Synchronization on Simplicial Complexes*

The synchronization of oscillators based on nodes of simplicial complexes --structures made by aggregation of generalized triangles (simplexes of different sizes) exhibits some exciting features. In our analysis [1,2], considering 4-dimensional simplicial complexes, the transition to a synchronized state can be abrupt or smooth depending on the geometry and spectral dimension of the underlying simplicial complex, the internal frequency distribution of the oscillators, and the initial conditions, i.e., the distribution of phase angles. The level of

synchrony is described by an order parameter that suitably quantifies both partial and complete synchronization. Adding interactions of higher order, e.g., three-point interactions based on triangle faces of the simplexes,  modifies these features further with new phenomena, especially where the interactions tend to be of opposite signs. Mainly, the hysteresis loop appears with an abrupt desynchronization transition and cluster synchronization, leading to partial order with multifractal oscillations of the order parameter [3]. Similar dynamical behaviours can be expected in realistic systems such as nanomaterials and the brain connectome.

Work done in collaboration with Samir Sahoo and Bosiljka  Tadic