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Nat Commun. 2015 Jul 21;6:7723. doi: 10.1038/ncomms8723.

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

### Author information

1
1] Statistical and Applied Mathematical Sciences Institute, Research Triangle Park, North Carolina 27709, USA [2] Carolina Center for Interdisciplinary Applied Mathematics, Department of Mathematics, University of North Carolina, Chapel Hill, North Carolina 27599, USA.
2
1] Potsdam Institute for Climate Impact Research, 14473 Potsdam, Germany [2] Department of Physics, Humboldt-Universität zu Berlin, 12489 Berlin, Germany [3] Mathematical Institute, University of Oxford, Oxford OX2 6GG, UK.
3
Mathematical Institute, University of Oxford, Oxford OX2 6GG, UK.
4
Department of Mathematics, Rutgers, The State University of New Jersey, Piscataway, New Jersey 08854, USA.
5
1] Department of Mathematics, Rutgers, The State University of New Jersey, Piscataway, New Jersey 08854, USA [2] BioMaPS Institute, Rutgers, The State University of New Jersey, Piscataway, New Jersey 08854, USA.
6
1] Mathematical Institute, University of Oxford, Oxford OX2 6GG, UK [2] CABDyN Complexity Centre, University of Oxford, Oxford OX1 1HP, UK.
7
Carolina Center for Interdisciplinary Applied Mathematics, Department of Mathematics, University of North Carolina, Chapel Hill, North Carolina 27599, USA.

### 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.

PMID:
26194875
PMCID:
PMC4566922
DOI:
10.1038/ncomms8723
[Indexed for MEDLINE]
Free PMC Article