Cosheaf Theoretical Constructions in Networks and Persistent Homology

TL;DR

This paper integrates cosheaf theory with persistent homology to analyze data flow stability and errors in hierarchical recurrent networks, providing new tools for network analysis and packet delivery applications.

Contribution

It introduces a novel framework combining cosheaves and persistent homology to study data flow errors in networks, linking topological and local data analysis.

Findings

Developed a cosheaf-based construction of persistence diagrams for network data errors

Enabled statistical analysis of data flow malfunctions using topological methods

Applied the framework to real network packet delivery systems

Abstract

Persistent homology has recently emerged as a powerful technique in topological data analysis for analyzing the emergence and disappearance of topological features throughout a filtered space, shown via persistence diagrams. Additionally, (co)sheaves have proven to be powerful instruments in tracking locally defined data across global systems, resulting in innovative applications to network science. In this paper, we combine the topological results of persistent homology and the quantitative data tracking capabilities of cosheaf theory to develop novel techniques in network data flow analysis. Specifically, we use cosheaf theory to construct persistent homology in a framework geared towards assessing data flow stability in hierarchical recurrent networks (HRNs). We use cosheaves to link topological information about a filtered network encoded in persistence diagrams with data associated locally to the network. From this construction, we use the homology of cosheaves as a framework to study the notion of "persistent data flow errors." That is, we generalize aspects of persistent homology to analyze the lifetime of local data flow malfunctions. We study an algorithmic construction of persistence diagrams parameterizing network data flow errors, thus enabling novel applications of statistical methods to study data flow malfunctions. We conclude with an application to network packet delivery systems.