Topological data analysis of contagion maps for examining spreading processes on networks

Dane Taylor (), Florian Klimm, Heather A. Harrington, Miroslav Kramár, Konstantin Mischaikow, Mason A. Porter and Peter J. Mucha
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Dane Taylor: Statistical and Applied Mathematical Sciences Institute, Research Triangle Park, North Carolina 27709, USA
Florian Klimm: Potsdam Institute for Climate Impact Research
Heather A. Harrington: Mathematical Institute, University of Oxford
Miroslav Kramár: Rutgers, The State University of New Jersey
Konstantin Mischaikow: Rutgers, The State University of New Jersey
Mason A. Porter: Mathematical Institute, University of Oxford
Peter J. Mucha: Carolina Center for Interdisciplinary Applied Mathematics, University of North Carolina, Chapel Hill

Nature Communications, 2015, vol. 6, issue 1, 1-11

Abstract: 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.

Date: 2015
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DOI: 10.1038/ncomms8723

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