A functorial Dowker theorem and persistent homology of asymmetric networks

Abstract

We study two methods for computing network features with topological underpinnings: the Rips and Dowker persistent homology diagrams. Our formulations work for general networks, which may be asymmetric and may have any real number as an edge weight. We study the sensitivity of Dowker persistence diagrams to asymmetry via numerous theoretical examples, including a family of highly asymmetric cycle networks that have interesting connections to the existing literature. In particular, we characterize the Dowker persistence diagrams arising from asymmetric cycle networks. We investigate the stability properties of both the Dowker and Rips persistence diagrams, and use these observations to run a classification task on a dataset comprising simulated hippocampal networks. Our theoretical and experimental results suggest that Dowker persistence diagrams are particularly suitable for studying asymmetric networks. As a stepping stone for our constructions, we prove a functorial generalization of a theorem of Dowker, after whom our constructions are named.