Horospherical random graphs and lockdown strategies

TL;DR

This paper introduces a model of sparse, well-connected graphs inspired by expander properties, and analyzes how different lockdown strategies affect the spread of random walks, with implications for controlling epidemics.

Contribution

It presents a novel random graph model combining expansion and clustering, and evaluates the impact of various lockdown strategies on spreading dynamics.

Findings

Complete closure of medium and long-distance travel more effectively slows spread.

The model can generate graphs with both high expansion and clustering.

Lockdown strategies significantly influence the speed of random walk dissemination.

Abstract

Expanders are sparse graph that are strongly connected, where {\it connectivity} is quantified using eigenvalues of the adjacency matrix, and {\it sparsity} in terms of vertex valency. We give a model of random graphs and study their connectivity and sparsity. This model is a particular case of soft geometric random graphs, and allows to construct sparse graphs with good expansion properties, as well as highly clustered ones. On those graphs, we study the speed at which random walks spread in the graph, and visit all vertices. As an illustration, we build a model for mainland France and study the spread of random walks under several types of lockdown. Our experiments show that completely closing medium and long distance travel to slow down the spread of a random walk is more efficient than than local restrictions.