Geometric characterisation of structural and regular equivalences in undirected (hyper)graphs

TL;DR

This paper generalizes and characterizes structural and regular equivalences in undirected hypergraphs and graphs using neighbourhood graphs and Ollivier-Ricci curvature, facilitating new clustering methods.

Contribution

It introduces a novel geometric framework for understanding similarity notions in hypergraphs and graphs, enabling the construction of regular partitions.

Findings

Characterization of equivalences via neighbourhood graphs and curvature

New insights into similarity notions in hypergraphs and graphs

Potential for improved clustering algorithms

Abstract

Similarity notions between vertices in a graph, such as structural and regular equivalence, are one of the main ingredients in clustering tools in complex network science. We generalise structural and regular equivalences for undirected hypergraphs and provide a characterisation of structural and regular equivalences of undirected graphs and hypergraphs through neighbourhood graphs and Ollivier-Ricci curvature. Our characterisation sheds new light on these similarity notions opening a new avenue for their exploration. These characterisations also enable the construction of a possibly wide family of regular partitions, thereby offering a new route to a task that has so far been computationally challenging.