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

TL;DR

This paper introduces a topological data analysis approach using contagion maps to understand and predict spreading processes on networks, capturing complex spatial and long-range interactions.

Contribution

It develops a novel methodology combining topological data analysis and nonlinear dimension reduction to analyze contagion spread on networks, revealing low-dimensional structures.

Findings

Contagion maps effectively capture the topology and geometry of spreading processes.

The method can infer low-dimensional structures in complex networks.

Insights gained can improve modeling and control of contagions.

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 analyzing the topology, geometry, and dimensionality of manifold structure in such point clouds, we reveal insights to aid in the modeling, forecast, and control of spreading processes. Our approach highlights contagion maps also as a viable tool for inferring low-dimensional structure in networks.