Network Geometry and Complexity

TL;DR

This paper explores the relationship between the geometric structure and complexity of higher order networks, using a non-equilibrium model to analyze emergent community structures, degree distributions, and hyperbolic geometry in cell-complexes built from regular polytopes.

Contribution

It extends the Network Geometry with Flavor model to cell-complexes formed by regular polytopes and analyzes their emergent geometric and complex network properties.

Findings

Emergent community structure varies with the polytope used.

Degree distribution depends on the regular polytope's properties.

The network exhibits hyperbolic geometric features.

Abstract

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.