Moderate Higher-Order Interactions Enhance Stability While Preserving Basin Structure

TL;DR

Problem: Higher-order interactions in Kuramoto oscillators on ring networks can reshape both basin structure and energetic stability, potentially altering synchronization difficulty. Approach: The authors study a ring network of $n=83$ oscillators with local radius $r=2$, incorporating pairwise coupling $\sigma$ and triadic coupling $\sigma_\Delta$, and analyze twisted states with winding number $q$ using mean first passage times under Gaussian noise and quasipotential theory. Findings: In a moderate coupling regime, twisted states dominate the basin and the distribution among twisted states remains fixed, while increasing $\sigma$ or $\sigma_\Delta$ deepens quasipotential wells, increasing MFPTs; higher $|q|$ states are less stable. Significance: The work reveals that higher-order interactions can enhance stabilization through energetic landscapes without major basin reshaping, offering guidance for robust synchronization in neural and infrastructural networks.

Abstract

Synchronization is a ubiquitous phenomenon in complex systems. The Kuramoto model serves as a paradigmatic framework for understanding how coupled oscillators achieve collective rhythm. Conventional approaches focus on pairwise interactions, but real-world systems frequently involve higher-order couplings among multiple elements. Previous studies have shown that higher-order interactions enrich dynamics but generally shrink the attraction basin of synchronized states, making synchronization harder to achieve. Here, we demonstrate this picture is incomplete. Through systematic analysis of twisted states on ring networks, we identify a moderate coupling regime where higher-order interactions enhance stability without altering basin structure. The relative distribution among twisted states remains constant, yet quasipotential barriers deepen as coupling strengths increase. By measuring mean first passage times, we show both pairwise and higher-order couplings contribute synergistically to enhance stability, consistent with large deviation theory. These findings provide new insights into the role of higher-order interactions in synchronization.

Figures (5)

• Figure 1: Proportion of state space occupied by twisted states ($R = 1$) as a function of triadic coupling strength $\sigma_\Delta$ for different pairwise coupling strengths. Numerical results based on $1.6 \times 10^5$ random initial conditions.
• Figure 2: Relative proportions of twisted states with different winding numbers in the regime where twisted states dominate the basin structure. (a) Distribution of twisted states as a function of pairwise coupling strength $\sigma$ with fixed $\sigma_\Delta = 1.0$. (b) Distribution as a function of triadic coupling strength $\sigma_\Delta$ with fixed $\sigma = 2.0$. (c-f) Representative phase configurations for twisted states with winding numbers $|q| = 0, 1, 2, 3$, respectively. Numerical results based on $1.6 \times 10^5$ random initial conditions uniformly distributed in $[-\pi, \pi)^n$.
• Figure 3: Linear fitting of mean first passage time versus noise intensity. (a) Transitions from synchronized state ($|q| = 0$) to non-zero winding states with varying $\sigma_\Delta$ at fixed $\sigma = 2.0$. (b) Transitions from synchronized state to non-synchronized twisted states with varying $\sigma$ at fixed $\sigma_\Delta = 1.0$. Here, $k$ (shown in legend) denotes the slope of the fitted line. The colored bars represent the standard deviations of $\ln(\tau_e)$ for each corresponding data series. Parameters: 1600 sample trajectories.
• Figure 4: Linear fitting of mean first passage time versus noise intensity for transitions from non-synchronized twisted states to synchronized state. (a) Varying triadic coupling strengths $\sigma_\Delta$ at fixed $\sigma = 2.0$. (b) Varying pairwise coupling strengths $\sigma$ at fixed $\sigma_\Delta = 1.0$. Here, $k$ (shown in legend) denotes the slope of the fitted line. The colored bars represent the standard deviations of $\ln(\tau_e)$ for each corresponding data series. Parameters: 1600 sample trajectories.
• Figure 5: Linear fitting of mean first passage time versus noise intensity for transitions from specific twisted states to all other states. Panels (a-c) show results at fixed $\sigma = 2.0$ for twisted states with winding numbers $|p| = 1, 2, 3$, respectively, with varying $\sigma_\Delta$. Panels (d-f) show results at fixed $\sigma_\Delta = 1.0$ for $|p| = 1, 2, 3$, respectively, with varying $\sigma$. Here, $k$ (shown in legend) denotes the slope of the fitted line. The colored bars represent the standard deviations of $\ln(\tau_e)$ for each corresponding data series. Parameters: 1600 sample trajectories.