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Higher-order Laplacian renormalization

Nature Physics (2025)Cite this article

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Abstract

The renormalization group is a pillar of the theory of scaling, scale invariance and universality in physics. Recently, this tool has been adapted to complex networks with pairwise interactions through a scheme based on diffusion dynamics. However, as the importance of polyadic interactions in complex systems becomes more evident, there is a pressing need to extend the renormalization group methods to higher-order networks. Here we fill this gap and propose a Laplacian renormalization group scheme for arbitrary higher-order networks. At the heart of our approach is the introduction of cross-order Laplacians, which generalize existing higher-order Laplacians by allowing the description of diffusion processes that can happen on hyperedges of any order via hyperedges of any other order. This approach enables us to probe higher-order structures, define scale invariance at various orders and propose a coarse-graining scheme. We validate our approach on controlled synthetic higher-order systems and then use it to detect the presence of order-specific scale-invariant profiles of real-world complex systems from multiple domains.

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Fig. 1: Cross-order Laplacian renormalization scheme: partition and coarse graining.
Fig. 2: Scale invariance and renormalization in pseudofractal simplicial complexes.
Fig. 3: Higher-order Laplacian renormalization scheme applied to NGF simplicial complexes.
Fig. 4: Higher-order scale invariance in real data.

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Data availability

The data supporting the results presented in this Article are available via GitHub at https://github.com/nplresearch/higher_order_LRG.

Code availability

The code supporting the results presented in this Article are available via GitHub at https://github.com/nplresearch/higher_order_LRG.

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Acknowledgements

M.N. acknowledges the project PNRR-NGEU, which has received funding from the MUR-DM 352/2022. M.L. is a postdoctoral researcher of the Fonds de la Recherche Scientifique–FNRS with project Under-Net 40016866. T.G. acknowledges financial support from the European Union NextGenerationEU–National Recovery and Resilience Plan (Piano Nazionale di Ripresa e Resilienza (PNRR)), project ‘SoBigData.it—Strengthening the Italian RI for Social Mining and Big Data Analytics’—grant IR0000013 (no. 3264, 28/12/2021). F.V. acknowledges financial support from the European Union NextGenerationEU (PIANO NAZIONALE DI RIPRESA E RESILIENZA (PNRR) – MISSIONE 4 COMPONENTE 2, INVESTIMENTO 1.3 – D.D. 1555 11/10/2022, PE00000013). This project was carried out within the project FAIR (Future Artificial Intelligence Research). We also thank P. Villegas, A. Gabrielli, C. Agostinelli, M. Neri and T. Robiglio for extremely valuable suggestions on preliminary versions of the manuscript.

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Authors and Affiliations

  1. Dipartimento di Scienze Matematiche, Politecnico di Torino, Turin, Italy

    Marco Nurisso & Francesco Vaccarino

  2. CENTAI Institute, Turin, Italy

    Marco Nurisso, Marta Morandini, Maxime Lucas & Giovanni Petri

  3. SmartData@PoliTO Center, Politecnico di Torino, Turin, Italy

    Marco Nurisso & Francesco Vaccarino

  4. Institut de Neurosciences de La Timone, UMR 7289, CNRS, Aix-Marseille Université, Marseille, France

    Marta Morandini

  5. Namur Institute for Complex Systems (naXys), Université de Namur, Namur, Belgium

    Maxime Lucas

  6. Laboratory of Mycology, Earth and Life Institute, Université catholique de Louvain, UCLouvain, Louvain-la-Neuve, Belgium

    Maxime Lucas

  7. Networks Unit, IMT Scuola Alti Studi Lucca, Lucca, Italy

    Tommaso Gili

  8. NPLab, Network Science Institute, Northeastern University London, London, UK

    Giovanni Petri

  9. Department of Physics, Northeastern University, Boston, MA, USA

    Giovanni Petri

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All authors designed the research and wrote the paper. M.N. and M.M. performed the computational simulations and data analysis.

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Correspondence to Tommaso Gili or Giovanni Petri.

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Nurisso, M., Morandini, M., Lucas, M. et al. Higher-order Laplacian renormalization. Nat. Phys. (2025). https://doi.org/10.1038/s41567-025-02784-1

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