Spectral theory of dense hypergraph limits

TL;DR

This paper develops a spectral theory for dense hypergraph limits, proving spectral convergence under the 1-cut metric and exploring continuity properties of associated matrix operators.

Contribution

It introduces a spectral framework for hypergraph limits, establishing spectral convergence results and analyzing operator continuity under different cut norms.

Findings

Spectra of adjacency and Laplacian matrices converge for hypergraph sequences in the 1-cut metric.

Certain matrix operators associated with hypergraphs are not spectrally continuous under the 1-cut metric.

These operators are continuous with respect to other cut norms.

Abstract

In this work, we develop a spectral theory for hypergraph limits. We prove the convergence of the spectra of adjacency and Laplacian matrices for hypergraph sequences converging in the $1$-cut metric. On the other hand, we give examples of matrix operators associated with hypergraphs whose spectra are not continuous with respect to the $1$-cut metric. Furthermore, we show that these operators are continuous with respect to other cut norms.