Generalizing the Hypergraph Laplacian via a Diffusion Process with Mediators

TL;DR

This paper introduces a generalized diffusion process on hypergraphs allowing vertices to act as mediators, leading to a new Laplacian operator that maintains Cheeger's inequality and broadens spectral hypergraph theory.

Contribution

It proposes a novel diffusion model with mediators in hypergraphs, extending spectral properties and practical applicability beyond previous maximum-minimum flow models.

Findings

The new Laplacian satisfies Cheeger's inequality.

The model introduces a family of operators related to hypergraph conductance.

Vertices can participate continuously, enabling wider practical applications.

Abstract

In a recent breakthrough STOC~2015 paper, a continuous diffusion process was considered on hypergraphs (which has been refined in a recent JACM 2018 paper) to define a Laplacian operator, whose spectral properties satisfy the celebrated Cheeger's inequality. However, one peculiar aspect of this diffusion process is that each hyperedge directs flow only from vertices with the maximum density to those with the minimum density, while ignoring vertices having strict in-beween densities. In this work, we consider a generalized diffusion process, in which vertices in a hyperedge can act as mediators to receive flow from vertices with maximum density and deliver flow to those with minimum density. We show that the resulting Laplacian operator still has a second eigenvalue satsifying the Cheeger's inequality. Our generalized diffusion model shows that there is a family of operators whose spectral properties are related to hypergraph conductance, and provides a powerful tool to enhance the development of spectral hypergraph theory. Moreover, since every vertex can participate in the new diffusion model at every instant, this can potentially have wider practical applications.