Spectral theory of dense hypergraph limits

TL;DR

This work develops a spectral theory for dense hypergraph limits by embedding hypergraphs into hypergraphons and using the $1$-cut norm, establishing the pointwise convergence of adjacency and Laplacian spectra under $\delta_{\square,1}$ for hypergraph sequences. It also demonstrates that certain matrix operators are not continuous under this metric and may require alternative cut norms (e.g., $2$-cut norm) to ensure spectral convergence, illustrated by a vertex-vertex intersection construction. The authors extend the theory to nonuniform hypergraphs via rank decompositions and $p$-weighted adjacency matrices, enabling spectral analysis of higher-order network dynamics. Overall, the paper provides a rigorous framework connecting hypergraph limit objects, contraction operations, and spectral convergence, with implications for understanding dynamics on dense, higher-order networks.

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.

Theorems & Definitions (46)

• Example 2.1
• Example 2.2
• Theorem 2.3
• Theorem 2.4
• Proposition 2.5
• proof
• Remark 2.7
• Lemma 2.8
• proof
• Remark 2.9