar 

  87. Courtney, O. T. & Bianconi, G. Weighted growing simplicial complexes. Phys. Rev. E 95, 062301 (2017).

    Article  ADS  Google Scholar 

  88. Kovalenko, K. et al. Growing scale-free simplices. Commun. Phys. 4, 43 (2021).

    Article  Google Scholar 

  89. Peixoto, T. P. Disentangling homophily, community structure and triadic closure in networks. Preprint at https://arxiv.org/pdf/2101.02510.pdf (2021).

  90. Cencetti, G., Battiston, F., Lepri, B. & Karsai, M. Temporal properties of higher-order interactions in social networks. Sci. Rep. 11, 7028 (2021).

    Article  ADS  Google Scholar 

  91. Bianconi, G. & Rahmede, C. Emergent hyperbolic network geometry. Sci. Rep. 7, 41974 (2017).

    Article  ADS  Google Scholar 

  92. Torres, L., Blevins, A. S., Bassett, D. & Eliassi-Rad, T. The why, how, and when of representations for complex systems. SIAM Rev. 63, 435–485 (2021).

    Article  MathSciNet  MATH  Google Scholar 

  93. Bick, C., Gross, E., Harrington, H. A. & Schaub, M. T. What are higher-order networks? Preprint at https://arxiv.org/pdf/2104.11329.pdf (2021).

  94. Benson, A. R., Gleich, D. F. & Higham, D. J. Higher-order network analysis takes off, fueled by classical ideas and new data. Preprint at https://arxiv.org/pdf/2103.05031.pdf (2021).

  95. Golubitsky, M. & Stewart, I. Nonlinear dynamics of networks: the groupoid formalism. Bull. Am. Math. Soc. 43, 305–364 (2006).

    Article  MathSciNet  MATH  Google Scholar 

  96. Stewart, I., Golubitsky, M. & Pivato, M. Symmetry groupoids and patterns of synchrony in coupled cell networks. SIAM J. Appl. Dyn. Syst. 2, 609–646 (2003).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  97. Golubitsky, M., Stewart, I. & Török, A. Patterns of synchrony in coupled cell networks with multiple arrows. SIAM J. Appl. Dyn. Syst. 4, 78–100 (2005).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  98. Nakao, H. Phase reduction approach to synchronisation of nonlinear oscillators. Contemp. Phys. 57, 188–214 (2016).

    Article  ADS  Google Scholar 

  99. Pietras, B. & Daffertshofer, A. Network dynamics of coupled oscillators and phase reduction techniques. Phys. Rep. 819, 1–105 (2019).

    Article  ADS  MathSciNet  Google Scholar 

  100. Matheny, M. H. et al. Exotic states in a simple network of nanoelectromechanical oscillators. Science 363, eaav7932 (2019).

    Article  MathSciNet  Google Scholar 

  101. Komarov, M. & Pikovsky, A. Dynamics of multifrequency oscillator communities. Phys. Rev. Lett. 110, 134101 (2013).

    Article  ADS  Google Scholar 

  102. Ashwin, P. & Rodrigues, A. Hopf normal form with SN symmetry and reduction to systems of nonlinearly coupled phase oscillators. Physica D. 325, 14–24 (2016).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  103. Ashwin, P., Bick, C. & Burylko, O. Identical phase oscillator networks: bifurcations, symmetry and reversibility for generalized coupling. Front. Appl. Math. Stat. 2, 7 (2016).

    Article  Google Scholar 

  104. León, I. & Pazó, D. Phase reduction beyond the first order: the case of the mean-field complex Ginzburg–Landau equation. Phys. Rev. E 100, 012211 (2019).

    Article  ADS  MathSciNet  Google Scholar 

  105. Bick, C., Ashwin, P. & Rodrigues, A. Chaos in generically coupled phase oscillator networks with nonpairwise interactions. Chaos 26, 094814 (2016).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  106. Bick, C. Heteroclinic switching between chimeras. Phys. Rev. E 97, 050201 (2018).

    Article  ADS  Google Scholar 

  107. Bick, C. Heteroclinic dynamics of localized frequency synchrony: heteroclinic cycles for small populations. J. Nonlin. Sci. 29, 2547–2570 (2019).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  108. Kirkpatrick, T. R. & Thirumalai, D. Dynamics of the structural glass transition and the p-spin—interaction spin-glass model. Phys. Rev. Lett. 58, 2091–2094 (1987).

    Article  ADS  MathSciNet  Google Scholar 

  109. De Domenico, M., Granell, C., Porter, M. A. & Arenas, A. The physics of spreading processes in multilayer networks. Nat. Phys. 12, 901–906 (2016).

    Article  Google Scholar 

  110. Lambiotte, R., Rosvall, M. & Scholtes, I. From networks to optimal higher-order models of complex systems. Nat. Phys. 15, 313–320 (2019).

    Article  Google Scholar 

  111. Patania, A., Vaccarino, F. & Petri, G. Topological analysis of data. EPJ Data Sci. 6, 7 (2017).

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Network and Data Science, Central European University, Vienna, Austria

    Federico Battiston, Iacopo Iacopini & Tiago P. Peixoto

  2. Institute of Bioengineering/Center for Neuroprosthetics, Ecole Polytechnique Fédérale de Lausanne, Lausanne, Switzerland

    Enrico Amico

  3. Department of Radiology and Medical Informatics, University of Geneva, Geneva, Switzerland

    Enrico Amico

  4. Aix Marseille Univ, Université de Toulon, CNRS, CPT, Marseille, France

    Alain Barrat

  5. Tokyo Tech World Research Hub Initiative (WRHI), Tokyo Institute of Technology, Tokyo, Japan

    Alain Barrat

  6. School of Mathematical Sciences, Queen Mary University of London, London, UK

    Ginestra Bianconi & Vito Latora

  7. The Alan Turing Institute, The British Library, London, UK

    Ginestra Bianconi

  8. Mathematics and Complex Systems Research Area, ISI Foundation, Turin, Italy

    Guilherme Ferraz de Arruda, Yamir Moreno & Giovanni Petri

  9. Laboratory for Investigative Neurophysiology (The LINE), Department of Radiology, Lausanne University Hospital and University of Lausanne, Lausanne, Switzerland

    Benedetta Franceschiello & Micah M. Murray

  10. CIBM Center for Biomedical Imaging, University of Lausanne (CHUV-UNIL), Lausanne, Switzerland

    Benedetta Franceschiello & Micah M. Murray

  11. ISEM, CNRS, Université Montpellier, IRD, EPHE, Montpellier, France

    Sonia Kéfi

  12. Santa Fe Institute, Santa Fe, NM, USA

    Sonia Kéfi

  13. Dipartimento di Fisica ed Astronomia, Università di Catania, Catania, Italy

    Vito Latora

  14. INFN Sezione di Catania, Catania, Italy

    Vito Latora

  15. Complexity Science Hub Vienna (CSHV), Vienna, Austria

    Vito Latora & Yamir Moreno

  16. Institute for Biocomputation and Physics of Complex Systems (BIFI), University of Zaragoza, Zaragoza, Spain

    Yamir Moreno

  17. Department of Theoretical Physics, University of Zaragoza, Zaragoza, Spain

    Yamir Moreno

  18. Department of Hearing and Speech Sciences, Vanderbilt University, Nashville, TN, USA

    Micah M. Murray

  19. Department of Mathematical Sciences, University of Bath, Bath, UK

    Tiago P. Peixoto

  20. Dipartimento di Scienze Matematiche, Politecnico di Torino, Turin, Italy & SmartData@PoliTO, Politecnico di Torino, Turin, Italy

    Francesco Vaccarino

  21. ISI Global Science Foundation, New York, NY, USA

    Giovanni Petri

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  1. Federico Battiston
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Correspondence to Federico Battiston or Giovanni Petri.

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Battiston, F., Amico, E., Barrat, A. et al. The physics of higher-order interactions in complex systems. Nat. Phys. 17, 1093–1098 (2021). https://doi.org/10.1038/s41567-021-01371-4

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