Finding the Cores of Higher Graphs Using Geometric and Topological Means: A Survey

TL;DR

This survey reviews recent methods for identifying cores of higher graphs like hypergraphs and simplicial complexes using geometric and topological techniques such as discrete curvature, resistance, and persistent homology.

Contribution

It synthesizes geometric and topological approaches to simplify higher graphs while preserving their essential geometric and topological features.

Findings

Connections between graph theory, geometry, and topology are used to identify graph cores.

Methods based on discrete curvature and resistance effectively capture core structures.

Persistent homology aids in understanding the topological essence of higher graphs.

Abstract

In this survey, we explore recent literature on finding the cores of higher graphs using geometric and topological means. We study graphs, hypergraphs, and simplicial complexes, all of which are models of higher graphs. We study the notion of a core, which is a minimalist representation of a higher graph that retains its geometric or topological information. We focus on geometric and topological methods based on discrete curvatures, effective resistance, and persistent homology. We aim to connect tools from graph theory, discrete geometry, and computational topology to inspire new research on the simplification of higher graphs.