Le Minh Quang, Taylor Dane
Department of Mathematics, University at Buffalo, State University of New York, Buffalo, New York 14260, USA.
Phys Rev E. 2022 Apr;105(4-1):044311. doi: 10.1103/PhysRevE.105.044311.
Convection is a well-studied topic in fluid dynamics, yet it is less understood in the context of network flows. Here, we incorporate techniques from topological data analysis (namely, persistent homology) to automate the detection and characterization of convective flows (also called cyclic or chiral flows) over networks, particularly those that arise for irreversible Markov chains. As two applications, we study convection cycles arising under the PageRank algorithm and we investigate chiral edge flows for a stochastic model of a bimonomer'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 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.
对流是流体动力学中一个研究充分的课题，但在网络流的背景下，人们对它的理解还不够深入。在这里，我们采用拓扑数据分析技术（即持久同调）来自动检测和刻画网络上的对流（也称为循环流或手性流），特别是那些由不可逆马尔可夫链产生的对流。作为两个应用实例，我们研究了PageRank算法下出现的对流循环，并研究了双单体构型动力学随机模型的手性边流。我们的实验突出了系统参数——例如，PageRank的跳转率以及单体外部和内部状态变化的转移率——如何能够作为对流的同调正则化器，我们用持久条形码和同调分岔图对其进行了总结。我们的方法在对流循环研究与同调之间建立了联系，同调是正式研究循环的数学分支，在整个科学和工程领域具有多种潜在应用。
