Cosheaf Theoretical Constructions in Networks and Persistent Homology

In this paper, we study data flows in directed networks with a hierarchical recurrent structure from a cosheaf theoretical perspective. We utilize the visual parametrization of directed recurrent programs provided in persistence diagrams for cosheaf theoretical constructions. In considering cosheaves on persistence diagrams, we link global network structure and local recurrent process data. An application of homology to analyze data transfer errors in recurrent processes within a hierarchical framework is established. Additionally, we generalize certain aspects of persistent homology to accommodate a homological description of network malfunctions. Our persistent homological analogs is further developed in a general categorical setting, which naturally gives rise to relations between recurrent subprocess representation and the homological description of data flow errors, permitting a statistical study that would not otherwise arise. Our results indicate that the interplay between cosheaves and persistent homology has fruitful applications in data flow analysis in networks. The link between topological network trends provided by persistent homology and the real information association yielded by cosheaves creates the framework for a more thorough study of data flows.