Persistent homology of convection cycles in network flows

TL;DR

This paper applies persistent homology, a topological data analysis technique, to detect and analyze convection cycles in network flows, revealing how system parameters influence these cycles in various applications.

Contribution

It introduces a novel method combining persistent homology with network flow analysis to characterize convection cycles, especially in irreversible Markov chains.

Findings

System parameters act as homology regularizers of convection.

Persistence barcodes effectively summarize convection dynamics.

Homological bifurcation diagrams reveal phase transitions in flows.

Abstract

Convection is a well-studied topic in fluid dynamics, yet it is less understood in the context of networks flows. Here, we incorporate techniques from topological data analysis (namely, persistent homology) to automate the detection and characterization of convective/cyclic/chiral flows over networks, particularly those that arise for irreversible Markov chains (MCs). As two applications, we study convection cycles arising under the PageRank algorithm, and we investigate chiral edges flows for a stochastic model of a bi-monomer's configuration dynamics. Our experiments highlight how system parameters -- e.g., the teleportation rate for PageRank and the transition rates of external and internal state changes for a monomer -- can act as homology regularizers of convection, which we summarize with persistence barcodes and homological bifurcation diagrams. Our approach establishes a new connection between the study of convection cycles and homology, the branch of mathematics that formally studies cycles, which has diverse potential applications throughout the sciences and engineering.