Directional Sheaf Hypergraph Networks: Unifying Learning on Directed and Undirected Hypergraphs

TL;DR

DSHN introduces Directed Hypergraph Cellular Sheaves and a complex-valued Directed Sheaf Hypergraph Laplacian to model directionality in higher-order interactions. By embedding diffusion in the complex domain and learning direction-aware restriction maps, the method unifies and extends Laplacians for graphs and hypergraphs, enabling spectral-style convolution on directed hypergraphs. Empirically, DSNH and its lightweight variant achieve 2–20% relative accuracy gains over 13 baselines across 7 real-world and several synthetic benchmarks, with performance depending on the dataset's homophily and directionality. The approach offers a principled, scalable framework for learning on directed hypergraphs, with potential extensions such as learning the charge parameter $q$ end-to-end and applying to large-scale biological networks.

Abstract

Hypergraphs provide a natural way to represent higher-order interactions among multiple entities. While undirected hypergraphs have been extensively studied, the case of directed hypergraphs, which can model oriented group interactions, remains largely under-explored despite its relevance for many applications. Recent approaches in this direction often exhibit an implicit bias toward homophily, which limits their effectiveness in heterophilic settings. Rooted in the algebraic topology notion of Cellular Sheaves, Sheaf Neural Networks (SNNs) were introduced as an effective solution to circumvent such a drawback. While a generalization to hypergraphs is known, it is only suitable for undirected hypergraphs, failing to tackle the directed case. In this work, we introduce Directional Sheaf Hypergraph Networks (DSHN), a framework integrating sheaf theory with a principled treatment of asymmetric relations within a hypergraph. From it, we construct the Directed Sheaf Hypergraph Laplacian, a complex-valued operator by which we unify and generalize many existing Laplacian matrices proposed in the graph- and hypergraph-learning literature. Across 7 real-world datasets and against 13 baselines, DSHN achieves relative accuracy gains from 2% up to 20%, showing how a principled treatment of directionality in hypergraphs, combined with the expressive power of sheaves, can substantially improve performance.

Figures (7)

• Figure 1: Visualization of sheaves over a directed hyperedge, illustrating the incidence relationship between nodes and the hyperedge, together with the restriction maps $\vec{\mathcal{F}}_{v \trianglelefteq e}$. The tail node $v_1$ is encoded via the $e^{-2\pi i q}$ coefficient which pre-multiplies the directionless restriction map $\mathcal{F}_{v \trianglelefteq e}$.
• Figure 2: Mean accuracy $\pm$ standard deviation on the synthetic datasets.
• Figure 3: Effect of the charge parameter $q$ on Telegram and Cora.
• Figure 4: Influence of architectural parameters on accuracy. (a) Effect of the number of layers on DSHN and HGNN. (b) Effect of stalk dimension $d$ on DSHN.
• Figure 5: Example of the creation of a directed hyperedge from the out-neighborhood of a node. Suppose we have a graph where node $n1$ connects to nodes $n2$, $n3$, and $n4$, so that $(n1,n2)$, $(n1,n3)$, and $(n1,n4)$ belong to $E$. The construction procedure yields a directed hyperedge $e_{1}$ with tail set $T(e_{1}) = \{n1\}$ and head set $H(e_{1}) = \{n2, n3, n4\}$.

Theorems & Definitions (28)

• Definition 1
• Theorem 1
• Theorem 2
• Corollary 1
• Theorem 3
• Theorem 4
• Theorem 5
• Theorem 6
• Theorem 7
• Lemma 1