Low-Dimensional Phase Diagram of Higher-Order Networked Systems

Jia-Jie Qin; Jack Murdoch Moore; Xiaozhu Zhang; Gang Yan

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arXiv · 2609.04692

Low-Dimensional Phase Diagram of Higher-Order Networked Systems

Jia-Jie Qin·Jack Murdoch Moore·Xiaozhu Zhang·Gang Yan

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Abstract

Higher-order networks exhibit rich critical phenomena that cannot be captured by traditional pairwise models. Here, we develop an analytical dimension-reduction framework that maps higher-order networked dynamics onto an effective low-dimensional system, allowing accurate prediction of tipping boundaries, bistability regions, and the nature of phase transitions. We demonstrate the power of this framework across a range of dynamical processes, revealing distinct effects of higher-order interactions on transition continuity and hysteresis. Furthermore, we find that system resilience exhibits a profound dependence on the alignment between pairwise and higher-order connectivity, with assortative mixing enhancing tipping toward active states. Our findings establish a general theory for understanding the critical transitions in higher-order networks, offering new insights for anticipating and managing systemic risk in complex systems.

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Jia-Jie Qin, Jack Murdoch Moore, Xiaozhu Zhang, Gang Yan. 2026-09-04. Low-Dimensional Phase Diagram of Higher-Order Networked Systems. https://arxiv.org/abs/2609.04692

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