Analyzing the relationship between synchronization dynamics in hypernetworks and their single-interaction counterparts

Hypernetworks provide a proper framework for representing systems in which units interact simultaneously through multiple layers of interactions. Synchronization is a fundamental collective behavior in such systems, yet its stability analysis remains challenging. Studies have often assumed hypernetworks with commuting Laplacians or purely linear interactions, conditions that rarely hold in real-world systems. This paper addresses this gap by investigating hypernetworks with non-commuting Laplacians, where oscillators interact through both linear and nonlinear diffusive coupling functions on random and ring topologies. Because the Master Stability Function (MSF) cannot be decoupled in such systems, direct stability analysis is intractable. To overcome this limitation, we establish a connection between the synchronization dynamics in hypernetworks and those of their derived single-interaction counterparts, enabling qualitative predictions of the MSF structure. Numerical simulations with Lorenz oscillators demonstrate that hypernetworks exhibit intermediate synchronization regions shaped by the dominant coupling function. Additionally, nonlinear interactions enhance synchronizability but increase coupling energy demands. An inverse relationship between pre-synchronization coupling energy and required time to achieve synchronization is consistently observed. These findings provide a systematic framework and offer novel insights for interpreting and predicting the dynamics and stability of synchronization in real-world complex networks with multiple interaction modes.