Topological data analysis of contagion maps for examining spreading processes on networks
Dane Taylor, Florian Klimm, Heather A. Harrington, Miroslav Kramár, Konstantin Mischaikow, Mason Alexander Porter, Peter J. Mucha · Nature Communications · 2015
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. The spreading dynamics of a contagion depend on the structure of an underlying network, and long-range edges due to airline transportation or media communication can significantly alter such dynamics. Here the authors use contagion dynamics on networks to produce point clouds for this analysis.