Community detection, pattern recognition, and hypergraph-based learning: approaches using metric geometry and persistent homology

TL;DR

This paper introduces a novel topological structure for hypergraph data, enabling new community detection and pattern recognition methods based on persistent homology and a generalized nearest neighbors approach, applicable to complex discrete data.

Contribution

It presents a new topological framework for hypergraphs and develops flexible algorithms for community detection and pattern recognition using persistent homology.

Findings

New topological structure for hypergraph data

Effective community detection using persistent homology

Modified nearest neighbors method applicable to hypergraphs

Abstract

Hypergraph data appear and are hidden in many places in the modern age. They are data structure that can be used to model many real data examples since their structures contain information about higher order relations among data points. One of the main contributions of our paper is to introduce a new topological structure to hypergraph data which bears a resemblance to a usual metric space structure. Using this new topological space structure of hypergraph data, we propose several approaches to study community detection problem, detecting persistent features arising from homological structure of hypergraph data. Also based on the topological space structure of hypergraph data introduced in our paper, we introduce a modified nearest neighbors methods which is a generalization of the classical nearest neighbors methods from machine learning. Our modified nearest neighbors methods have an advantage of being very flexible and applicable even for discrete structures as in hypergraphs. We then apply our modified nearest neighbors methods to study sign prediction problem in hypegraph data constructed using our method.