| Shrnutí: | A spreading process on a network can be influenced by the network’s underlying spatial structure. In particular, if a network’s nodes are embedded in a manifold and connected by both ‘geometric edges’, which respect the geometry of the underlying manifold, and ‘non-geometric edges’ that are not constrained by that geometry, a threshold contagion can either propagate as a wave along the network or jump via long non-geometric edges to remote areas of the network. In this thesis, we study so-called ‘contagion maps’, which are defined via spreading dynamics on such ‘noisy geometric networks’. We take two complementary perspectives: In the first part of this thesis, we use contagion maps to examine the spreading of a threshold contagion on a class of networks that are embedded in a torus. We analyse the dimensionality, geometry, and topology of point clouds produced by contagion maps to examine qualitative properties of this spreading process. We conduct a bifurcation analysis to identify a region in parameter space in which the contagion propagates predominantly via wavefront propagation and find that this region aligns with our computational analysis of the point clouds produced by contagion maps. We also consider different probability distributions for constructing non-geometric edges — reflecting different decay rates with respect to the distance between nodes in the underlying manifold — and examine how qualitative properties of the spreading dynamics are affected by these choices. Moreover, we incorporate an approach for calibrating persistent homology barcodes of different scales. This approach is applicable in various other applications that use persistent homology, and the Wasserstein distance between barcodes in particular. In the second part of this thesis, we test contagion maps as a manifold learning tool on a variety of data sets. In particular, we compare the performance of contagion maps on these data sets to that of Isomap and find that, under certain conditions, contagion maps are able to reliably detect underlying manifold structure in noisy data, when Isomap is prone to noise-induced error. Our work consolidates contagion maps both as a tool for investigating spreading behaviour on spatial networks and as a technique for manifold learning.
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