The role of hyperedge overlap in reshaping dynamics of neural networks with higher-order interactions.

Abstract

Existing neural network models predominantly rely on pairwise interactions, often neglecting the higher-order interactions inherent in real-world neural systems. Even among studies that do incorporate higher-order structures, the dynamical mechanisms by which hyperedge overlap governs stability and catastrophe behavior remain poorly understood. To bridge this critical gap, this paper proposes a generalized star-topology neural network framework that explicitly incorporates higher-order interactions via hypergraphs, with a specific focus on the dynamical consequences of hyperedge overlap. By distinguishing between low-overlap and high-overlap topological configurations, we rigorously analyze the local stability and the existence of Hopf bifurcations through the derivation of characteristic equations and critical time delay thresholds. Theoretical analysis and extensive numerical simulations reveal a fundamental structural insight: compared to the low-overlap counterpart, the high-overlap configuration functions as a structural stabilizer. It significantly expands the stability domain by elevating bifurcation thresholds and effectively suppresses the amplitude of post-bifurcation limit cycles. Furthermore, scalability and robustness analyses demonstrate that these stabilizing effects persist across varying network scales and provide superior resilience against stochastic perturbations. By contrasting hypergraph dynamics with traditional pairwise frameworks, this work provides a novel mathematical perspective on how higher-order topological redundancy fundamentally shapes and stabilizes the collective dynamics of complex neural networks.