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The hidden geometry of simplicial complexes can influence the collectivedynamics of nodes in different ways depending on the simplex-based interactionsof various orders and competition between local and global structural features.We study a system of phase oscillators attached to nodes of 4-dimensionalsimplicial complexes and interacting via positive/negative edges-based pairwise$K_1$ and triangle-based triple $K_2\geq 0$ couplings. Three prototypalsimplicial complexes are grown by aggregation of 5-cliques, controlled by thechemical affinity parameter $\nu$, resulting in sparse, mixed, and compactarchitecture, all of which have 1-hyperbolic graphs but different spectraldimensions. By changing the interaction strength $K_1\in[-4,2]$ along theforward and backward sweeps, we numerically determine individual phases of eachoscillator and a global order parameter to measure the level ofsynchronisation. Our results reveal how different architectures of simplicialcomplexes, in conjunction with the interactions and internal-frequencydistributions, impact the shape of the hysteresis loop and lead to patterns oflocally synchronised groups that hinder global network synchronisation.

Effect of hidden geometry and higher-order interactions on the  synchronization and hysteresis behaviour of phase oscillators on 5-cliques  simplicial assemblies