Random walks and Laplacians on hypergraphs: When do they match?

TL;DR

This paper develops a comprehensive theory of random walks on hypergraphs, comparing different Laplacian models and analyzing their spectral properties and applications to hypergraph dynamical systems.

Contribution

It introduces and analyzes general random walk Laplacians for hypergraphs, highlighting differences from non-random walk Laplacians and their implications.

Findings

Random walk and non-random walk hypergraph Laplacians coincide in graphs.

Spectral properties differ significantly between the two classes in hypergraphs.

Applications to hypergraph dynamical systems reveal distinct behaviors.

Abstract

We develop a general theory of random walks on hypergraphs which includes, as special cases, the different models that are found in literature. In particular, we introduce and analyze general random walk Laplacians for hypergraphs, and we compare them to hypergraph normalized Laplacians that are not necessarily related to random walks, but which are motivated by biological and chemical networks. We show that, although these two classes of Laplacians coincide in the case of graphs, they appear to have important conceptual differences in the general case. We study the spectral properties of both classes, as well as their applications to Coupled Hypergraph Maps: discrete-time dynamical systems that generalize the well-known Coupled Map Lattices on graphs. Our results also show why for some hypergraph Laplacian variants one expects more classical results from (weighted) graphs to generalize directly, while these results must fail for other hypergraph Laplacians.