Sheaf Neural Networks (SNNs) generalize message passing by replacing scalar edge weights of standard Graph Neural Networks (GNNs) with learnable, edge-dependent restriction maps between node stalks. Despite their strong theoretical foundations and promising transductive results, SNNs have been evaluated almost exclusively on transductive node classification, leaving their behaviour under inductive protocols unknown. We address this gap through the first systematic benchmark of the sheaf design space, evaluating three diffusion mechanisms (neural sheaf diffusion, sheaf attention, and sheaf attention with Graph Attention Network v2), three restriction-map parameterizations, three stalk dimensions, and six modern GNN architectural components, within a message-passing reformulation that never assembles the heavy sheaf Laplacian, making the full design space trainable under cross-graph batching. Across 1,890 controlled experiments on 14 inductive datasets, multiple insights emerge: restriction maps are the dominant design choice and general maps are preferable, larger stalks add capacity but not long-range reach, architectural components explain more performance variation than the entire sheaf-specific design space itself. Under a matched protocol, SNNs transfer to inductive settings but do not reach the strongest baselines, with gaps being dataset-dependent. Practically, a single sheaf configuration can generalize across datasets, so effort is better spent tuning the surrounding architectural recipe than the sheaf operator itself.
Deep Graph Neural Networks (GNNs) are essential for capturing complex dependencies in graph-structured data. However, scaling GNNs to depth remains challenging, as stacking layers leads to representation collapse and diminishing sensitivity due to repeated aggregation. While Neural Sheaf Diffusion (NSD) provides strong theoretical guarantees against such collapse, these guarantees do not translate to practice: as depth increases, the disagreement signal of the sheaf Laplacian vanishes, limiting the contribution of deeper layers. We identify mechanisms that hinder NSD effectiveness at depth and propose \emph{Deep Neural Sheaf Diffusion} (DNSD), which replaces the sheaf Laplacian with a sheaf adjacency operator to maintain informative signals across layers. This is complemented by normalization, odd nonlinearities, and gating. To provide a principled explanation of the expected performance improvement, we contrast sheaf diffusion to graph attention mechanisms, highlighting that DNSD replaces scalar attention scores with matrix-valued edge functions and normalizes node representations rather than attention scores. We demonstrate empirically that DNSD effectively utilizes deep aggregation in graph tasks, outperforming GNN and NSD baselines with up to 30pp accuracy on synthetic long-range datasets, and consistently outperforming them on real-world benchmarks. These results position sheaf-based architectures as a promising building block for graph foundation models by supporting effective deep architectures.
Sheaf Neural Networks (SNNs) generalize Graph Neural Networks (GNNs) by replacing scalar node signals with stalk-valued signals and by using restriction maps to measure compatibility across edges. Unlike standard graph diffusion, which encourages neighboring node features to become similar, sheaf diffusion promotes consistency through the restriction maps and can therefore model more general relationships between neighboring nodes. However, existing sheaf neural architectures mainly operate at a fixed graph resolution and do not provide a principled pooling mechanism for building hierarchical representations. In this paper, we introduce Hierarchical Sheaf Pool (HiSP), a sheaf-aware pooling framework based on local spectral coarsening. Given a partition of the graph, HiSP constructs each coarse stalk by projecting fine stalk-valued features onto the low-frequency eigenmodes of the cluster-internal sheaf Laplacian. These local modes define a cochain-level prolongation map, which allows the fine sheaf energy to be represented on the coarse space through a Galerkin operator. We further analyze the approximation induced by coarsening by separating truncation loss, due to discarded local modes, from realization loss, due to representing the projected operator as a coarse sheaf. Finally, we implement HiSP as a GNN pooling layer compatible with SNNs and provide a PyG implementation supporting batching, lifted sheaf Laplacians, and hierarchical architectures.
Learned restriction maps in sheaf graph neural networks are often treated as proof that the model has discovered useful edge geometry. That conclusion does not follow from parameter movement or from a post-hoc ablation: both can show how one checkpoint is organized while leaving open whether learned transport still helps after the rest of the model adapts. We separate these claims with two estimands. Checkpoint reliance intervenes on the maps of a fixed predictor; protocol-relative replacement retrains matched families that remove map capacity, edge variation, or persistent edge assignment. A task-null theorem shows why the claims can diverge: labels identify only the transported classifier directions, leaving d2−d invisible degrees of freedom in every full d×d map. An exact frame model then gives the boundary at which reliance becomes unreplaced task value. Label-only training realizes the predicted separation, while audits of public NSD, DNSD, and Directed Sheaf Neural Network (DSNN) implementations recover both replaceable and unreplaced transport regimes on real graphs. All five DNSD benchmarks exhibit fixed-checkpoint reliance. After retraining, assignment-breaking or shared-map controls recover Full performance on four; Roman-Empire retains a .0675 advantage over continually resampled assignment and a .0391 advantage over a parameter-matched shared map across ten official splits. Thus, a learned map can govern a fitted computation without constituting indispensable edge geometry. Claims of learned transport should pair checkpoint interventions with matched retraining.