Forman's Ricci curvature - From networks to hypernetworks

TL;DR

This paper introduces a geometric framework using Forman Ricci curvature to analyze complex network structures like hypernetworks, enabling new insights and practical embedding methods.

Contribution

It presents a unifying geometric approach for hypernetworks using Forman Ricci curvature, including modeling, embedding, and curvature computation methods.

Findings

Hypernetworks can be modeled as polyhedral complexes with curvature.

A simple method for embedding hypernetworks in Euclidean space is proposed.

An efficient way to compute Ollivier-Ricci curvature for hypernetworks is introduced.

Abstract

Networks and their higher order generalizations, such as hypernetworks or multiplex networks are ever more popular models in the applied sciences. However, methods developed for the study of their structural properties go little beyond the common name and the heavy reliance of combinatorial tools. We show that, in fact, a geometric unifying approach is possible, by viewing them as polyhedral complexes endowed with a simple, yet, the powerful notion of curvature - the Forman Ricci curvature. We systematically explore some aspects related to the modeling of weighted and directed hypernetworks and present expressive and natural choices involved in their definitions. A benefit of this approach is a simple method of structure-preserving embedding of hypernetworks in Euclidean N-space. Furthermore, we introduce a simple and efficient manner of computing the well established Ollivier-Ricci curvature of a hypernetwork.