Network Geometry and Complexity.

Recently higher order networks describing the interactions between two or more nodes are attracting large attention. Most notably higher order networks include simplicial complexes formed not only by nodes and links but also by triangles, tetrahedra, etc. glued along their faces. Simplicial complexes and in general higher order networks are able to characterize data as different as functional brain networks or collaboration networks beyond the framework of pairwise interactions. Interestingly higher order networks have a natural geometric interpretation and therefore constitute the 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 in order to capture higher order networks formed not only by simplices but by any type of regular polytope. We reveal the interplay between complexity and geometry of the higher-order networks by studying the emergent community structure and the dependence of the degree distribution on the dimension and the nature of the regular polytope forming the building blocks of the networked structure. 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.