Recently higher order networks describing the interactions between two or more nodes are attracting large attention [1] because they describe systems as different as functional brain networks or collaboration 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. glued along their faces. Simplicial complexes are structures that can be generalized by cell complexes formed by gluing “motifs” having a polytope structure.

Interestingly higher order networks have a natural geometric interpretation and therefore constitute the natural way to explore the discrete network geometry of complex networks. In a large variety of datasets it has been claimed that actually the underlying network geometry of complex networks is hyperbolic (see [1] and references there). However how this geometry can emerge from a dynamics that is purely combinatorial has been a long standing problem.

Network Geometry with Flavor (NGF) [2-4] is a comprehensive theoretical framework that provides a main avenue to explore emergent hyperbolic geometry [3,4] . This model uses a non-equilibrium evolution of simplicial complexes and cell complexes that is purely combinatorial, i.e. it makes no assumptions on the underlying geometry of the cell complex. Therefore the hyperbolic network geometry of the resulting structure is not a priori assumed but instead it is an emergent property of the network evolution.

In this talk we will reveal the interplay between complexity and network geometry [4] of the higher-order networks by characterizing the emergent hyperbolic nature of Network Geometry with Flavor (see Figure 1) for cell complexes of any dimension. We will explore 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 spectral dimension of the higher-order network to the dimension and nature of its building blocks.

This work show evidence that higher-order networks can be strongly affected by their local structure and their dimension and that complexity of these structure is tightly linked to their geometry.