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

## TL;DR

This paper investigates how contagion spreads on networks embedded in a torus, analyzing wavefront versus jump propagation, and introduces contagion maps as a tool for understanding spreading dynamics and manifold learning.

## Contribution

It extends previous work by analyzing contagion on noisy geometric networks embedded in a torus, and demonstrates the use of contagion maps for manifold learning and network analysis.

## Key findings

- Identifies parameter regions with wavefront propagation.
- Shows how nongeometric edges influence spreading behavior.
- Demonstrates contagion maps as a tool for manifold learning.

## Abstract

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. \cite{Taylor2015}, 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.