Hypernetworks: From Posets to Geometry

TL;DR

This paper introduces a geometric framework for hypernetworks by interpreting them as posets and simplicial complexes, enabling the use of Forman Ricci curvature to analyze their topological properties and simplifying existing methods.

Contribution

It provides a novel geometric perspective on hypernetworks, linking posets, simplicial complexes, and curvature, and streamlines the application of Persistent Homology.

Findings

Hypernetworks can be modeled as posets and simplicial complexes.

Forman Ricci curvature correlates with the Euler characteristic.

The approach simplifies previous geometric Persistent Homology methods.

Abstract

We show that hypernetworks can be regarded as posets which, in their turn, have a natural interpretation as simplicial complexes and, as such, are endowed with an intrinsic notion of curvature, namely the Forman Ricci curvature, that strongly correlates with the Euler characteristic of the simplicial complex. This approach, inspired by the work of E. Bloch, allows us to canonically associate a simplicial complex structure to a hypernetwork, directed or undirected. In particular, this greatly simplifying the geometric Persistent Homology method we previously proposed.