  Higher order networks are able to characterize data as different as
functional brain networks, protein interaction networks and social networks
beyond the framework of pairwise interactions. Most notably higher order
networks include simplicial complexes formed not only by nodes and links but
also by triangles, tetrahedra, etc. More in general, higher-order networks can
be cell-complexes formed by gluing convex polytopes along their faces.
Interestingly, higher order networks have a natural geometric interpretation
and therefore constitute a natural way to explore the discrete network geometry
of complex networks. Here we investigate the rich interplay between emergent
network geometry of higher order networks and their complexity in the framework
of a non-equilibrium model called Network Geometry with Flavor. This model,
originally proposed for capturing the evolution of simplicial complexes, is
here extended to cell-complexes formed by subsequently gluing different copies
of an arbitrary regular polytope. We reveal the interplay between complexity
and geometry of the higher order networks generated by the model by studying
the emergent community structure and the degree distribution as a function of
the regular polytope forming its building blocks. Additionally we discuss the
underlying hyperbolic nature of the emergent geometry and we relate the
spectral dimension of the higher-order network to the dimension and nature of
its building blocks.
