🍩 Database of Original & Non-Theoretical Uses of Topology

1. Analysis of Kolmogorov Flow and Rayleigh–Bénard Convection Using Persistent Homology (2016)

   Miroslav Kramár, Rachel Levanger, Jeffrey Tithof, Balachandra Suri, Mu Xu, Mark Paul, Michael F. Schatz, Konstantin Mischaikow

   Abstract

   We use persistent homology to build a quantitative understanding of large complex systems that are driven far-from-equilibrium. In particular, we analyze image time series of flow field patterns from numerical simulations of two important problems in fluid dynamics: Kolmogorov flow and Rayleigh–Bénard convection. For each image we compute a persistence diagram to yield a reduced description of the flow field; by applying different metrics to the space of persistence diagrams, we relate characteristic features in persistence diagrams to the geometry of the corresponding flow patterns. We also examine the dynamics of the flow patterns by a second application of persistent homology to the time series of persistence diagrams. We demonstrate that persistent homology provides an effective method both for quotienting out symmetries in families of solutions and for identifying multiscale recurrent dynamics. Our approach is quite general and it is anticipated to be applicable to a broad range of open problems exhibiting complex spatio-temporal behavior.

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   • Fluid dynamics
   • Point cloud
   • Temperature
   • Persistence diagrams
   • Persistent homology
   • Wasserstein distance

2. Topological Data Analysis of Contagion Maps for Examining Spreading Processes on Networks (2015)

   Dane Taylor, Florian Klimm, Heather A. Harrington, Miroslav Kramár, Konstantin Mischaikow, Mason A. Porter, Peter J. Mucha

   Abstract

   Social and biological contagions are influenced by the spatial embeddedness of networks. Historically, many epidemics spread as a wave across part of the Earth’s surface; however, in modern contagions long-range edges—for example, due to airline transportation or communication media—allow clusters of a contagion to appear in distant locations. Here we study the spread of contagions on networks through a methodology grounded in topological data analysis and nonlinear dimension reduction. We construct ‘contagion maps’ that use multiple contagions on a network to map the nodes as a point cloud. By analysing the topology, geometry and dimensionality of manifold structure in such point clouds, we reveal insights to aid in the modelling, forecast and control of spreading processes. Our approach highlights contagion maps also as a viable tool for inferring low-dimensional structure in networks.

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   • Contagion maps
   • Epidemiology
   • Geometric network
   • Maps
   • Networks
   • Clustering
   • Nonlinear dimension reduction
   • Persistent homology