Directional Sheaf Hypergraph Networks

• The paper introduces DSHN, a sheaf-theoretic framework that unifies learning on both directed and undirected hypergraphs using complex phase factors to encode directionality.
• DSHN mitigates homophily bias by employing learnable restriction maps that leverage full d×d matrices, enabling robust feature propagation in heterophilic settings.
• Empirical results on 7 datasets show accuracy gains of 2% to 20%, highlighting the framework’s effectiveness in tuning directional flows via the charge parameter.

Directional Sheaf Hypergraph Networks (DSHN) are a sheaf-theoretic neural framework for learning on directed and undirected hypergraphs, introduced to model higher-order interactions with explicit orientation while avoiding the implicit homophily bias that often constrains earlier directed-hypergraph methods. The framework combines cellular sheaves with a principled encoding of asymmetry through complex phases, yielding a complex-valued Directed Sheaf Hypergraph Laplacian that unifies several Laplacian constructions previously used in graph and hypergraph learning. In the formulation reported in "Directional Sheaf Hypergraph Networks: Unifying Learning on Directed and Undirected Hypergraphs," DSHN is evaluated on 7 real-world datasets against 13 baselines and is reported to achieve relative accuracy gains from 2% up to 20% (Mule et al., 6 Oct 2025).

1. Problem setting and motivation

Hypergraphs represent interactions among more than two entities by allowing a hyperedge to connect a set of vertices rather than a single pair. The directed case assigns orientation to these higher-order relations, making it possible to model oriented group interactions. The motivating premise of DSHN is that directed hypergraphs remain substantially less explored than undirected hypergraphs, despite their relevance for applications in which head–tail structure is semantically meaningful (Mule et al., 6 Oct 2025).

The framework is positioned against a recurring limitation of recent approaches to directed hypergraph learning: an implicit bias toward homophily. In heterophilic regimes, this bias can degrade performance because message passing that primarily averages neighboring information tends to suppress discriminative local structure. DSHN inherits from Sheaf Neural Networks the idea that learnable restriction maps can transport features across incidence relations without forcing simple agreement. In this setting, directionality is not appended as an external correction, but is built directly into the sheaf structure through complex phase factors.

A central claim of the framework is unification. The same operator family is constructed to subsume undirected hypergraph learning, directed graph learning, and directed hypergraph learning within a single spectral object. This positioning matters because it treats directionality, higher-order incidence, and sheaf transport as parts of the same algebraic mechanism rather than as independent design choices.

2. Directed hypergraph cellular sheaf and Laplacian

Let $\mathcal H=(V,E)$ be a directed hypergraph, with stalk dimension $d$, working over $\mathbb C^d$. DSHN assigns a vector space $\mathcal F(u)\cong\mathbb C^d$ to each vertex $u\in V$ and a vector space $\mathcal F(e)\cong\mathbb C^d$ to each hyperedge $e\in E$ (Mule et al., 6 Oct 2025).

Orientation is encoded by a complex phase matrix $S^{(q)}\in\mathbb C^{m\times n}$ with entries

$S^{(q)}_{u\trianglelefteq e}= \begin{cases} 1, & u\in H(e)\ (\text{head})\[4pt] e^{-2\pi i q}, & u\in T(e)\ (\text{tail})\[4pt] 0, & \text{otherwise.} \end{cases}$

The scalar $q\in[0,1)$ is the charge parameter. The reported interpretation is that $d$0 collapses directions, whereas $d$1 yields pure imaginary factors for tail–head interactions.

The model then introduces real-valued, learnable, direction-agnostic restriction maps

$d$2

and defines the complex restriction map

$d$3

The tuple $d$4 is termed a Directed Hypergraph Cellular Sheaf.

From these components, the framework assembles a complex-valued incidence operator

$d$5

It also defines

$d$6

and

$d$7

The reported unnormalized Directed Sheaf Laplacian is

$d$8

where

$d$9

The same presentation also writes, in boundary–co-boundary notation,

$\mathbb C^d$0

For numerical stability, the normalized form is used:

$\mathbb C^d$1

For $\mathbb C^d$2, the off-diagonal entries satisfy

$\mathbb C^d$3

The reported interpretation is that matching head–head or tail–tail incidences yield real negative weights, while tail–head incidences yield the complex phase $\mathbb C^d$4; at $\mathbb C^d$5, this becomes purely imaginary. The paper further states that $\mathbb C^d$6 is Hermitian, positive semidefinite, diagonalizable with real spectrum in $\mathbb C^d$7, and that its Dirichlet energy is

$\mathbb C^d$8

3. Recovery of existing operators and spectral interpretation

A major feature of DSHN is that specific parameter choices recover several established Laplacian constructions (Mule et al., 6 Oct 2025). For a 2-uniform undirected graph with trivial sheaf, meaning all restriction maps are $\mathbb C^d$9, $\mathcal F(u)\cong\mathbb C^d$0 reduces, up to scaling by $\mathcal F(u)\cong\mathbb C^d$1, to the classical graph Laplacian $\mathcal F(u)\cong\mathbb C^d$2. For a 2-uniform directed graph with $\mathcal F(u)\cong\mathbb C^d$3, it becomes the Magnetic Laplacian of Zhang et al., or the Sigmanet Laplacian when $\mathcal F(u)\cong\mathbb C^d$4. For general undirected hypergraphs with trivial sheaf, it recovers the Zhou et al. normalized hypergraph Laplacian. For directed hypergraphs with unit weights and trivial sheaf, it yields the Generalized Directed Laplacian of Fiorini et al.

This recovery behavior is the formal basis for the framework’s claim to unify learning on directed and undirected hypergraphs. It means that classical diffusion on pairwise undirected structure, magnetic phase-based diffusion on directed graphs, and hypergraph spectral propagation can all be treated as special cases of a single complex sheaf operator. A plausible implication is that comparisons among these models can be reframed as comparisons among particular choices of charge, stalk dimension, and restriction-map parameterization rather than as fundamentally separate architectures.

The spectral viewpoint also clarifies why the model is positioned for heterophilic settings. In ordinary averaging-based hypergraph propagation, alignment is encouraged by scalar or sign-limited edge couplings. In DSHN, each incidence relation is mediated by a full $\mathcal F(u)\cong\mathbb C^d$5 map together with a complex phase, so consistency is transported through learned linear correspondences rather than through direct feature averaging.

4. Network architecture and feature propagation

The neural architecture is derived from the sheaf-diffusion ODE

$\mathcal F(u)\cong\mathbb C^d$6

A forward Euler step yields

$\mathcal F(u)\cong\mathbb C^d$7

The $\mathcal F(u)\cong\mathbb C^d$8-th DSHN layer then inserts learnable linear maps and a nonlinearity:

$\mathcal F(u)\cong\mathbb C^d$9

where $u\in V$0 and $u\in V$1 are trainable, $u\in V$2 applies $u\in V$3 on each stalk, and $u\in V$4 is a complex activation such as complex-ReLU (Mule et al., 6 Oct 2025). At the final layer, complex features are unwound into real features by concatenating real and imaginary parts.

The input node features $u\in V$5 are first projected to $u\in V$6 and then reshaped to $u\in V$7. Restriction maps $u\in V$8 are predicted by an MLP $u\in V$9 applied to concatenated real-valued node and hyperedge features. When explicit hyperedge features are unavailable, they are aggregated from incident nodes by mean or sum. The reported implementation further applies layer normalization separately to $\mathcal F(e)\cong\mathbb C^d$0 and $\mathcal F(e)\cong\mathbb C^d$1 in the complex domain, jointly normalizing real–imaginary pairs, and allows residual connections together with multiple dropout options.

The framework attributes two main modeling effects to these design choices. First, because the restriction maps are full $\mathcal F(e)\cong\mathbb C^d$2 matrices, DSHN is reported to resist oversmoothing even at large depths and to naturally handle heterophily. Second, the charge parameter $\mathcal F(e)\cong\mathbb C^d$3 controls how strongly orientation affects propagation: $\mathcal F(e)\cong\mathbb C^d$4 recovers the undirected case, whereas $\mathcal F(e)\cong\mathbb C^d$5 or tuned values in $\mathcal F(e)\cong\mathbb C^d$6 amplify oriented flows.

5. Optimization protocol and computational complexity

The reported task setting is node classification with $\mathcal F(e)\cong\mathbb C^d$7 classes, using standard cross-entropy loss on output logits after unwinding (Mule et al., 6 Oct 2025). Optimization uses Adam with learning rate in $\mathcal F(e)\cong\mathbb C^d$8 and weight decay in $\mathcal F(e)\cong\mathbb C^d$9. Training uses early stopping after 200 no-improvement epochs and a maximum of 500 epochs. Regularization includes dropout on restriction maps, residual toggles, and layer normalization on $e\in E$0 and $e\in E$1.

For per-layer inference complexity, let $e\in E$2, $e\in E$3, $e\in E$4 be average hyperedge size, and $e\in E$5 be total channel size. Linear transforms cost

$e\in E$6

Message passing through $e\in E$7 costs

$e\in E$8

in the diagonal-restriction case, or

$e\in E$9

in general, since each hyperedge induces $S^{(q)}\in\mathbb C^{m\times n}$0 block interactions of size $S^{(q)}\in\mathbb C^{m\times n}$1. Predicting sheaf maps costs

$S^{(q)}\in\mathbb C^{m\times n}$2

or $S^{(q)}\in\mathbb C^{m\times n}$3 if the MLP is diagonal. Computing $S^{(q)}\in\mathbb C^{m\times n}$4, $S^{(q)}\in\mathbb C^{m\times n}$5, and assembling $S^{(q)}\in\mathbb C^{m\times n}$6 similarly costs $S^{(q)}\in\mathbb C^{m\times n}$7. The dominant per-layer term is therefore reported as

$S^{(q)}\in\mathbb C^{m\times n}$8

This complexity motivates the lightweight variant DSHNLight, which mitigates the cost by freezing the restriction-model MLP and detaching Laplacian assembly from backprop. The stated trade-off is therefore not between directionality and nondirectionality, but between full end-to-end sheaf adaptation and a more economical approximation of the same operator family.

6. Empirical behavior, ablations, and limitations

The experimental protocol converts seven standard directed graph benchmarks into directed hypergraphs by creating, for each node $S^{(q)}\in\mathbb C^{m\times n}$9, a hyperedge $S^{(q)}_{u\trianglelefteq e}= \begin{cases} 1, & u\in H(e)\ (\text{head})\[4pt] e^{-2\pi i q}, & u\in T(e)\ (\text{tail})\[4pt] 0, & \text{otherwise.} \end{cases}$0 with $S^{(q)}_{u\trianglelefteq e}= \begin{cases} 1, & u\in H(e)\ (\text{head})\[4pt] e^{-2\pi i q}, & u\in T(e)\ (\text{tail})\[4pt] 0, & \text{otherwise.} \end{cases}$1 and $S^{(q)}_{u\trianglelefteq e}= \begin{cases} 1, & u\in H(e)\ (\text{head})\[4pt] e^{-2\pi i q}, & u\in T(e)\ (\text{tail})\[4pt] 0, & \text{otherwise.} \end{cases}$2, thereby preserving original orientation as forward hyperedges (Mule et al., 6 Oct 2025). The datasets are Roman-empire, Squirrel, email-EU, Telegram, Chameleon, email-Enron, and Cora. Their reported statistics are as follows: Roman-empire has $S^{(q)}_{u\trianglelefteq e}= \begin{cases} 1, & u\in H(e)\ (\text{head})\[4pt] e^{-2\pi i q}, & u\in T(e)\ (\text{tail})\[4pt] 0, & \text{otherwise.} \end{cases}$3 nodes, $S^{(q)}_{u\trianglelefteq e}= \begin{cases} 1, & u\in H(e)\ (\text{head})\[4pt] e^{-2\pi i q}, & u\in T(e)\ (\text{tail})\[4pt] 0, & \text{otherwise.} \end{cases}$4 features, $S^{(q)}_{u\trianglelefteq e}= \begin{cases} 1, & u\in H(e)\ (\text{head})\[4pt] e^{-2\pi i q}, & u\in T(e)\ (\text{tail})\[4pt] 0, & \text{otherwise.} \end{cases}$5 hyperedges, $S^{(q)}_{u\trianglelefteq e}= \begin{cases} 1, & u\in H(e)\ (\text{head})\[4pt] e^{-2\pi i q}, & u\in T(e)\ (\text{tail})\[4pt] 0, & \text{otherwise.} \end{cases}$6 classes, average $S^{(q)}_{u\trianglelefteq e}= \begin{cases} 1, & u\in H(e)\ (\text{head})\[4pt] e^{-2\pi i q}, & u\in T(e)\ (\text{tail})\[4pt] 0, & \text{otherwise.} \end{cases}$7, average degree $S^{(q)}_{u\trianglelefteq e}= \begin{cases} 1, & u\in H(e)\ (\text{head})\[4pt] e^{-2\pi i q}, & u\in T(e)\ (\text{tail})\[4pt] 0, & \text{otherwise.} \end{cases}$8, and CE homophily $S^{(q)}_{u\trianglelefteq e}= \begin{cases} 1, & u\in H(e)\ (\text{head})\[4pt] e^{-2\pi i q}, & u\in T(e)\ (\text{tail})\[4pt] 0, & \text{otherwise.} \end{cases}$9; Squirrel has $q\in[0,1)$0 nodes, $q\in[0,1)$1 features, $q\in[0,1)$2 hyperedges, $q\in[0,1)$3 classes, average $q\in[0,1)$4, average degree $q\in[0,1)$5, and CE homophily $q\in[0,1)$6; email-EU has $q\in[0,1)$7 nodes, no explicit features, $q\in[0,1)$8 hyperedges, $q\in[0,1)$9 classes, average $d$00, average degree $d$01, and CE homophily $d$02; Telegram has $d$03 nodes, $d$04 feature, $d$05 hyperedges, $d$06 classes, average $d$07, average degree $d$08, and CE homophily $d$09; Chameleon has $d$10 nodes, $d$11 features, $d$12 hyperedges, $d$13 classes, average $d$14, average degree $d$15, and CE homophily $d$16; email-Enron has $d$17 nodes, no explicit features, $d$18 hyperedges, $d$19 classes, average $d$20, average degree $d$21, and CE homophily $d$22; Cora has $d$23 nodes, $d$24 features, $d$25 hyperedges, $d$26 classes, average $d$27, average degree $d$28, and CE homophily $d$29. Hyperparameters are tuned over stalk dimension $d$30, number of convolutional layers in $d$31, charge $d$32, activation in $d$33, dropout in $d$34, and residual on/off.

The baselines comprise undirected hypergraph GNNs including HGNN, HNHN, UniGCNII, LEGCN, HyperND, AllSet variants, ED-HNN, SheafHyperGNN, and PhenomNN, together with directed methods GeDi-HNN and DHGNN. The reported summary results are:

Dataset	Best baseline	DSHN / DSHNLight
Roman-empire	83.9 (GeDi)	89.2 (DSHNLight)
Squirrel	43.6 (PhenomNN)	44.1 (DSHNLight)
email-EU	68.9 (ED-HNN)	82.7 (DSHNLight)
Telegram	80.6 (DHGNN-emb)	88.7 (DSHN) / 81.2 (DSHNLight)
Chameleon	44.7 (ED-HNN)	47.0 (DSHN) / 46.5 (DSHNLight)
email-Enron	63.8 (AllSetTr)	75.7 (DSHN) / 76.8 (DSHNLight)
Cora	88.1 (PhenomNN)	87.8 (DSHN) / 88.0 (DSHNLight)

The accompanying gains are reported as $d$35 pt / $d$36 on Roman-empire, $d$37 pt / $d$38 on Squirrel, $d$39 pt / $d$40 on email-EU, $d$41 pt / $d$42 on Telegram, $d$43 pt / $d$44 on Chameleon, $d$45 pt / $d$46 on email-Enron, and approximately $d$47 on Cora. The contrast between Telegram and Cora is especially informative. On Telegram, described as highly directional, tuning $d$48 from $d$49 to $d$50 raises accuracy from $d$51 to $d$52. On Cora, described as homophilic, $d$53 is best. This suggests that the charge parameter functions as an explicit bias control: it can amplify oriented flows when directionality is semantically useful, but may act as directional noise in strongly homophilic settings.

The ablation analysis reports that DSHN accuracy improves with up to 7 layers, while standard HGNN degrades, and that accuracy rises with larger stalk dimension $d$54. It also reports that removing either the sheaf restriction maps or the complex phase causes performance to drop back to baseline levels. These observations are presented as evidence that the framework’s gains do not arise from directionality alone or from sheaf parameterization alone, but from their combination.

The reported limitations are equally specific. Scalability is constrained by the $d$55 cost when hyperedges are extremely large. Large-scale directed hypergraphs such as biochemical reaction networks and multi-agent flows remain to be tested. Further engineering of sparse GPU kernels for complex Laplacians is stated as needed. Future work is suggested in the form of a learnable charge $d$56, either as a trainable scalar or a per-hyperedge parameter, to automatically balance homophilic and heterophilic bias.