Topological Community Detection: A Sheaf-Theoretic Approach

TL;DR

This paper introduces a novel topological approach using cellular sheaves for community detection in networks, demonstrating effective algorithms on real social data and pioneering the application of sheaves in this context.

Contribution

It presents three new algorithms based on sheaf theory for vertex clustering, the first implementation of sheaves on social network data, and a proof-of-concept for topological data analysis in network science.

Findings

Achieved near-optimal modularity in social network community detection

First application of sheaves to real social network data

Demonstrated effectiveness of topological methods in complex systems

Abstract

We propose a model for network community detection using topological data analysis, a branch of modern data science that leverages theory from algebraic topology to statistical analysis and machine learning. Specifically, we use cellular sheaves, which relate local to global properties of various algebraic topological constructions, to propose three new algorithms for vertex clustering over networks to detect communities. We apply our algorithms to real social network data in numerical experiments and obtain near optimal results in terms of modularity. Our work is the first implementation of sheaves on real social network data and provides a solid proof-of-concept for future work using sheaves as tools to study complex systems captured by networks and simplicial complexes.