The combined impact of structure and coupling type on synchronization dynamics in hypernetworks

Hypernetworks, which consist of simultaneous multiple layers of interactions between nodes, with differing network structures for each interaction layer, provide a suitable framework for describing many real-world systems. This study investigates the relationship between the synchronization patterns in hypernetworks involving two distinct interactions and the corresponding single-interaction networks derived from them. The analysis considers the combined effects of the diffusive-linear coupling functions and the associated graphs, which feature ring and random structures. Two classes of coupling functions are explored: Type I, which produces unbounded synchronization regions where increased coupling strength reliably enhances synchronization, and Type II, which yields bounded regions where synchronization can deteriorate if coupling becomes too strong. The results reveal that when both coupling functions are of the same type (either Type I or Type II), the synchronization pattern of the hypernetwork falls between those of the corresponding single-interaction networks. In hypernetworks, where the coupling functions are of different types, the synchronization pattern resembles that of the single-interaction network associated with the coupling function that contributes more links in the hypernetwork. This study's findings contribute to a deeper understanding of the expected synchronization patterns in networks with multiple interaction layers, offering potential applications in predicting collective behaviors of complex real-world systems such as brain networks and ecological systems.