Simplicial complexes naturally describe discrete topological spaces. When their links are assigned a length they describe discrete geometries. As such simplicial complexes have been widely used in quantum gravity approaches that involve a discretization of spacetime. Recently they are becoming increasingly popular to describe complex interacting systems such a brain networks or social networks. In this talk we present non-equilibrium statistical mechanics approaches to model large simplicial complexes. We propose the simplicial complex model of Network Geometry with Flavor (NGF), we explore the hyperbolic nature of its emergent geometry and their relation with quantum statistics. Finally we reveal the rich interplay between Network Geometry with Flavor and dynamics. We investigate the percolation properties of NGF using the renormalization group finding BKT and discontinuous phase transitions depending on the dimensionality simplex. We also comment on the synchronization properties of NGF and the emergence of frustrated synchronization.
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