Geometry, Topology and Simplicial Synchronization

Simplicial synchronization reveals the role that topology and geometry have in determining the dynamical properties of simplicial complexes. Simplicial network geometry and topology are naturally encoded in the spectral properties of the graph Laplacian and of the higher-order Laplacians of simplicial complexes. Here we show how the geometry of simplicial complexes induces spectral dimensions of the simplicial complex Laplacians that are responsible for changing the phase diagram of the Kuramoto model. In particular, simplicial complexes displaying a non-trivial simplicial network geometry cannot sustain a synchronized state in the infinite network limit if their spectral dimension is smaller or equal to four. This theoretical result is here verified on the Network Geometry with Flavor simplicial complex generative model displaying emergent hyperbolic geometry. On its turn simplicial topology is shown to determine the dynamical properties of the higher-order Kuramoto model. The higher-order Kuramoto model describes synchronization of topological signals, i.e., phases not only associated to the nodes of a simplicial complexes but associated also to higher-order simplices, including links, triangles and so on. This model displays discontinuous synchronization transitions when topological signals of different dimension and/or their solenoidal and irrotational projections are coupled in an adaptive way.