Generalizing Lattice Structures to Hypergraphs: Spectra of Clique and Hyperedge-based Laplacians

TL;DR

The paper generalizes lattice models to hypergraphs by introducing hyperlattices on discrete tori and analyzing two Laplacian formulations: the clique-based and the hyperedge-based (incidence) Laplacians. It provides closed-form spectral characterizations for periodic and open-boundary variants, revealing how higher-order interactions (via hyperedge size and directional families) shape diffusion spectra. The main results include explicit eigenvalue expressions dependent on lattice size $\ell$, dimension $m$, and hyperedge size $r$, and a Toeplitz-type open-boundary spectrum with separable trigonometric modes. These hyperlattices offer analytically tractable benchmarks for higher-order diffusion, synchronization, and spin-models, linking spectral structure to the geometry of multiway interactions.

Abstract

Lattice structures play a central role in spectral graph theory, offering analytical insight into diffusion, synchronization, and transport processes on regular discrete spaces. While their spectral properties are completely characterized in the classical graph setting, an extension to hypergraphs, where interactions involve more than two nodes, remains largely unexplored in the matrix-based formulation. In this work, we generalize the notion of a lattice to the hypergraph framework and study its Laplacian spectra under two alternative definitions: the clique Laplacian, obtained through pairwise projection, and the hyperedge-based Laplacian, defined via normalized hyperedge incidences. For both definitions, we derive the corresponding Laplacian matrices, analyze their eigenvalue spectra, and discuss how they reflect the underlying topological and dynamical structure of the hyperlattice. Our main result is a theorem giving a full spectral characterization in the periodic case, together with a Toeplitz-type open analogue whose spectrum retains a separable trigonometric structure. The obtained eigenvalues are expressed explicitly in terms of the hyperedge size, the number of directional families, and the lattice side length, thereby capturing how the geometry of higher-order interactions shapes the spectral structure.

Figures (3)

• Figure 1: The $5$-lattice hypergraph $\mathcal{H}=(V,H)$. nodes are arranged on a $5\times5$ grid. Each horizontal (red) and vertical (blue) region represents a hyperedge containing all nodes in a row or column, respectively. Every node belongs to exactly two hyperedges: one horizontal and one vertical.
• Figure 2: $\ell$-hyperlattice in $\mathbb{Z}_\ell^3$ with $\ell=3$. The three hyperedges adjacent to vertex 0 are shown. Each hyperedge contains $\ell=3$ nodes aligned along one coordinate axis and is emphasized by a colored elliptical region.
• Figure 3: The three panels illustrate the hyperlattice in $\mathbb{Z}_\ell^m$ with hyperedges of size $r$, for $m = 2$, $\ell = 6$, and $r = 3$. The structure is decomposed into its $r$ directional sub-families, corresponding to the three possible sliding offsets ($\mathrm{shift}=0,1,2$). Each sub-figure displays the horizontal (red) and vertical (blue) hyperedges sharing the same offset, so that every node belongs to exactly two hyperedges (one per direction) in each panel. Together, the three sub-families account for the full set of $mr = 6$ hyperedges incident to every node in the complete hyperlattice.

Theorems & Definitions (26)

• Remark 1
• Proposition 1: Fundamental spectral properties
• proof : Sketch of the proof
• proof : Sketch of the proof
• Proposition 3: Spectrum of the hyperedge-based Laplacian on the $l$-lattice
• proof : Sketch of the proof
• Remark 2: Comparison with the classical lattice graph
• Remark 3: Effect of hyperedge normalization
• Definition 1: Directional family in $\mathbb{Z}^2_l$
• Definition 2: $\ell$-hyperlattice in $\mathbb{Z}_\ell^2$