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Triadic percolation on multilayer networks

Hanlin Sun1, Filippo Radicchi2, and Ginestra Bianconi3

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Phys. Rev. E 113, 014313 – Published 21 January, 2026

DOI: https://doi.org/10.1103/yvtg-wnn4

Abstract

Triadic interactions are special types of higher-order interactions that occur when regulator nodes modulate the interactions between other two or more nodes. In the presence of triadic interactions, a percolation process occurring on a single-layer network becomes a full fledged dynamical system, characterized by period doubling and a route to chaos. Here we generalize the model to multilayer networks and name it as the multilayer triadic percolation (MTP) model. We find a much richer dynamical behavior of the MTP model than its single-layer counterpart. MTP displays a Neimark-Sacker bifurcation, leading to oscillations of arbitrarily large period or pseudoperiodic oscillations. Moreover, MTP admits period-two oscillations without negative regulatory interactions, whereas single-layer systems only display discontinuous hybrid transitions. This comprehensive model offers new insights on the importance of regulatory interactions in real-world systems such as brain networks, climate, and ecological systems.

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Phys. Rev. E 113, 014313– Published 21 January, 2026
  • Received 13 October 2025
  • Accepted 22 December 2025
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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. Funded by Bibsam.

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