A one-dimensional reduction for dynamical systems on networks with purely higher-order interactions is developed, formulated through an effective higher-order interaction strength, associated with the triangular interactions of the underlying network and the dynamical system's effective state.
Abstract
Low-dimensional reductions provide a useful framework for studying high-dimensional dynamics on complex networks, but most existing approaches are restricted to pairwise interactions. Here, we develop a one-dimensional reduction for dynamical systems on networks with purely higher-order interactions. The reduction is formulated through an effective higher-order interaction strength ($\beta_{\Delta}$), associated with the triangular interactions of the underlying network and the dynamical system's effective state. We present a theoretical framework for the dimension-reduction approach and validate it across three dynamical models with exclusively higher-order interactions. We find that the reduction accuracy is mainly determined by the homogeneity of node states, i.e., the deviations in state values become very small. Numerical results on synthetic and real networks show that the reduced model captures the effective steady states and transitions of the full system with good accuracy.
We derive a one-dimensional reduction for nonlinear dynamics on simplicial complexes containing both pairwise and triangular (higher-order) interactions. The effective state is defined using a mixed weight determined by the pairwise and triangular degrees of each node. The resulting reduced equation retains two structural coefficients, associated separately with the pairwise and higher-order coupling channels. A fluctuation expansion identifies the closure assumptions underlying the reduction and shows how deviations of individual node states from the effective state contribute to the approximation error. We numerically validate the proposed framework on Gene-regulatory dynamics, the double-well system, and SIS spreading. The states of the reduced model are compared with full-network simulations through coupling-parameter sweeps, steady-state branch calculations, and progressive node-removal experiments on synthetic and real-world networks. The reduced model successfully reproduces the principal transitions and steady-state branches in all three dynamical systems considered. Agreement is strongest for relatively homogeneous networks and deteriorates when structural heterogeneity produces a broader distribution of node states. The closure diagnostics account for this loss of accuracy and indicate when a single effective state is no longer sufficient. The reduction therefore provides a tractable description of resilience in systems with coexisting pairwise and higher-order interactions.
Amit Tiwari, C. Hens, Prosenjit Kundu· 0 citations
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.
Across all the realizations, SSM/gSSM consistently outperform classical spectral and mean-field methods in modeling critical transitions at both microscopic and macroscopic scales, establishing SSM-based reduction as a robust, interpretable tool for nonlinear networked systems with broad applicability to epidemiology, ecology, and engineered networks.
A novel large-scale delayed neural network with dual-star architecture, incorporating both second-order and third-order interactions, and the accurate function fitting achieved by the model provides new insights for the design and control of neural networks.
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The Kuramoto model is a canonical paradigm for characterizing the synchronization dynamics of coupled systems. Recently, Kuramoto systems with higher-order coupling have emerged as a growing research hotspot, yet most studies have largely focused on dynamical behavior analysis, with little attention on the control of global network behaviors. Building on the network stochastic resonance theory proposed by Wang et al (2026 Proc. R. Soc. A: Math. Phys. Eng. Sci. 482 20250945), this study extends the control channel from three-body coupling modulation to pairwise coupling modulation, and proposes an open-loop dynamical regulation strategy for global collective behaviors. This strategy leverages network stochastic resonance to achieve effective control without the need for real-time monitoring of the network. The findings of this research provide useful insights for the regulation of collective dynamics in higher-order coupled Kuramoto networks and the open-loop dynamic control of complex oscillator networks.
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A bottom-up derivation of higher-order SIS dynamics, building systematically from node-level equations to pairs, triplets, and three-body interactions and provides a principled method of generating new models.
Kevin Teo, Péter L. Simon, I. Z. Kiss· 0 citations
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