Random Walks on Hypergraphs with Edge-Dependent Vertex Weights

TL;DR

This paper develops a spectral theory for hypergraphs with edge-dependent vertex weights using random walks, deriving a hypergraph Laplacian, analyzing mixing times, and demonstrating advantages in ranking tasks.

Contribution

It introduces a new spectral framework for hypergraphs with edge-dependent weights, including a hypergraph Laplacian and analysis of random walk properties.

Findings

Derived a hypergraph Laplacian based on random walks

Bounded the mixing time of random walks on such hypergraphs

Showed advantages of edge-dependent weights in ranking applications

Abstract

Hypergraphs are used in machine learning to model higher-order relationships in data. While spectral methods for graphs are well-established, spectral theory for hypergraphs remains an active area of research. In this paper, we use random walks to develop a spectral theory for hypergraphs with edge-dependent vertex weights: hypergraphs where every vertex $v$ has a weight $\gamma_e(v)$ for each incident hyperedge $e$ that describes the contribution of $v$ to the hyperedge $e$. We derive a random walk-based hypergraph Laplacian, and bound the mixing time of random walks on such hypergraphs. Moreover, we give conditions under which random walks on such hypergraphs are equivalent to random walks on graphs. As a corollary, we show that current machine learning methods that rely on Laplacians derived from random walks on hypergraphs with edge-independent vertex weights do not utilize higher-order relationships in the data. Finally, we demonstrate the advantages of hypergraphs with edge-dependent vertex weights on ranking applications using real-world datasets.