Multiscale fractality in partial phase synchronisation on simplicial complexes around brain hubs

Brain hubs are best connected central nodes in the human connectome that play a critical role in integrated brain dynamics. How the hubs perform their function vs different dynamical processes and the role of their higher-order connections involving different brain regions remains elusive. Here we simulate the phase synchronisation processes on the human connectome core network consisting of the eight brain hubs and all attached simplexes of different sizes. The leading pairwise interactions among neighbouring nodes are assumed, taking into account the natural weights of edges. Our results reveal that increasing the positive pairwise couplings promotes a continuous synchronisation while a weak partial synchronisation occurs for a wide range of negative couplings. The weights of edges stabilise the synchronisation process supporting the absence of hysteresis. Furthermore, the time evolution of the order parameter shows cyclic fluctuations induced by the concurrent evolution of phases associated with different groups of nodes. We show that these oscillations exhibit long-range temporal correlations and multifractality. The asymmetrical singularity spectra are determined, which vary with the time scale and depend on the weights of edges. These findings suggest a possible way that the brain functional geometry maintains a desirable low-level synchrony through complex patterns of phase fluctuations.