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  • Perspective
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Topology shapes dynamics of higher-order networks

Nature Physics volume 21, pages 353–361 (2025)Cite this article

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Abstract

Higher-order networks capture the many-body interactions present in complex systems, shedding light on the interplay between topology and dynamics. The theory of higher-order topological dynamics, which combines higher-order interactions with discrete topology and nonlinear dynamics, has the potential to enhance our understanding of complex systems, such as the brain and the climate, and to advance the development of next-generation AI algorithms. This theoretical framework, which goes beyond traditional node-centric descriptions, encodes the dynamics of a network through topological signals—variables assigned not only to nodes but also to edges, triangles and other higher-order cells. Recent findings show that topological signals lead to the emergence of distinct types of dynamical state and collective phenomena, including topological and Dirac synchronization, pattern formation and triadic percolation. These results offer insights into how topology shapes dynamics, how dynamics learns topology and how topology evolves dynamically. This Perspective primarily aims to guide physicists, mathematicians, computer scientists and network scientists through the emerging field of higher-order topological dynamics, while also outlining future research challenges.

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Fig. 1: The emerging field of higher-order topological dynamics of complex systems.
Fig. 2: The dynamical state of a higher-order network.
Fig. 3: The topological Kuramoto model and global synchronization.
Fig. 4: The properties of the topological Dirac operator and the topological Dirac equation.
Fig. 5: Signed triadic interactions and the phase diagram of triadic percolation.

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Acknowledgements

This work was partially supported by a grant from the Simons Foundation (to G.B.). A.P.M., H.S., T. C. and G.B. thank the Isaac Newton Institute for Mathematical Sciences, Cambridge, for support and hospitality during the programme Hypergraphs: Theory and Applications, where work on this paper was undertaken. This work was supported by EPSRC grant number EP/V521929/1. A.P.M. acknowledges financial support from the ‘Ramón y Cajal’ programme of the Spanish Ministry of Science and Innovation (grant number RYC2021-031241-I). A.P.M. and J.J.T. acknowledge financial support under grant number PID2023-149174NB-I00 financed by the Spanish Ministry and Agencia Estatal de Investigación MICIU/AEI/10.13039/501100011033 and EDRF funds (European Union). H.S. is supported by the Wallenberg Initiative on Networks and Quantum Information (WINQ). R.M. acknowledges JSPS, Japan KAKENHI JP22K11919, JP22H00516, and JST, Japan CREST JP-MJCR1913 for financial support. F.R. acknowledges support from the Army Research Office under contract number W911NF-21-1-0194 and the Air Force Office of Scientific Research under award numbers FA9550-21-1-0446 and FA9550-24-1-0039. G.B. acknowledges discussions with M. Niedostatek, R. Wang and A. A. A. Zaid.

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Authors and Affiliations

  1. Institute ‘Carlos I’ for Theoretical and Computational Physics and Electromagnetism and Matter Physics Department, University of Granada, Granada, Spain

    Ana P. Millán & Joaquín J. Torres

  2. Nordita, KTH Royal Institute of Technology and Stockholm University, Stockholm, Sweden

    Hanlin Sun

  3. Department of Physics, Freie Universiät Berlin, Berlin, Germany

    Lorenzo Giambagli

  4. Department of Physics and Astronomy, University of Florence, INFN & CSDC, Sesto Fiorentino, Italy

    Lorenzo Giambagli

  5. Department of Systems and Control Engineering, Institute of Science Tokyo (former Tokyo Tech), Tokyo, Japan

    Riccardo Muolo

  6. Department of Mathematics and NaXys, Namur Institute for Complex Systems, University of Namur, Namur, Belgium

    Timoteo Carletti

  7. Center for Complex Networks and Systems Research, Luddy School of Informatics, Computing, and Engineering, Indiana University, Bloomington, IN, USA

    Filippo Radicchi

  8. Potsdam Institute for Climate Impact Research, Potsdam, Germany

    Jürgen Kurths

  9. Department of Physics, Humboldt University of Berlin, Berlin, Germany

    Jürgen Kurths

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

    Ginestra Bianconi

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

    Ginestra Bianconi

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  1. Ana P. Millán
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  2. Hanlin Sun
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  3. Lorenzo Giambagli
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G.B. designed the project. A.P.M., H.S., L.G., R.M., T.C., J.J.T. and G.B. prepared the figures and wrote the codes. All authors wrote the manuscript.

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Correspondence to Ginestra Bianconi.

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Millán, A.P., Sun, H., Giambagli, L. et al. Topology shapes dynamics of higher-order networks. Nat. Phys. 21, 353–361 (2025). https://doi.org/10.1038/s41567-024-02757-w

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