  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.
