Horospherical random graphs and lockdown strategies

TL;DR

The paper investigates how the geometry of travel networks affects diffusion by introducing horospherical random graphs, a layered construction that interpolates between sparse, highly clustered and long-range connected topologies. By analyzing the first nonzero eigenvalue $\lambda_1$ and conductance, the authors show that long-range connections substantially accelerate mixing of simple random walks, while suppressing these connections can dramatically slow spread. They apply the framework to a mainland France model, comparing four mobility regimes and demonstrating that complete long-distance travel shutdown reduces $\lambda_1$ and slows diffusion more effectively than purely local restrictions. The work provides a principled, graph-theoretic approach to evaluating lockdown strategies and highlights the critical role of long-range connectivity in epidemic spreading on sparse networks.

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.

Figures (6)

• Figure 1: The first picture on the left shows the scaled down layers added. In the middle we added the behavior of geodesics traveling up and creating thin triangles, while on the right we see the underlying tree-like geometry.
• Figure 2: Four plots of graph properties over 5 different connection functions. For each connection function and each property, the given property was averaged over 20 iterations on a uniformly distributed set, then plotted with a band of one standard deviation.
• Figure 3: The vertex set $X$, obtained with a simulation of major metropolitan areas of mainland France along with a grid of points representing the rest of the population.
• Figure 4: A simple random walk on $\Gamma_U, \Gamma_S, \Gamma_C$ and $\Gamma_I$.
• Figure 5: A replicating random walk simulation on $\Gamma_U, \Gamma_S, \Gamma_C$ and $\Gamma_I$.

Theorems & Definitions (10)

• Definition 2.1
• Theorem 2.2: Cheeger inequality
• Remark 2.3
• Definition 2.4
• Definition 2.5
• Remark 2.6
• Definition 3.1: Horospherical random graphs
• Remark 3.2
• Remark 3.3
• Remark 3.4