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

arXiv: https://arxiv.org/abs/1812.09806 · HTML: https://mlsanity.com/paper/1812.09806/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 24 Dec 2018.

## Built on by (most cited first, 1 of 1)

- [Topological data analysis of truncated contagion maps](https://mlsanity.com/paper/2203.01720/) (2022)

## Abstract (arXiv)

A spreading process on a network is influenced by the network's underlying spatial structure, and it is insightful to study the extent to which a spreading process follows such structure. We consider a threshold contagion model 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 on ideas from Taylor et al. citeTaylor2015, we examine when a contagion propagates as a wave along a network whose nodes are embedded in a torus and when it jumps via long nongeometric edges to remote areas of the network. We build a `contagion map' for a contagion spreading on such a `noisy geometric network' to produce a point cloud; and we study the dimensionality, geometry, and topology of this point cloud to examine qualitative properties of this spreading process. We identify a region in parameter space in which the contagion propagates predominantly via wavefront propagation. We consider different probability distributions for constructing nongeometric edges -- reflecting different decay rates with respect to the distance between nodes in the underlying manifold -- and examine the effect of such choices on the qualitative properties of the spreading dynamics. Our work generalizes the analysis in Taylor et al. and consolidates contagion maps both as a tool for investigating spreading behavior on spatial networks and as a technique for manifold learning.