Hypergraph assortativity: a dynamical systems perspective

TL;DR

This paper introduces the expansion eigenvalue for hypergraphs, linking spectral properties to dynamical processes, and demonstrates how manipulating hypergraph assortativity can control epidemic spread.

Contribution

It extends spectral analysis to hypergraphs, introduces a dynamical assortativity measure, and shows how rewiring hypergraphs affects epidemic dynamics.

Findings

Expansion eigenvalue approximated for uncorrelated hypergraphs

Dynamical assortativity defined for hypergraphs

Rewiring hypergraphs can extinguish epidemics

Abstract

The largest eigenvalue of the matrix describing a network's contact structure is often important in predicting the behavior of dynamical processes. We extend this notion to hypergraphs and motivate the importance of an analogous eigenvalue, the expansion eigenvalue, for hypergraph dynamical processes. Using a mean-field approach, we derive an approximation to the expansion eigenvalue in terms of the degree sequence for uncorrelated hypergraphs. We introduce a generative model for hypergraphs that includes degree assortativity, and use a perturbation approach to derive an approximation to the expansion eigenvalue for assortative hypergraphs. We define the dynamical assortativity, a dynamically sensible definition of assortativity for uniform hypergraphs, and describe how reducing the dynamical assortativity of hypergraphs through preferential rewiring can extinguish epidemics. We validate our results with both synthetic and empirical datasets.