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A spreading process on a network is influenced by the network's underlyingspatial structure, and it is insightful to study the extent to which aspreading process follows such structure. We consider a threshold contagionmodel on a network whose nodes are embedded in a manifold and which has both`geometric edges', which respect the geometry of the underlying manifold, and`nongeometric edges' that are not constrained by that geometry. Building onideas from Taylor et al. \cite{Taylor2015}, we examine when a contagionpropagates as a wave along a network whose nodes are embedded in a torus andwhen it jumps via long nongeometric edges to remote areas of the network. Webuild a `contagion map' for a contagion spreading on such a `noisy geometricnetwork' to produce a point cloud; and we study the dimensionality, geometry,and topology of this point cloud to examine qualitative properties of thisspreading process. We identify a region in parameter space in which thecontagion propagates predominantly via wavefront propagation. We considerdifferent probability distributions for constructing nongeometric edges --reflecting different decay rates with respect to the distance between nodes inthe underlying manifold -- and examine the effect of such choices on thequalitative properties of the spreading dynamics. Our work generalizes theanalysis in Taylor et al. and consolidates contagion maps both as a tool forinvestigating spreading behavior on spatial networks and as a technique formanifold learning.

Analysis of contagion maps on a class of networks that are spatially embedded in a torus