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Inhibition of explosive transitions in spreading dynamics by nonlinear higher-order adaptation

Longzhao Liu1,2,3,4,5, Hongwei Zheng6, Zhihao Han1,2, Xin Wang1,2,3,4,5,7,*, and Shaoting Tang1,8,2,9,3,4,5,†

  • 1Institute of Artificial Intelligence, Beihang University, Beijing 100191, China
  • 2Key laboratory of Mathematics, Informatics and Behavioral Semantics, Beihang University, Beijing 100191, China
  • 3Zhongguancun Laboratory, Beijing 100094, China
  • 4Beijing Advanced Innovation Center for Future Blockchain and Privacy Computing, Beihang University, Beijing 100191, China
  • 5State Key Laboratory of Complex & Critical Software Environment, Beihang University, Beijing 100191, China
  • 6Beijing Academy of Blockchain and Edge Computing, Beijing 100085, China
  • 7State Key Laboratory of General Artificial Intelligence, BIGAI, Beijing 100080, China
  • 8Hangzhou International Innovation Institute, Beihang University, Hangzhou 311115, China
  • 9Institute of Medical Artificial Intelligence, Binzhou Medical University, Yantai 264003, China

  • *Contact author: wangxin_1993@buaa.edu.cn
  • †Contact author: tangshaoting@buaa.edu.cn
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Phys. Rev. Research 8, 033068 – Published 16 July, 2026

DOI: https://doi.org/10.1103/5vff-syxb

Abstract

The coevolution of structure and dynamics, known as adaptation, is a fundamental property in various systems and drives diverse emergent behaviors. However, the adaptation in most works primarily stems from pairwise situations, which is insufficient to capture ubiquitous higher-order characteristics of real systems. Here, we introduce higher-order adaptation, characterized by baseline rewiring rate and nonlinear higher-order exponent, to model the coevolution of higher-order structure and spreading dynamics. We develop a hyperedge-based theoretical framework that well predicts critical behaviors and outbreak sizes. Notably, the inherent nonlinearity of higher-order adaptation inhibits bistability and explosive transitions, which is qualitatively opposite to the classical findings associated with pairwise adaptation or higher-order contagion. This inhibitory effect is enhanced by rewiring accuracy and remains robust across various adaptive spreading models. Our work establishes higher-order adaptation as a fundamentally distinct mechanism from pairwise adaptation, shedding light on adaptive systems.

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Phys. Rev. Research 8, 033068– Published 16 July, 2026
  • Received 21 August 2025
  • Accepted 1 July 2026
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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.

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