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Benchmarking Sheaf Neural Networks for Inductive Tasks

2026 · arxiv_cs

arXiv CS · Papers · 2026

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Benchmarking Sheaf Neural Networks for Inductive Tasks Stefano Fiorini1∗† , Edoardo Coppola2∗† , Pietro Liò2 1

Independent Researcher Department of Computer Science and Technology, University of Cambridge S.fiorini1994@gmail.com, ec2013@cam.ac.uk, pl219@cam.ac.uk

arXiv:2608.02558v1 [cs.LG] 3 Aug 2026

Abstract 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 messagepassing reformulation that never assembles the heavy sheaf Laplacian, making the full design space trainable under crossgraph 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.

†

Diffusion Mechanisms (NSD, SAN, [new] SANv2)

Laplacian-free Inductive Message Passing xj

mᵢⱼ xk

Restriction Map Parameterization (DIAG, O(d), GEN)

ℱ

update

mik

⊕

mil

aggregation of neighboring messages

ℱ

x'i

messages

Modern GNN Components (Dropout, Residual, FFN, Edge Features, PE, Normalization)

(message) mig = ϕ(xi, xj , eij ) (aggregation) ai = ⊕j∈N(i) mig (update) x′i = γ(xi, ai)

Benchmarks and Datasets ■ Molecules

■ Vision

■ Synthetic

■ Proteins

■ Code

GNN Benchmark

LRGB

OGB

■ ZINC ■ MNIST ■ CIFAR1 ■ PATTERN (node) ■ CLUSTER (node)

■ Peptides-func ■ Peptides-struct ■ PascalVOC-SP (node) ■ COCO-SP (node) ■ MalNet-Tiny

■ OGBG - Molhiv ■ OGBG - Molpcba ■ OGBG - PPA ■ OGBG - Code2

Figure 1: Benchmarking workflow overview. (Top-left) sheaf design space spanning 3 diffusion mechanisms, 3 restrictionmap parameterizations, and 6 GNN components; (top-right) our Laplacian-free inductive message passing; (bottom) evaluation benchmarks, datasets, and domain taxonomy.

Introduction

Graph Neural Networks (GNNs) have become the dominant paradigm for learning from graph-structured data, achieving remarkable success across a wide range of applications, including molecular prediction (Abramson et al. 2024), biological networks (Traversa et al. 2023), and recommender systems (Wu et al. 2022). Despite their success, conventional message-passing GNNs suffer from fundamental limitations, including oversmoothing, oversquashing, and reduced expressiveness on heterophilous graphs, motivating the development of increasingly expressive graph learning architectures (Li, Han, and Wu 2018; Alon and Yahav 2020). Among these, Sheaf Neural Networks (SNNs) represent one of the most principled extensions of message passing, replacing the classical graph Laplacian with a learnable sheaf Laplacian and propagating information through edge-dependent restriction maps rather than scalar weights (Hansen and Gebhart ∗

Sheaf Design Space

These authors contributed equally. Corresponding author.

2020). This richer diffusion mechanism has been shown, both theoretically and empirically, to improve robustness to heterophily and mitigate oversmoothing relative to standard message passing (Bodnar et al. 2022). However, despite a rapidly growing body of work on sheafbased graph learning (Barbero et al. 2022b,a; Duta et al. 2023; Fiorini et al. 2025), our empirical understanding of these models remains almost exclusively confined to transductive node classification, where training and inference are performed on a single, fixed graph. While such benchmarks have historically been central to graph representation learning, they capture only a narrow slice of real-world applications: molecular learning, protein modelling, graph classification, and dynamic networks all require inductive generalization, where models must operate on previously unseen graphs or unseen regions of a graph. Reflecting this, graph

learning research has progressively shifted its focus from transductive citation networks toward large-scale inductive benchmarks that better reflect realistic deployment scenarios, a shift that sheaf-based models have yet to follow. This shift has also highlighted how sensitive architectural comparisons are to the evaluation protocol. Once model selection, encoders, and tuning budgets are standardized, much of the performance gap between classical GNNs and more sophisticated architectures disappears (Errica et al. 2019; Dwivedi et al. 2023; Tönshoff et al. 2023). In particular, augmenting classical GNNs with modern architectural components enables them to match or even outperform substantially more expensive Graph Transformers (GTs) across 14 inductive benchmarks (Luo, Shi, and Wu 2025), establishing strong baselines for inductive graph learning. SNNs have remained largely absent from this re-evaluation, with prior applications limited to narrow settings (Ribeiro et al. 2025). More broadly, it remains unclear whether restriction-map type, diffusion mechanism, or their interaction drives inductive performance, and whether sheaf diffusion benefits from the modern architectural components that have proven decisive for classic GNNs. Addressing these gaps requires a systematic evaluation of SNNs under contemporary inductive protocols. This paper addresses this gap through the first systematic study of SNNs under a modern inductive protocol. Since existing sheaf-diffusion implementations target a single fixed graph and cannot operate on cross-graphs batches, we first reformulate sheaf diffusion as a Laplacian-free, edge-wise message-passing operator trainable across batched graphs. We then systematically evaluate the sheaf design space across three diffusion mechanisms, three restriction-map parameterizations, three stalk dimensions, and leave-one-out ablations of six modern GNN architectural components on 14 graphand node-level datasets, confirming the selected configurations at full budget. The resulting picture yields both mechanistic insight into what drives inductive sheaf performance and actionable guidance for where future effort should go. Our contributions are summarized as follows: • An inductive, Laplacian-free, message-passing reformulation of sheaf diffusion that enables the first systematic characterization and evaluation of SNNs under a modern inductive protocol. The same reformulation introduces SANv2, a new sheaf-attention layer with Graph Attention (GAT) v2-style dynamic attention. • A characterization of the sheaf design space over 1,890 controlled experiments, revealing, among others, that restriction-map parameterization is the dominant sheafspecific design choice, increased stalk dimension adds capacity rather than long-range reach, and the diffusion mechanism has negligible aggregate effect. • Actionable design guidance for inductive SNNs: the surrounding GNN+ architecture explains more performance variation than the entire sheaf design space and a single sheaf configuration generalizes across datasets, so tuning is better spent on the recipe than on the sheaf. We further quantify, dataset by dataset, the remaining gap to the strongest baselines.

Preliminaries

Given an undirected graph G = (V, E) with n = |V | nodes and m = |E| edges, let N (i) = {j : {i, j} ∈ E} denote the neighborhood of node i. We use Ik for the k × k identity, ⊗ for the Kronecker product, and ⊙ for the Hadamard product.

2.1

Cellular Sheaves and the Sheaf Laplacian

Definition 1. A cellular sheaf (G, F ) over an undirected graph G associates a vector space F(i) (the node stalk) to each node i ∈ V , a vector space F(e) (the edge stalk) to each edge e ∈ E, and a linear restriction map Fi⊴e : F (i) → F(e) for each incident node–edge pair i ⊴ e. Following Bodnar et al. (2022), we take all stalks isomorphic to Rd for a fixed stalk dimension d, so each restriction map is a d × d matrix. In the opinion-dynamics interpretation (Hansen and Gebhart 2020), node stalks F(i) represent a node’s private space, and Fi⊴e xi describes how that private state is expressed in the shared L edge space F(e). The space of 0-cochains C 0 (G, F) = i∈V F(i) stacks the node-stalk L vectors; the space of 1-cochains C 1 (G, F) = e∈E F(e) stacks the edge-stalk vectors. Fixing an arbitrary orientation e = i → j per edge, the coboundary operator δ : C 0 (G, F) → C 1 (G, F) is (δX)e = Fj⊴e xj − Fi⊴e xi , and the sheaf Laplacian is LF = δ ⊤ δ. Definition 2. The sheaf Laplacian acts on a 0-cochain X node-wise as X ⊤ LF (X)i = Fi⊴e (Fi⊴e xi − Fj⊴e xj ) . (1) i,j⊴e

As a matrix, LF is symmetric positive semi-definite with P ⊤ diagonal blocks Lii = i⊴e Fi⊴e Fi⊴e and off-diagonal ⊤ blocks Lij = −Fi⊴e Fj⊴e . The normalized sheaf Laplacian −1/2 is ∆F = D LF D−1/2 , with D the block diagonal of LF . For the trivial sheaf, d = 1, ∆F reduces to the normalized graph Laplacian, so the sheaf Laplacian strictly generalizes the operator underlying graph convolution (Kipf and Welling 2017). Its kernel ker(LF ) = {X : Fi⊴e xi = Fj⊴e xj ∀e = {i, j}} is the space of signals in agreement across every edge, and the asymptotics of sheaf diffusion are governed by it.

2.2

Neural Sheaf Diffusion

Bodnar et al. (2022) build Neural Sheaf Diffusion (NSD) by discretizing the sheaf diffusion PDE Ẋ(t) = −∆F X(t). After lifting the input features into h channels per stalk dimension, an NSD layer applies Xt+1 = Xt − σ ∆F (t) (In ⊗ W1,t ) Xt W2,t , (2) where σ is a point-wise nonlinearity and W1,t ∈ Rd×d , W2,t ∈ Rf ×f mix the stalk and channel dimensions. Crucially, the sheaf F(t) is itself learned and evolves with depth: each restriction map is produced by a parametric function of the incident node features, Fi⊴e:=(i,j) = Φ(xi , xj ), with Φ typically an MLP on the concatenation, i.e. (xi ∥xj ). Bodnar et al. (2022) show that, in the infinite-depth limit, sheaf diffusion with sufficiently rich restriction maps can linearly separate classes in heterophilic settings where standard graph diffusion provably oversmooths. The relevant axis is the structure imposed on the restriction maps.

2.3

Sheaf Attention

Sheaf Attention Networks (SAN) (Barbero et al. 2022b) augment sheaf diffusion with input-dependent attention, generalizing the GAT architecture (Veličković et al. 2018) to SNNs. Let AF denote the sheaf adjacency with selfloops, whose (i, j)-th block is given by the transport map ⊤ Pij = Fi⊴e Fj⊴e . A scalar attention matrix Λ is computed as in GAT, Λij = softmaxj LeakyReLU(a⊤ [W xi ∥W xj ]) , and lifted to stalk dimension as Λ̂ = Λ ⊗ 1d . The SAN layer is defined as Xt+1 = σ Λ̂(Xt ) ⊙ AF (In ⊗ W1,t ) Xt W2,t , (3) with a residual variant (Res-SAN) inserting an (Λ̂ ⊙ AF − I) update around a skip connection. GAT is recovered when d = 1 and the restriction maps are equal to 1. Since Pij and Λij are learned independently, feature transport is decoupled from neighbor weighting. Barbero et al. (2022b) parameterize O(d) maps and let attention coefficients determine neighbors relative contributions.

2.4

Restriction Map Families

The restriction maps Fi⊴e ∈ Rd×d can be organized into a hierarchy of increasingly general parameterizations (Bodnar et al. 2022): Diagonal (Diag). Fi⊴e is diagonal, scaling each stalk coordinate independently. This parametrization has fewest parameters and stalk dimensions interact only through the left multiplication by W1 . Orthogonal (O(d)). Fi⊴e ∈ O(d), realizing a discrete O(d)-bundle. Orthogonal maps mix stalk dimensions while preserving norms, regularize via fewer free parameters (d(d − 1)/2), and yield Laplacians that are numerically convenient to normalize. General (Gen). Fi⊴e is an unconstrained d × d matrix ensuring maximal flexibility and more parameters at the cost of greater overfitting risk and a Laplacian that is harder to normalize numerically (requiring SVD).

Related Works

Sheaf Neural Networks. SNNs equip a graph with a cellular sheaf, attaching vector-space stalks to nodes and edges and linear restriction maps to incident pairs, thereby generalizing the graph Laplacian to a sheaf Laplacian (Hansen and Ghrist 2019). While Hansen and Gebhart (2020) introduced the first sheaf network with a hand-crafted Laplacian, Bodnar et al. (2022) made the approach practical with Neural Sheaf Diffusion (NSD), learning the restriction maps end-to-end and establishing that sheaf diffusion can linearly separate classes in heterophilic regimes where graph diffusion oversmooths. Subsequent work extended NSD along several axes: attention (Barbero et al. 2022b), connection Laplacians obtained via Riemannian preprocessing (Barbero et al. 2022a), hypergraphs (Duta et al. 2023), heterogeneous graphs (Braithwaite et al. 2024), and directionality (Fiorini et al. 2025; Ribeiro et al. 2025). Across this line, evaluation is almost entirely transductive node classification on a fixed set of heterophilic

benchmarks and, notably, the reported best model family is inconsistent across papers (O(d) maps for Bodnar et al. (2022), GEN maps for Braithwaite et al. (2024)), indicating the optimal diffusion–map parameterization design is taskdependent and, to date, uncharacterized in inductive settings. GNN Benchmarking. A parallel literature has shown GNN gains are fragile to evaluation protocol: once model selection and budgets are standardized, much of the reported gap between classic GNNs and sophisticated architectures (e.g. GTs) disappears (Errica et al. 2019; Dwivedi et al. 2023; Tönshoff et al. 2023). Most recently, Luo, Shi, and Wu (2025) pushed this further, showing component-augmented classic GNNs (GCN, GIN, GatedGCN) match or surpass state-of-the-art GTs across 14 inductive graph- and nodelevel datasets at a fraction of their cost, and establishing the standard evaluation protocol and baselines we adopt.

Sheaf Models and Inductive Implementation

Throughout, d is the stalk dimension, f the channels per stalk, c = df the per-node representation size. Node-level tasks feed the node-stalk representations to a per-node head, while graph-level tasks first pool them with a permutation-invariant readout before the head. Our study is organized around a twodimensional design space: three diffusion mechanisms that propagate information and three restriction-map parameterizations (and dimensions) that transport features between stalks. The two axes are orthogonal, so their combination forms a 3 × 3 grid, one row of which is a new sheaf-attention layer we introduce. We first define the grid, then propose a Laplacian-free implementation for any diffusion mechanism that makes it trainable under cross-graph batching, introduce our dynamic sheaf-attention mechanism, derive the complexity, and define how modern GNN components are combined with the inductive sheaf operator (Fig. 1-Top-left). The Sheaf Design Space. The three diffusion mechanisms analyzed in this study share the sheaf transport block ⊤ Pij = Fi⊴e Fj⊴e , the mixing matrices W1 ∈ Rd×d and W2 ∈ Rf ×f , the intrinsic nonlinearity σ, and a residual connection. They differ in how neighbor messages are weighted. NSD propagates through the learnable normalized sheaf Laplacian of Eq. (2), so every neighbor contributes through the transport alone. SAN and our newly proposed SANv2 instead modulate the transport with input-dependent attention, and differ from each other in how the attention scores are computed: statically for SAN, dynamically for SANv2. Because the restriction maps set Pij while the attention coefficients set the weighting, how features are transported between stalks stays decoupled from how strongly each neighbor contributes. Each mechanism is paired with one of the three restriction-map families seen in Sec. 2 (Diag, O(d), Gen), giving nine models. With all else fixed, this grid isolates two questions prior work conflates: (1) whether restriction-map type or diffusion mechanism dominates, and (2) whether they interact, that is, whether the best map family depends on the mechanism and the reverse.

Laplacian-Free Inductive Implementation. The reference NSD implementation (Bodnar et al. 2022) targets a single fixed graph and cannot batch across graphs without restructuring. We instead implement every grid cell as a message-passing operator that never materializes the heavy nd × nd sheaf Laplacian and evaluates its action edge-wise, with the restriction maps Fi⊴e predicted from edge_index and its (Fig. 1-Top-right). For each edge {i, j} ∈ E with edge features eij , each layer computes three steps: (message) (aggregation)

mij = ϕ(xi , xj , eij ); M X ai = mij = mij ; j∈N (i)

(update)

(4) (5)

j∈N (i)

x′i = γ(xi , ai ).

(6)

In every case the message ϕ routes the neighbor’s repre⊤ sentation through the per-edge transport Pij = Fi⊴e Fj⊴e , while an optional mechanism-specific edge gate Γij and an optional node-wise normalization determine the exact form of ϕ and of the update γ. NSD applies no attention mechanism, Γij = If , and normalizes each stalk by its block degree Di = P ⊤ e∈E(i) Fi⊴e Fi⊴e , where E(i) is the set of edges inci−1/2

dent to i. Writing zi = Di xi for the normalized representation, the aggregation of Eq. (5) becomes the transported difference between a node and its neighbors, ai = P Di zi − j∈N (i) Pij zj , and the update function, Eq. (6), is defined as follows: −1/2 γ(xi , ai ) = xi − σ Di ai . (7) −1/2

For the Gen parameterization we obtain Di through a batched, deterministic Newton–Schulz iteration rather than the reference’s random-noise regularization, which would otherwise be a source of numerical instability. SAN and SANv2 use the mixed representation x̃ = W1 xW2⊤ directly, without degree normalization, and set Γij = diag(αij ) with an attention gate αij ∈ Rf , the channel-wise counterpart of the scalar coefficient Λij of Eq. (3): X ai = Pij x̃j diag (αij ) , j∈N (i) (8) ′ xi = xi + σ(ai − x̃i ) , where x̃i makes the bracket a high-pass sheaf update (Di Giovanni et al. 2022) distinct from the outer residual skip xi . The weights are produced by multi-head attention over peredge scores αij . SAN computes them with the static scoring of GAT (Veličković et al. 2018), ⊤ αij = LeakyReLU a⊤ (9) s W xi + ad W xj . This scoring is static: the ranking a node induces over its neighbors is fixed by the learned parameters and does not depend on the query node, a limitation that Brody, Alon, and Yahav (2021) identified for GAT and resolved with GATv2. We therefore introduce SANv2, which keeps the sheaf update of Eq. (8) unchanged, same transport Pij , same channel-wise

gate, same high-pass self term, and replaces only the scoring with its dynamic counterpart, αij = a⊤ LeakyReLU (Wl xi + Wr xj ) .

(10)

By applying the nonlinearity after the combined projection and moving the attention vector outside it, GATv2 reordering makes neighbor ranking query-dependent rather than fixed (Brody, Alon, and Yahav 2021). The change is confined to how each edge is scored, so it leaves the sheaf geometry, the complexity, and the batching properties of the layer untouched, and it composes with all three restriction-map families. To our knowledge SANv2 is the first sheaf-attention layer with dynamic attention. Scoring is multi-head with H heads (head-specific parameters in Eqs. (9)–(10)), giving (h) (h) per-head coefficients αij = softmaxj (eij ). By default the heads are combined by concatenation: head h weights its own contiguous block of the f channels, so αij is block-constant h(c) across channels ([αij ]c = αij for the head h(c) owning channel c), while the transport Pij stays a single d × d block per edge. Complexity. Per layer per graph, restriction-map prediction dominates at O(m d3 f ) for O(d)/Gen and O(m d2 f ) for Diag; transport and aggregation add O(m d2 f ); degree blocks and their inverse square roots add O((n + m) d3 ). In total, Time: O (n + m) d2 f + m d3 f (11) Space: O m d2 + (n + m) df , both linear in n and m. No operator is assembled or factorized, and the edge-wise pass composes with PyTorch Geometric batching, so a batch of graphs is one disjoint union processed without a per-graph loop. This is what makes our sheaf models trainable in the inductive, graph-batched regime the reference implementation does not support. Composition with Modern GNN Components. Two placements are specific to the sheaf operator. The activation σ is intrinsic to the sheaf operator (Eqs. (2) and (8)), so no post-convolution activation is applied, and normalization and dropout act on the convolution increment before the residual add. With Nz a normalization and Dp a dropout (both reduced to the identity when their component is switched off, i.e. gnorm =0 or gdrop =0), g• ∈ {0, 1} the component selectors, and input encodings concatenated upstream when gpe =1, layer l maps a(l) = Convl h(l) , E, gedge ·eij , (12) b(l) = Dp Nz a(l) , (13) c(l) = gres h(l) + b(l) , h(l+1) = gffn FFNl c(l) + (1 − gffn ) c(l) ,

(14) (15)

with the feed-forward block FFNl (z) = Nz2 Nz1 z) + Dp W2ff Dp σ(W1ff Nz1 (z)

The residual in Eq. (15) is the simple additive skip used by GNN+ . Edge features are integrated inside Convl by

conditioning the restriction maps rather than the messages or attention coefficients. The sheaf learner predicts each map from the concatenated endpoint representations, optionally augmented by an encoded edge feature, h i (l) (l) (l) Fv⊴e = Φl hi ∥ hj ∥ gedge ·We eij , where Φl denotes the map-predicting linear projection followed by the family-specific construction (DIAG, O(d), or Gen), and We a learned edge encoder. When gedge = 1 the transport blocks Fi⊤ Fj and the degree normalization are thus conditioned on eij . When gedge = 0 the maps depend on the node features alone as in the reference implementation. The selectors g• parametrize the component configurations outlined in the following section.

Experimental Setup

Datasets and Tasks. Datasets and Tasks. We evaluate sheaf-based models on 14 inductive datasets spanning graph- and node-level classification and regression tasks (Fig. 1-Bottom). The evaluation includes five datasets from the GNN Benchmark (Dwivedi et al. 2023) (ZINC, MNIST, CIFAR10, PATTERN, and CLUSTER), four from the Open Graph Benchmark (OGB) (Hu et al. 2020) (ogbg-molhiv, ogbg-molpcba, ogbg-code2, and ogbg-ppa), and five from the Long-Range Graph Benchmark (LRGB) (Dwivedi et al. 2022; Freitas and Dong 2021) (Peptides-func, Peptides-struct, PascalVOC-SP, COCO-SP, and MalNet-Tiny). All 14 datasets are used in the screening stage, while the confirmation stage covers 12 for conciseness. We report the two remaining datasets, ogbg-ppa and COCO-SP, in the Supplementary Material. We follow the official train/validation/test splits and evaluation protocols of each benchmark (Supplementary Material for details). We follow the official train/validation/test splits and evaluation protocols of each benchmark (Supplementary Material for details). Evaluation Protocol. Following Luo, Shi, and Wu (2025), we adopt their standardized GNN+ training pipeline1 , including the same feature and positional/structural encoders, optimizer, learning-rate schedule, gradient clipping, and evaluation protocol (see Supplementary Material for details). We replace only the message-passing layer with the proposed sheaf operator, ensuring that performance differences are attributable solely to the architecture. We further adopt a two-stage screen-then-confirm protocol. Candidate configurations are first screened using a single random seed and a reduced training budget, with model capacity constrained to the per-dataset parameter budget of Luo, Shi, and Wu (2025). During the screening stage, we evaluate the two primary sheaf diffusion mechanisms, NSD and SAN, across the full factorial of datasets, restriction-map parameterizations, and stalk dimensions (d ∈ {2, 3, 4}), together with leave-one-out ablations of each GNN+ component and GNN+ augments classic GNNs with normalization, residual connections, dropout, feed-forward layers, edge features, and positional/structural encodings (Luo, Shi, and Wu 2025). 1

an all-components-off baseline. Each component is evaluated across all restriction-map parameterizations and stalk dimensions, allowing conclusions to be drawn from aggregate trends over the full factorial design. For each diffusion mechanism and dataset, we select the best-performing configuration. In the final confirmation stage, we evaluate NSD, SAN, and SANv2, where the latter inherits the best SAN configuration and is therefore not screened independently. All selected models are then trained using the full epoch and budget of Luo et al. Baselines. Our comparisons differ between the two stages. During screening, leave-one-out ablations are evaluated against the corresponding all-off sheaf baseline to isolate the contribution of each GNN+ component. During confirmation, the selected sheaf models are compared against strong GNNs equipped with the GNN+ recipe, Graph Transformers, and prior specialized architectures, using the results reported in (Luo, Shi, and Wu 2025). The best, second-, and third-best results are highlighted. We report the mean performance and standard deviation over five random seeds.

6 6.1

Results

Screening: Understanding the Sheaf Design Space

The screening stage explores the sheaf design space across diffusion mechanisms, restriction-map parameterizations, stalk dimensions, and GNN+ components to determine which design choices consistently govern the performance of inductive SNNs. Of its 14 × 6 × 3 × 8 = 2,016 cells, 108 collapse onto the full recipe because six datasets carry no positional encoding and 18 are infeasible at the parameter budget, leaving 1,890 runs. Because the 14 datasets carry six metrics with opposite optimization directions, per-cell values cannot be pooled directly: we map each run to a within-dataset ranknormalized score in [0, 1] (Fig. 2) and treat the dataset as the unit of analysis, assessing every axis by paired comparisons across the 14 datasets with exact Wilcoxon signed-rank tests (Table 1). Conclusions therefore hold across benchmarks rather than within individual cells. Since ranking discards the size of the gaps it orders, we repeated every comparison under a metric-linear score that retains them: 12 of the 13 are unchanged in direction and significance. Several insights are revealed. Restriction-map parameterization is the dominant design choice, accounting for 12.2% of the score variance against 0.2% for the choice of diffusion mechanism, with the ordering Gen > Diag > O(d) holding in all six mechanism × d slices. The Gen–O(d) gap is 0.242, 0.244 and 0.224 at d={2, 3, 4}, hence independent of how many free parameters an orthogonal map is given. This suggests that orthogonal transport is limited by its norm-preserving constraint rather than by its smaller parameterization, since preserving feature norms limits the ability to adaptively attenuate neighboring messages. Consistently, O(d) fares worst where edges are most numerous (0.316 on the four superpixel datasets against 0.395 on OGB). Increasing the stalk dimension improves performance monotonically on 13/14 datasets, with every successive step significant (except ogbg-ppa, discussed below). Setting

Diag O(d) Gen

A Restriction map

d=2

B Stalk dim.

d=3

d=4

C GNN + component full recipe − edge feat. − BatchNorm − dropout − residual − FFN − PE all removed

ZINC ∼ 500K MAE↓

MNIST ∼ 100K Accuracy↑

CIFAR10 ∼ 100K Accuracy↑

PATTERN ∼ 500K Accuracy↑

CLUSTER ∼ 500K Accuracy↑

EGT (2022) Graph ViT/MLP-Mixer (2023) GRIT (2023) GRED (2024) GEAET (2024) TIGT (2024) GMN (2024) GIN+ (2025) GatedGCN+ (2025)

0.108±0.009 0.073±0.001 0.059±0.002 0.077±0.002 – 0.057±0.002 – 0.065±0.004 0.077±0.005

0.9817±0.0009 0.9846±0.0009 0.9811±0.0011 0.9838±0.0001 0.9851±0.0009 0.9823±0.0013 0.9839±0.0018 0.9829±0.0010 0.9871±0.0014

0.6870±0.0041 0.7396±0.0033 0.7647±0.0088 0.7685±0.0019 0.7663±0.0043 0.7396±0.0036 0.7456±0.0038 0.6959±0.0029 0.7722±0.0038

0.8682±0.0002 – 0.8720±0.0008 0.8676±0.0002 0.8699±0.0003 0.8668±0.0006 0.8709±0.0126 0.8684±0.0005 0.8703±0.0004

0.7923±0.0035 – 0.8003±0.0028 0.7850±0.0010 – 0.7802±0.0022 – 0.7479±0.0021 0.7913±0.0024

NSD SAN SANv2

0.089±0.008 0.088±0.003 0.092±0.004

0.9751±0.0037 0.9753±0.0012 0.9742±0.0015

0.6661±0.0051 0.6636±0.0059 0.6750±0.0063

0.8674±0.0001 0.8677±0.0003 0.8684±0.0002

0.7836±0.0071 0.7666±0.0041 0.6805±0.0678†

Param. budget Metric

Table 2: Performance on GNN Benchmark datasets.

GNN Benchmark LRGB OGB 0

mean within-dataset rank-normalized score (1 = best)

Figure 2: Screening marginal effects. Each point is the mean within-dataset rank-normalized score (1 = best cell on that dataset) at one level of a screening axis, averaged over the datasets of a benchmark suite. A: restriction-map parameterization. B: stalk dimension. C: GNN+ component, where each row removes one component from the full recipe. Directions are consistent across all three suites. Comparison

Mean ∆

Restriction map Gen > O(d) Diag > O(d) Gen > Diag

(η 2 = 12.2%; Friedman χ22 = 17.71, p < 0.001) +0.237 13/14 0.0004 +0.186 13/14 0.0002 +0.052 11/14 0.0052

Stalk dimension d=4 > d=2 d=3 > d=2 d=4 > d=3

(η 2 = 8.5%; Friedman χ22 = 20.57, p < 0.001) +0.204 13/14 0.0002 +0.132 13/14 0.0002 +0.071 13/14 0.0017

Datasets favoring

Diffusion mechanism (η 2 = 0.2%) NSD vs SAN +0.023 8/14

0.502

GNN+ component, removed vs. full recipe − dropout +0.211 14/14 − BatchNorm† +0.124 12/14 − FFN −0.045 5/14 − PE −0.109 2/8 − edge feat. −0.087 2/14 − residual −0.139 0/14 all removed −0.294 1/14

(η 2 = 26.7%) 0.0001 0.0085 0.268 0.148 0.0005 0.0001 0.0006

Table 1: Paired comparisons over 1,890 completed screening runs. ∆ is the difference in within-dataset rank-normalized score, averaged over 12 datasets; p is an exact two-sided Wilcoxon signed-rank test on paired per-dataset differences. η 2 is the share of total score variance attributable to the axis. † The only row whose significance depends on normalization: under a score linear in the native metric it falls to p = 0.058, so we treat it as suggestive.

larger d, each node receives extra channel capacity to route different neighbors’ contributions. Such added capacity is what sheaf diffusion is argued to gain over scalar-weighted

Param. budget Metric

Peptides-func ∼ 500K Avg. Precision↑

Peptides-struct ∼ 500K MAE↓

PascalVOC-SP ∼ 500K F1 score↑

MalNet-Tiny ∼ 500K Accuracy↑

GraphGPS (2022) Exphormer (2023) GRED (2024) Graph-Mamba (2024) GSSC (2024) GCN+ (2025) GIN+ (2025) GatedGCN+ (2025)

0.6534±0.0091 0.6258±0.0092 0.7133±0.0011 0.6739±0.0087 0.7081±0.0062 0.7261±0.0067 0.7059±0.0089 0.7006±0.0033

0.2509±0.0014 0.2512±0.0025 0.2455±0.0013 0.2478±0.0016 0.2459±0.0020 0.2421±0.0016 0.2429±0.0019 0.2431±0.0020

0.4440±0.0065 0.3446±0.0064 – 0.4191±0.0126 0.4561±0.0039 0.3357±0.0087 0.3189±0.0105 0.4263±0.0057

0.9350±0.0041 0.9402±0.002 – 0.9340±0.0027 0.9406±0.0064 0.9354±0.0045 0.9325±0.0040 0.9460±0.0057

NSD SAN SANv2

0.6374±0.0071 0.6376±0.0075 0.6228±0.0093

0.2577±0.0021 0.2583±0.0021 0.2546±0.0037

0.3317±0.0069 0.3251±0.0035 0.3052±0.0008

0.9300±0.0014 0.9235±0.0035 0.9275±0.0078

Table 3: Performance on LRGB datasets.

Metric

ogbg-molhiv AUROC↑

ogbg-molpcba Avg. Precision↑

ogbg-code2 F1 score↑

SAT (2022) Specformer (2023) Subgraphormer (2024) GECO (2024) GSSC (2024) GCN+ (2025) GIN+ (2025) GatedGCN+ (2025)

– 0.7889 ± 0.0124 0.8038 ± 0.0192 0.7980 ± 0.0200 0.8035 ± 0.0142 0.8012 ± 0.0124 0.7928 ± 0.0099 0.8040 ± 0.0164

– 0.2972 ± 0.0023 – 0.2961 ± 0.0008 – 0.2721 ± 0.0046 0.2703 ± 0.0024 0.2981 ± 0.0024

0.1937 ± 0.0028 – – 0.1915 ± 0.0020 – 0.1787 ± 0.0026 0.1803 ± 0.0019 0.1896 ± 0.0024

NSD SAN SANv2

0.7606±0.0021 0.7563±0.0275 0.7241±0.0202

0.2333±0.0014 0.2349±0.0031 0.2362±0.0008

0.1559±0.0197 0.1592±0.0219 0.1594±0.0205

Table 4: Performance on OGB datasets.

message passing. However, if observed gains came from that mechanism, they should be concentrated where information must travel farthest, or where many neighbors compete for the same channels, but we observe no such concentration. Over the 13 datasets on which the trend is monotone, the d=4 over d=2 improvement is uncorrelated with the average number of nodes per graph (Spearman ρ = +0.08) and with the average node degree (ρ = −0.15), and its mean over the five LRGB datasets (+0.234), which are constructed to require long-range dependencies, is close to its mean over the remaining eight (+0.218). Therefore, larger stalks behave as a generic increase in model capacity rather than as a targeted remedy for long-range reach, although with 13 datasets only a strong association would be detectable. Searching the sheaf design space per dataset recovers almost nothing once the GNN+ recipe is tuned. Selecting the best of the 18 architectures separately on every dataset attains a rank-normalized score of 1.0 there by construction, yet fixing the architecture globally to NSD-Gen at d=4 and tuning only the recipe per dataset still averages 0.991, and is the single best cell outright on five of the 14 datasets. A per-

dataset tuning budget is therefore better spent on the GNN+ components than on the sheaf architecture. The surrounding GNN+ architecture explains more performance variation (26.7%) than the entire 18-cell sheaf design space (21.1%). In particular, residual connections and edge features are essential, with only 0/14 and 2/14 datasets improving when they are removed, whereas dropout is detrimental on all 14 datasets. A plausible explanation is that the learned restriction maps already act as a per-edge bottleneck, providing an implicit form of regularization that reduces the benefit of dropout. Finally, while NSD and SAN achieve comparable average performance (p = 0.50), they exhibit different behaviors: SAN is more robust to the removal of GNN+ components, losing 0.169 score points against 0.422 for NSD when all are removed, and it leads on 5/8 graph-level datasets, whereas NSD is ahead on all four node-level datasets (mean +0.126). However, the two mechanisms do not merely degrade at different rates: their ranking reverses with the component stack. Under the full recipe NSD leads, 0.564 against 0.515; with every component removed SAN leads by a wider margin, 0.346 against 0.142. The aggregate parity reported above is thus an average over two opposite regimes, and a comparison between diffusion mechanisms is interpretable only with respect to the component stack in which it was measured. One dataset, ogbg-ppa, deviates from the overall trends. Under every recipe retaining BatchNorm, as prescribed by the evaluation protocol, the models fit the training set while validation accuracy collapses to at most 11%. Removing BatchNorm leaves training essentially unchanged but restores validation performance (see Supplementary Material), suggesting that estimated batch statistics do not transfer to evaluation. This sensitivity appears specific to sheaf diffusion rather than to the dataset itself, as GNN+ baselines achieve 0.81–0.83 under the same protocol. We therefore retain the reference normalization for a fair comparison and simply note that default ranking compares near-chance models. This also highlights a constraint of the inherited protocol that prevents from using alternative normalizations that behave identically during training and evaluation (e.g. LayerNorm). While we cannot relax it, we leave this exploration to future studies. Overall, these findings identify expressive restriction maps, larger stalk dimensions, and a carefully designed GNN+ architecture as the primary factors behind SNN’s inductive performance. Performance of individual cells is provided in the Supplementary Material.

6.2

Confirmation: Overall Performance

Following the screening stage, Tables 2–4 report the selected NSD, SAN, and SANv2 configurations alongside strong GNNs, GTs, and specialized baselines under the standardized evaluation protocol of Luo, Shi, and Wu (2025). Performance is reported as the mean and standard deviation over five random seeds. Only top-3 entries from the work of Luo, Shi, and Wu (2025) are reported for conciseness. Overall, SNNs can transfer to inductive settings, although a gap with the strongest modern architectures remains. That gap is highly dataset dependent, spanning two orders of mag-

nitude across the twelve datasets reported here, and it tracks neither the benchmark suite nor the task level: the smallest and largest gaps both occur within the GNN Benchmark suite, and node- and graph-level datasets are interleaved throughout the ranking. The closest results are obtained on PATTERN, MNIST, MalNet-Tiny and CLUSTER, where sheaf models average 1.4% from the best-performing method, with NSD outperforming TIGT and GIN+ on CLUSTER. The largest gaps, averaging 30%, are on ZINC, PascalVOC-SP, ogbg-molpcba and ogbg-code2. Notably, the four datasets on which sheaf models come closest are also those on which the baselines themselves differ least, so near-parity there suggests limited headroom rather than a property of sheaf diffusion. Comparing the three mechanisms reveals complementary behavior rather than a universally superior architecture. NSD attains the best sheaf result on two of the three node-classification datasets, in line with transductive findings (Barbero et al. 2022b; Bodnar et al. 2021), whereas the attention-based mechanisms take seven of the nine graphlevel datasets. Although SANv2 introduces dynamic GATv2style attention, it does not consistently outperform the original SAN, suggesting that attention design alone is not the primary bottleneck for inductive sheaf learning. Overall, these observations closely mirror the conclusions of the screening study. Restriction-map parameterization and the surrounding GNN+ architecture account for substantially more performance variation than the choice of diffusion mechanism, while NSD and SAN exhibit complementary inductive biases that persist under full training. These results indicate that the principal limitations of current SNNs lie less in the diffusion mechanism itself than in the architectural design surrounding it, providing a clear direction for future sheaf-based models.

Conclusions

We explored the sheaf design space across diffusion mechanisms, restriction-map parameterizations, stalk dimensions, and modern GNN components on 14 inductive graph- and node-level datasets under a standardized protocol, screening 1,890 configurations and confirming the best ones at full budget. SNNs can transfer to inductive settings, but do not reach the strongest published baselines on any dataset considered, and the gap varies by two orders of magnitude across datasets without tracking benchmark suite or task level. Screening identifies where performance is decided: restriction-map parameterizations govern the sheaf operator, GEN maps are preferable at every stalk dimension, larger stalks contribute capacity rather than long-range reach, and the surrounding component recipe explains more variance than the entire sheaf-specific design. Practically, a single sheaf architecture can generalize across datasets while the component recipe does not, so tuning is better directed at the latter. This suggests where future SNNs should invest: expressive maps and component recipe. Limitations and Future Works. Despite our extensive experiments, our factorial exploration is only near-complete, and additional interactions among the GNN+ components may provide further insights. Likewise, although we evaluate a broad collection of datasets, stronger correlations may emerge only from a larger and more diverse benchmark suite. We leave both directions for future work.

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Experimental Setup and Reproducibility

8.1

Code

The code associated with this work will be released upon acceptance for fostering research on Sheaf Neural Networks.

8.2

Computing Environment

All experiments are run, one per GPU, on NVIDIA DGX A100 (with 8 GPUs of 80GB memory, optionally partitioned in MIG instances of 40GB for lighter experiments), a double 64-core AMD EPYC 7742 and 2TB of RAM. The software environment is fully specified by the environment.yml file shipped with the code and can be recreated with a single conda env create invocation. The versions of the components that determine numerical behavior are listed in Table 5. We do not use mixed precision, and all reported numbers are produced in float32. Component

Version

Python PyTorch CUDA toolkit PyTorch Geometric pyg-lib torch-scatter torch-sparse torch-cluster torchmetrics GPU

3.10.20 2.2.0 (+cu118) 11.8 2.3.1 0.4.0 2.1.2 0.6.18 1.6.3 1.9.0 NVIDIA A100 40–80 GB

Table 5: Software and hardware environment. The complete dependency set, including transitive pins, is given in environment.yml in the released code.

8.3

What Is Inherited and What Is Replaced

Our implementation reuses, without modification, the GraphGym-based training and evaluation pipeline released by Luo, Shi, and Wu (2025). Only the message-passing block is replaced by the sheaf operator. Specifically, we inherit verbatim the per-dataset node and edge feature encoders, Random-Walk Structural Encoding (RWSE) where applicable, the AdamW optimizer, the cosine learning-rate schedule with warmup, gradient clipping, weight decay, batch sizes, loss functions, metric implementations, and the train/validation/test splits. Any difference in performance is therefore attributable to the architecture rather than to the training recipe, so that the results we report are directly comparable to those of Luo, Shi, and Wu (2025). We acknowledge that this inheritance may also represent a constraint in some cases. For example, the normalization layer is exposed by the pipeline as a binary switch rather than as a choice of scheme, which is why the BatchNorm behavior on ogbg-ppa discussed in the main paper could be diagnosed but not altered within the protocol.

8.4

Seeding and Determinism

Every run is seeded through a single entry point before any model, data loader, or shuffling is constructed, using torch_geometric.seed_everything, which sets the Python, NumPy, and PyTorch (CPU and CUDA) generators. A run with index i uses seed s0 +i, with s0 = 0 throughout, so the seeds of a k-seed confirmation are 0, . . . , k−1 and are identical across architectures and datasets. Seeds control weight initialization, data-loader shuffling, dropout masks, and any stochastic component of the restriction-map learner. Notably, screening uses a single - design (Section 8.5), since its purpose is to identify aggregate trends and consistent ranking across many configurations rather than to support claims about individual cells. Nonetheless, the insights reported in the main paper are supported by solid statistical tests and analysis. Confirmation stage adopts 5 seeds, consistently with prior literature.

8.5

Screen-then-Confirm Protocol

The evaluation proceeds in two stages: 1. Screening. A near-complete factorial over diffusion mechanism, restriction-map family, stalk dimension, and GNN+ component ablations is run with a single seed and the reduced per-dataset epoch budget of Table 6. The purpose of this stage is to rank candidate configurations consistently and aggregate trends rather than to estimate their final performance. 2. Confirmation. For each dataset and each diffusion mechanism, the configuration with the best validation score is retrained from scratch at the full reference epoch budget and matched per-dataset parameter budget of Luo, Shi, and Wu (2025), over five random seeds. SANv2 inherits the SAN configuration rather than being screened independently, so that the comparison between the two isolates the attention mechanism. Model selection. In both stages, the reported test metric is the one recorded at the epoch of best validation performance (test-at-best-validation). The test split is never used for model selection or hyperparameter tuning.

8.6

Fair-Benchmarking Choices

We list the decisions that most affect comparability, so that they can be audited rather than inferred. • Matched pipeline. Every component outside the message-passing block is inherited from Luo, Shi, and Wu (2025) and is identical across all architectures compared, including the baselines whose numbers we quote. • Matched parameter budget. Confirmation widths are chosen so that the parameter count matches as closely as possible the values reported by Luo, Shi, and Wu (2025). This ensures the fairest comparison across architectures. • Official splits. All datasets use their prescribed train/validation/test splits and evaluation metrics. No split is re-partitioned and no metric is redefined. • No test-set selection. Model selection is performed exclusively on the validation set in both screening and confirmation.

Dataset

Screen epochs

ZINC MNIST CIFAR10 PATTERN CLUSTER Peptides-func Peptides-struct PascalVOC-SP COCO-SP MalNet-Tiny ogbg-molhiv ogbg-molpcba ogbg-ppa ogbg-code2

200 20 20 20 10 50 30 20 15 15 30 10 10 5

Table 6: Reduced per-dataset epoch budget used in the screening stage. The confirmation stage uses the full reference budget of Luo, Shi, and Wu (2025). • Baselines are quoted, not re-run. All baseline numbers are taken from the original publications. Comparability rests on the shared evaluation pipeline and the matched parameter budget rather than on re-training. • Negative and anomalous results are reported. For transparency, we report all completed experiments, including unsuccessful configurations and anomalous behaviors such as the BatchNorm sensitivity on ogbg-ppa and the infeasible SAN configurations on PascalVOC-SP (Figs 3-5).

9 9.1

Datasets

Selection Rationale

We evaluate on the same 14 inductive benchmarks used by Luo, Shi, and Wu (2025). We adopt an existing, widely used suite in full and remove the possibility of favorable dataset selection. This makes every number we report directly comparable to a published set of strong baselines evaluated under an identical pipeline. The suite spans: graph- and node-level tasks, classification and regression, molecular, image, synthetic, program, and biological graphs, and graph sizes from 23 to 1,410 nodes on average.

9.2

Descriptions

• GNN Benchmark (Dwivedi et al. 2023). ZINC contains molecular graphs (atom node features, bond edge features); the task is to regress the constrained solubility (logP). MNIST and CIFAR10 are image-classification datasets recast as 8-nearest-neighbor graphs of SLIC superpixels, retaining the original 10-class tasks. PATTERN and CLUSTER are synthetic Stochastic Block Model graphs for inductive node classification (pattern recognition and cluster-ID inference, respectively). • Long-Range Graph Benchmark (LRGB) (Dwivedi et al. 2022; Freitas and Dong 2021). Peptides-func

and Peptides-struct are atomic peptide graphs, with 10way multi-label graph classification and 11-target structural regression. PascalVOC-SP and COCO-SP are superpixel node-classification datasets derived from Pascal VOC and MS COCO. MalNet-Tiny is a subset of function-call graphs from Android APKs, with softwaretype classification from structure alone. • Open Graph Benchmark (OGB) (Hu et al. 2020). ogbg-molhiv and ogbg-molpcba are molecular property prediction datasets (binary HIV-inhibition classification and a 128-task bioassay panel). ogbg-ppa classifies protein-association networks into 37 taxonomic groups, and ogbg-code2 predicts the first five subtokens of a function’s name from its abstract syntax tree. Dataset

# graphs

Avg. # nodes

Avg. # edges

Task type

Metric

ZINC MNIST CIFAR10 PATTERN CLUSTER

12,000 70,000 60,000 14,000 12,000

23.2 70.6 117.6 118.9 117.2

24.9 564.5 941.1 3,039.3 2,150.9

Graph regr. Graph cls. Graph cls. Node cls. Node cls.

MAE Accuracy Accuracy Accuracy Accuracy

Peptides-func Peptides-struct PascalVOC-SP COCO-SP MalNet-Tiny

15,535 15,535 11,355 123,286 5,000

150.9 150.9 479.4 476.9 1,410.3

307.3 307.3 2,710.5 2,693.7 2,859.9

Graph cls. Graph regr. Node cls. Node cls. Graph cls.

Avg. Precision MAE F1 score F1 score Accuracy

ogbg-molhiv ogbg-molpcba ogbg-ppa ogbg-code2

41,127 437,929 158,100 452,741

25.5 26.0 243.4 125.2

27.5 28.1 2,266.1 124.2

Graph cls. Graph cls. Graph cls. Graph cls.

AUROC Avg. Precision Accuracy F1 score

Table 7: Statistics of the 14 inductive datasets used in this paper.

9.3

Licenses and Availability

Table 8 lists the license of every dataset and code resource used in this work. All datasets are publicly available and are obtained through their official benchmark loaders, and each remains subject to the license of its original provider. We do not redistribute any dataset as part of our code release. Our implementation builds upon the GNN+ codebase of Luo, Shi, and Wu (2025), released under the MIT license and based in turn on GraphGPS (Rampášek et al. 2022). The released code contains only our own implementation and requires users to download all datasets from their official sources.

Sheaf-Specific Hyperparameters

The message-passing block is the sheaf operator, parametrized along three axes: i) the diffusion mechanism ∈ {NSD, SAN, SANv2}; ii) the restriction-map family ∈ {Diag, O(d), Gen}, with O(d) maps produced by Householder reflections; and iii) the stalk dimension d ∈ {2, 3, 4}. The channels-per-stalk f are chosen so that the per-node representation size c = df does not exceed the reference width budget for each dataset. Following Bodnar et al. (2022), restriction maps are predicted by a small learner with a bounded tanh activation, and the diffusion nonlinearity is σ = elu. For the Gen parameterization, the block-diagonal inverse square root D−1/2 is computed with a deterministic Newton–Schulz

Resource

License

Source

GNN Benchmark suite

MIT

github.com/graphdeeplearning/benchmarking-gnns

LRGB release PascalVOC-SP COCO-SP Peptides-func/struct

CC BY 4.0 Custom (Flickr ToU) CC BY 4.0 Public-domain sources

doi.org/10.5281/zenodo.6975830 Pascal VOC 2011 MS COCO SATPdb

MalNet-Tiny

CC BY 4.0

mal-net.org

OGB datasets

MIT

ogb.stanford.edu

MIT

github.com/LUOyk1999/GNNPlus

GNN codebase

Table 8: Licenses of the datasets and code used in this work.

COCO-SP F1 score ↑

ogbg-ppa Accuracy ↑

Graph-Mamba GraphGPS GCN+ GIN+ GatedGCN+

0.3960 ± 0.0175 0.3884 ± 0.0050 0.2733 ± 0.0041 0.2483 ± 0.0046 0.3802 ± 0.0015

0.8077 ± 0.0041 0.8107 ± 0.0053 0.8258 ± 0.0055

NSD SAN SANv2

0.3004 ± 0.0027 0.2979 ± 0.0034 0.2971 ± 0.0029

0.6752 ± 0.0047 0.6744 ± 0.0374 0.6535 ± 0.0474

Metric

Table 9: Confirmation results on COCO-SP and ogbg-ppa. All results except the sheaf models are taken from Luo, Shi, and Wu (2025)

eter budget and are shaded separately. The two positionalencoding-free ablations are unaffected and therefore run normally. Reading the figures column-wise highlights the effect of the restriction-map parameterization: the O(d) columns are visibly darker than the Diag and Gen columns on nearly every dataset, and the ordering is stable across both diffusion mechanisms and all three stalk dimensions. Reading the figures row-wise highlights the contribution of the GNN+ components: the -dropout row is brighter than the full recipe almost everywhere, whereas -residual and all-removed are consistently among the worst-performing configurations. ogbg-ppa is the only dataset that systematically deviates from these trends, for the reasons discussed in the main paper.

12 iteration rather than random-noise regularization to improve numerical stability. SAN and SANv2 use multi-head attention (H = 8) combined by channel-wise concatenation. The two models differ only in the attention scoring function: SANv2 replaces the static GAT scoring of SAN with dynamic GATv2 scoring. The GNN+ components toggled in the ablation are edgefeature conditioning, batch normalization, dropout, the residual connection, the post-message FFN (hidden multiplier 2), and RWSE positional encoding.

Complete Screening Results

Figures 3, 4, 5 report the complete screening grid for all 14 datasets: for each dataset, the 8 GNN+ ablations are evaluated across the 18 architecture cells (diffusion mechanism × restriction-map parameterization × stalk dimension). Colors indicate within-dataset rank-normalized performance, with brighter colors corresponding to better-performing configurations regardless of the optimization direction, and each cell is annotated with the corresponding test-at-best-validation value. These figures report the 1,890 screening runs analyzed in the main paper. On the six datasets that use no positional encoding, the no_pe row is empty, since it coincides with the full recipe, and is marked =on rather than being run. The SAN cells at d = 2 on PascalVOC-SP are infeasible at the dataset’s param-

Additional Confirmation Results

The main paper reports 12 of the 14 datasets for conciseness. Table 9 completes the picture with COCO-SP and ogbg-ppa. Hyperparameter of confirmation sheaf models are reported in Table 10. These results confirm what we observe in the main paper, with sheaf models unable to close the gap to the strongest baselines. On COCO-SP, a node-classification task, the gap is narrower, with NSD 0.10 F1 points away from the bestperforming method and leading the three mechanisms. On the graph-level ogbg-ppa the gap widens to roughly 15 accuracy points. On both datasets, dynamic attention (SANv2) brings no visible benefit over static attention (SAN).

Dataset

Map

NSD GNN+

width

Map

SAN GNN+

width

Map

SANv2 GNN+ L

ZINC MNIST CIFAR10 PATTERN CLUSTER

diag gen gen gen gen

4 4 4 3 4

−drop −drop −drop −drop −norm

9 10 10 12 16

96 40 40 81 68

diag diag diag diag gen

4 4 4 4 4

−drop −drop −drop −drop −drop

9 10 10 12 16

60 40 40 48 36

diag diag diag diag gen

4 4 4 4 4

−drop −drop −drop −drop −drop

9 10 10 12 16

40 32 36 32 24

Peptides-func Peptides-struct PascalVOC-SP COCO-SP MalNet-Tiny

gen gen gen gen gen

3 4 4 4 4

−drop −drop −drop −drop −drop

5 4 12 20 6

120 132 84 60 120

diag gen diag diag diag

4 4 4 4 4

−drop −norm −drop −drop −drop

5 4 12 20 6

88 100 52 36 88

diag gen diag diag diag

4 4 4 4 4

−drop −norm −drop −drop −drop

5 4 12 20 6

64 80 36 20 64

ogbg-molhiv ogbg-molpcba ogbg-ppa ogbg-code2

gen gen gen gen

3 4 4 4

−res −drop −norm −norm

3 10 4 5

255 256 512 512

gen gen gen gen

2 4 4 4

−PE −drop −norm −norm

3 10 4 5

256 256 512 512

gen gen gen gen

2 4 4 4

−PE −drop −norm −norm

3 10 4 5

256 256 512 512

width

Table 10: Configurations promoted from screening to confirmation, selected per dataset and per diffusion mechanism on validation performance. “Map” is the restriction-map parameterization, d the stalk dimension, “GNN+ ” is the architectural component removal that ranked best (−drop = dropout removed, −norm = BatchNorm removed, −res = residual removed, −PE = positional encoding removed), L the number of message-passing layers, and “width” the hidden dimension. SANv2 inherits SAN’s configuration rather than being screened independently.

Screening on GNN Benchmark ZINC

all_on no_edge no_norm no_dropout no_residual no_ffn no_pe all_off

(MAE,

0.223 0.225 0.166 0.141 0.806 0.266 0.364 1.547 MNIST

all_on no_edge no_norm no_dropout no_residual no_ffn no_pe all_off

0.192 0.230 0.133 0.112 0.448 0.221 0.325 1.547

0.944 0.935 0.938 0.957 0.687 0.866

0.962 0.944 0.958 0.972 0.931 0.922

0.222 0.432 0.162 0.118 0.648 0.233 0.391 1.376

0.209 0.371 0.152 0.126 0.791 0.245 0.357 1.235

0.210 0.225 0.154 0.132 0.463 0.219 0.366 1.547

0.178 0.215 0.141 0.116 0.389 0.184 0.301 1.547

0.177 0.211 0.134 0.106 0.241 0.204 0.267 1.547

0.244 0.234 0.180 0.135 0.285 0.294 0.439 0.482

0.177 0.220 0.162 0.113 0.200 0.223 0.411 0.437

0.191 0.213 0.152 0.101 0.199 0.263 0.401 0.437

1.057 1.435 1.109 1.125 1.132 1.029 0.871 1.383

0.223 1.309 0.168 0.125 1.077 0.423 0.489 1.213

0.226 0.712 0.181 0.147 0.706 0.416 0.505 1.218

0.195 0.270 0.164 0.129 0.238 0.209 0.403 0.442

0.153 0.210 0.151 0.108 0.239 0.171 0.379 0.397

0.166 0.213 0.131 0.105 0.221 0.167 0.377 0.361

0.924 0.713 0.936 0.958 0.615 0.868

0.940 0.892 0.949 0.956 0.644 0.899

0.952 0.939 0.949 0.963 0.833 0.893

0.969 0.960 0.969 0.976 0.947 0.944

0.974 0.965 0.974 0.979 0.958 0.961

0.936 0.925 0.925 0.966 0.902 0.897

0.956 0.954 0.956 0.974 0.947 0.919

0.966 0.955 0.959 0.975 0.958 0.937

0.530 0.122 0.114 0.954 0.112 0.596

0.929 0.785 0.954 0.956 0.114 0.889

0.933 0.893 0.949 0.962 0.297 0.860

0.939 0.934 0.942 0.968 0.892 0.935

0.970 0.955 0.964 0.977 0.937 0.953

0.974 0.965 0.301 0.976 0.945 0.963

↑ better)

0.964 0.957 0.971 0.976 0.920 0.940

0.781 0.366 0.457 0.939 0.296 0.693

=on

0.114

0.274

0.221

0.316

0.114

0.879

0.926

0.943

0.183

0.186

0.337

0.914

0.943

0.957

0.438 0.409 0.487 0.580 0.248 0.429

0.502 0.464 0.540 0.600 0.274 0.420

0.531 0.515 0.523 0.604 0.442 0.465

0.564 0.561 0.561 0.636 0.506 0.526

0.602 0.595 0.619 0.667 0.541 0.543

0.510 0.502 0.498 0.605 0.437 0.453

0.571 0.561 0.526 0.634 0.525 0.500

0.589 0.585 0.588 0.655 0.564 0.545

0.187 0.163 0.320 0.473 0.135 0.368

0.514 0.278 0.513 0.596 0.146 0.375

0.467 0.347 0.534 0.610 0.155 0.422

0.540 0.527 0.515 0.604 0.431 0.496

0.596 0.569 0.575 0.637 0.503 0.547

0.606 0.531 0.621 0.090 0.391 0.554

(Accuracy,

0.480 0.465 0.506 0.590 0.209 0.440

0.575 0.541 0.580 0.648 0.461 0.467

↑ better)

0.574 0.568 0.598 0.661 0.492 0.506

0.311 0.154 0.415 0.502 0.136 0.353

=on

0.100

0.178

0.290

0.328

0.100

0.465

0.510

0.529

0.167

0.207

0.189

0.457

0.530

0.559

0.866 0.864 0.866 0.868 0.866 0.866 0.851 0.849

0.867 0.866 0.867 0.868 0.867 0.867 0.852 0.843

0.865 0.865 0.868 0.868 0.867 0.865 0.855 0.501

0.868 0.867 0.867 0.868 0.867 0.867 0.854 0.500

0.868 0.867 0.868 0.868 0.866 0.867 0.856 0.500

0.867 0.867 0.867 0.868 0.866 0.865 0.512 0.651

0.867 0.866 0.867 0.869 0.867 0.867 0.508 0.774

0.867 0.867 0.867 0.868 0.867 0.867 0.500 0.778

0.867 0.696 0.866 0.867 0.860 0.859 0.509 0.551

0.866 0.865 0.867 0.867 0.865 0.865 0.540 0.690

0.866 0.866 0.867 0.868 0.866 0.866 0.516 0.570

0.867 0.866 0.867 0.868 0.867 0.866 0.538 0.731

0.867 0.867 0.867 0.868 0.868 0.867 0.636 0.781

0.866 0.866 0.867 0.869 0.867 0.868 0.518 0.810

0.867 0.867 0.867 0.868 0.866 0.867 0.850 0.501

(Accuracy,

0.867 0.867 0.867 0.868 0.867 0.868 0.851 0.500

↑ better)

0.868 0.867 0.867 0.868 0.868 0.867 0.854 0.500

0.379 0.489 0.455 0.677 0.440 0.625 0.452 0.167

0.596 0.632 0.612 0.735 0.561 0.607 0.580 0.167

0.725 0.699 0.661 0.745 0.582 0.656 0.711 0.167

0.227 0.166 0.231 0.404 0.172 0.356 0.495 0.368

0.551 0.347 0.292 0.684 0.321 0.494 0.585 0.463

0.472 0.461 0.458 0.716 0.531 0.579 0.654 0.490

0.277 0.345 0.354 0.607 0.259 0.662 0.613 0.167

0.733 0.486 0.620 0.743 0.598 0.707 0.602 0.167

0.398 0.609 0.754 0.751 0.612 0.714 0.736 0.167

0.167 0.235 0.182 0.515 0.316 0.454 0.464 0.466

0.461 0.347 0.409 0.691 0.368 0.597 0.594 0.493

0.582 0.494 0.414 0.713 0.512 0.601 0.644 0.507

0.169 0.169 0.168 0.169 0.168 0.285 0.336 0.355

0.169 0.174 0.166 0.337 0.173 0.372 0.451 0.460

0.607 0.330 0.184 0.665 0.283 0.394 0.498 0.468

0.355 0.236 0.168 0.513 0.254 0.463 0.467 0.484

0.476 0.409 0.168 0.708 0.381 0.578 0.606 0.505

0.737 0.345 0.167 0.742 0.238 0.728 0.738 0.545

nsd_diag d4

nsd_orth d2

nsd_orth d3

nsd_orth d4

nsd_gen d2

nsd_gen d3

nsd_gen d4

san_diag d2

san_diag d3

san_diag d4

san_orth d2

san_orth d3

san_orth d4

san_gen d2

san_gen d3

san_gen d4

↑ better)

nsd_diag d3

(Accuracy,

0.866 0.842 0.866 0.867 0.864 0.863 0.843 0.805

nsd_diag d2

CLUSTER

all_on no_edge no_norm no_dropout no_residual no_ffn no_pe all_off

1.031 1.394 0.194 0.142 1.083 0.256 0.401 1.406

0.114

PATTERN

all_on no_edge no_norm no_dropout no_residual no_ffn no_pe all_off

0.186 0.220 0.125 0.109 0.384 0.222 0.322 1.547

(Accuracy,

CIFAR10

all_on no_edge no_norm no_dropout no_residual no_ffn no_pe all_off

↓ better)

better

worse better

worse

Diffusion x Map Parameterization x Stalk dimension d Figure 3: Complete screening grid, GNN Benchmark. Rows are GNN+ ablations, columns are the 18 architecture cells, color is within-dataset goodness (brighter is better, direction-aware), and the printed value is test-at-best-val.

Screening on LRGB peptides-func

all_on no_edge no_norm no_dropout no_residual no_ffn no_pe all_off

0.496 0.478 0.505 0.580 0.480 0.499 0.484 0.165

0.513 0.516 0.558 0.572 0.498 0.515 0.516 0.165

peptides-struct

all_on no_edge no_norm no_dropout no_residual no_ffn no_pe all_off

0.293 0.293 0.298 0.279 0.290 0.336 0.291 0.642

0.282 0.287 0.287 0.268 0.293 0.314 0.288 0.642

PascalVOC-SP

all_on no_edge no_norm no_dropout no_residual no_ffn no_pe all_off

0.164 0.130 0.195 0.227 0.111 0.154 0.161 0.039

0.209 0.182 0.257 0.270 0.121 0.209 0.218 0.039

COCO-SP

all_on no_edge no_norm no_dropout no_residual no_ffn no_pe all_off

0.000 0.058 0.104 0.131 0.019 0.067

0.435 0.394 0.476 0.504 0.260 0.442 0.388 0.227

(MAE,

↓ better)

0.280 0.280 0.275 0.272 0.284 0.312 0.290 0.642

0.312 0.342 0.293 0.287 0.314 0.313 0.308 0.299

(F1 score,

0.246 0.224 0.269 0.314 0.152 0.220 0.274 0.039

0.106 0.061 0.135 0.002 0.037 0.070

n/f: PE width > budgeted model width

↑ better)

0.535 0.539 0.583 0.592 0.520 0.539 0.509 0.165

(F1 score,

0.459 0.463 0.534 0.497 0.436 0.498 0.467 0.435

0.472 0.469 0.574 0.515 0.460 0.517 0.473 0.421

0.500 0.483 0.522 0.582 0.484 0.486 0.485 0.165

0.524 0.516 0.557 0.612 0.522 0.522 0.505 0.165

0.532 0.537 0.578 0.598 0.536 0.556 0.512 0.165

0.473 0.466 0.491 0.548 0.479 0.468 0.453 0.456

0.494 0.493 0.532 0.578 0.498 0.507 0.497 0.484

0.504 0.510 0.545 0.592 0.506 0.514 0.507 0.528

0.428 0.347 0.458 0.473 0.276 0.415 0.412 0.396

0.472 0.433 0.506 0.535 0.445 0.485 0.454 0.210

0.463 0.437 0.526 0.512 0.432 0.500 0.460 0.462

0.479 0.473 0.490 0.555 0.464 0.477 0.473 0.490

0.508 0.507 0.552 0.578 0.504 0.516 0.501 0.501

0.526 0.528 0.571 0.586 0.498 0.536 0.523 0.549

0.297 0.306 0.284 0.284 0.319 0.314 0.294 0.294

0.294 0.299 0.280 0.283 0.306 0.304 0.297 0.297

0.289 0.290 0.290 0.275 0.284 0.333 0.294 0.642

0.287 0.286 0.276 0.269 0.294 0.301 0.284 0.643

0.280 0.279 0.269 0.265 0.281 0.301 0.287 0.643

0.327 0.301 0.314 0.280 0.310 0.304 0.307 0.302

0.295 0.294 0.287 0.272 0.293 0.318 0.300 0.295

0.294 0.288 0.271 0.271 0.289 0.306 0.295 0.280

0.321 0.326 0.308 0.285 0.348 0.319 0.308 0.296

0.299 0.304 0.277 0.281 0.307 0.308 0.299 0.298

0.295 0.311 0.273 0.279 0.298 0.304 0.300 0.273

0.301 0.299 0.284 0.276 0.298 0.298 0.292 0.289

0.283 0.291 0.276 0.269 0.292 0.291 0.290 0.263

0.278 0.280 0.265 0.269 0.284 0.282 0.280 0.261

0.063 0.072 0.214 0.101 0.061 0.064 0.120 0.052

0.127 0.106 0.234 0.195 0.064 0.159 0.160 0.056

0.182 0.144 0.207 0.275 0.138 0.166 0.218 0.039

0.209 0.211 0.262 0.291 0.191 0.217 0.264 0.039

0.281 0.239 0.286 0.323 0.244 0.244 0.263 0.039

0.128 0.083

0.131 0.118 0.161 0.202 0.110 0.137 0.142 0.109

0.183 0.149 0.204 0.257 0.143 0.157 0.167 0.123

0.041 0.044

0.065 0.048 0.138 0.065 0.046 0.079 0.042 0.042

0.066 0.068 0.192 0.089 0.066 0.087 0.089 0.042

0.165 0.101

0.175 0.140 0.186 0.230 0.073 0.154 0.197 0.109

0.189 0.002 0.216 0.243 0.090 0.180 0.216 0.095

0.014 0.010 0.130 0.016 0.013 0.063

0.017 0.012 0.166 0.038 0.014 0.075

0.089 0.088 0.112 0.137 0.047 0.069

0.137 0.093 0.148 0.167 0.059 0.103

0.160 0.138 0.184 0.211 0.099 0.125

0.042 0.013 0.068 0.001 0.014 0.014

0.062 0.035 0.001 0.022 0.015 0.015

0.001 0.063 0.117 0.126 0.015 0.074

0.011 0.001 0.011 0.011 0.010 0.012

0.002 0.010 0.081 0.015 0.010 0.047

0.013 0.010 0.011 0.020 0.011 0.050

0.016 0.031 0.001 0.004 0.013 0.066

0.052 0.010 0.012 0.012 0.014 0.073

0.055 0.027 0.012 0.117 0.018 0.078

↑ better)

0.057 0.040 0.145 0.051 0.046 0.059 0.048 0.044

n/f n/f n/f n/f n/f n/f

↑ better)

0.136 0.117 0.163 0.193 0.038 0.083

0.012 0.010 0.091 0.013 0.011 0.016

=on

0.010

0.000

0.010

0.011

0.010

0.000

0.048

0.000

0.011

0.013

0.024

0.066

0.776 0.630 0.818 0.895 0.723 0.731

0.783 0.764 0.783 0.913 0.774 0.780

0.864 0.804 0.790 0.925 0.820 0.810

0.880 0.834 0.865 0.923 0.866 0.829

0.731 0.686 0.676 0.920 0.772 0.701

0.799 0.794 0.766 0.921 0.777 0.783

0.852 0.790 0.791 0.918 0.821 0.735

0.694 0.446 0.786 0.787 0.661 0.641

0.744 0.639 0.801 0.889 0.605 0.704

0.782 0.688 0.809 0.892 0.726 0.739

0.758 0.634 0.713 0.914 0.750 0.689

0.801 0.717 0.771 0.935 0.687 0.769

0.844 0.727 0.853 0.928 0.212 0.807

MalNet-Tiny

=on

0.200

0.663

0.729

0.754

0.200

0.744

0.749

0.883

0.610

0.694

0.767

0.777

0.807

0.863

nsd_orth d4

nsd_gen d2

nsd_gen d3

nsd_gen d4

san_diag d2

san_diag d3

san_diag d4

san_orth d2

san_orth d3

san_orth d4

san_gen d2

san_gen d3

san_gen d4

0.691 0.616 0.782 0.854 0.636 0.662

nsd_orth d3

0.642 0.576 0.744 0.773 0.555 0.556

nsd_orth d2

0.861 0.830 0.795 0.918 0.811 0.786

nsd_diag d4

0.850 0.778 0.819 0.922 0.799 0.770

↑ better)

nsd_diag d3

0.755 0.658 0.760 0.923 0.770 0.773

(Accuracy,

nsd_diag d2

all_on no_edge no_norm no_dropout no_residual no_ffn no_pe all_off

(Avg. Precision,

better

worse better

worse

Diffusion x Map Parameterization x Stalk dimension d Figure 4: Complete screening grid, LRGB. Rows are GNN+ ablations, columns are the 18 architecture cells, color is withindataset goodness (brighter is better, direction-aware), and the printed value is test-at-best-val. The shaded SAN d=2 cells on PascalVOC-SP are infeasible at the parameter budget, as also reported in the main paper.

Screening on OGB 0.692 0.758 0.749 0.691 0.752 0.741 0.706 0.581

0.770 0.709 0.766 0.765 0.755 0.772 0.718 0.702

0.780 0.774 0.758 0.783 0.753 0.771 0.788 0.719

0.766 0.763 0.764 0.758 0.762 0.775 0.748 0.502

0.769 0.732 0.766 0.737 0.787 0.762 0.761 0.502

0.790 0.765 0.771 0.797 0.763 0.759 0.768 0.502

0.719 0.740 0.765 0.751 0.746 0.746 0.739 0.745

0.774 0.763 0.741 0.784 0.744 0.767 0.744 0.731

0.768 0.749 0.777 0.765 0.753 0.765 0.772 0.734

0.734 0.757 0.748 0.737 0.713 0.737 0.737 0.686

0.772 0.757 0.753 0.765 0.720 0.780 0.746 0.679

0.750 0.798 0.783 0.771 0.774 0.773 0.771 0.709

0.748 0.756 0.743 0.742 0.737 0.769 0.751 0.736

0.739 0.760 0.746 0.761 0.748 0.745 0.717 0.753

0.755 0.775 0.781 0.784 0.745 0.734 0.745 0.684

0.100 0.044 0.164 0.152 0.051 0.085 0.088 0.028

0.126 0.080 0.173 0.147 0.050 0.117 0.105 0.029

0.145 0.127 0.162 0.171 0.123 0.140 0.119 0.025

0.170 0.169 0.183 0.196 0.144 0.145 0.144 0.025

0.189 0.180 0.201 0.209 0.169 0.152 0.159 0.025

0.146 0.146 0.152 0.179 0.148 0.134 0.127 0.095

0.155 0.159 0.174 0.202 0.156 0.142 0.143 0.102

0.171 0.174 0.181 0.208 0.165 0.152 0.147 0.111

0.047 0.033 0.128 0.043 0.038 0.055 0.045 0.031

0.049 0.041 0.167 0.056 0.048 0.087 0.092 0.026

0.117 0.091 0.179 0.152 0.047 0.113 0.106 0.033

0.168 0.145 0.161 0.173 0.141 0.136 0.135 0.102

0.191 0.162 0.171 0.202 0.133 0.148 0.155 0.116

0.190 0.178 0.195 0.214 0.146 0.155 0.167 0.136

0.095 0.095 0.464 0.095 0.095 0.124

0.095 0.095 0.547 0.095 0.095 0.028

0.091 0.095 0.532 0.095 0.027 0.059

0.095 0.095 0.563 0.123 0.083 0.106

0.020 0.095 0.613 0.024 0.017 0.095

0.098 0.095 0.442 0.097 0.124 0.095

0.068 0.095 0.534 0.095 0.120 0.123

0.123 0.095 0.570 0.110 0.088 0.124

0.120 0.095 0.095 0.095 0.158 0.097

0.095 0.095 0.095 0.095 0.379 0.094

0.093 0.095 0.095 0.095 0.155 0.091

0.124 0.095 0.545 0.083 0.124 0.096

0.124 0.095 0.610 0.094 0.068 0.097

0.057 0.095 0.622 0.095 0.023 0.095

molpcba (Avg. Precision, ↑ better) 0.139 0.137 0.154 0.173 0.113 0.127 0.115 0.025

0.150 0.141 0.173 0.186 0.136 0.137 0.133 0.025

0.173 0.149 0.191 0.205 0.154 0.141 0.143 0.025

0.041 0.032 0.139 0.044 0.040 0.081 0.047 0.026

ppa (Accuracy, ↑ better) 0.124 0.095 0.477 0.091 0.035 0.084

0.074 0.095 0.513 0.096 0.080 0.021

0.045 0.095 0.500 0.095 0.095 0.041

0.095 0.095 0.354 0.095 0.095 0.095

=on

0.134

0.158

0.169

0.133

0.145

0.151

0.145

0.164

0.162

0.095

0.117 0.087 0.129 0.124 0.105 0.121

0.122 0.084 0.148 0.131 0.108 0.128

0.120 0.115 0.123 0.122 0.111 0.114

0.128 0.130 0.140 0.133 0.129 0.132

0.140 0.129 0.153 0.146 0.134 0.144

0.121 0.119 0.131 0.125 0.119 0.120

0.133 0.135 0.141 0.138 0.130 0.130

0.080 0.143 0.000 0.144 0.132 0.137

0.120 0.064 0.130 0.127 0.104 0.113

0.131 0.088 0.137 0.134 0.113 0.130

0.133 0.104 0.000 0.093 0.117 0.133

0.132 0.127 0.134 0.130 0.122 0.130

0.146 0.142 0.146 0.142 0.135 0.141

0.153 0.149 0.157 0.148 0.144 0.148

code2 (F1 score, ↑ better)

=on

0.068

0.102

0.100

0.061

0.075

0.090

0.104

0.123

0.096

0.120

0.116

0.061

0.084

0.100

0.121

0.131

0.143

nsd_gen d3

nsd_gen d4

san_diag d2

san_diag d3

san_diag d4

san_orth d2

san_orth d3

san_orth d4

san_gen d2

san_gen d3

san_gen d4

0.110 0.062 0.117 0.116 0.098 0.104

nsd_gen d2

0.132 0.134 0.139 0.140 0.119 0.126

nsd_orth d4

0.121 0.125 0.134 0.130 0.116 0.122

nsd_orth d3

0.118 0.110 0.124 0.121 0.103 0.108

nsd_orth d2

all_on no_edge no_norm no_dropout no_residual no_ffn no_pe all_off

0.762 0.782 0.766 0.761 0.762 0.760 0.766 0.502

nsd_diag d4

all_on no_edge no_norm no_dropout no_residual no_ffn no_pe all_off

0.729 0.757 0.780 0.784 0.752 0.761 0.739 0.502

nsd_diag d3

all_on no_edge no_norm no_dropout no_residual no_ffn no_pe all_off

0.779 0.760 0.705 0.781 0.760 0.761 0.762 0.502

nsd_diag d2

all_on no_edge no_norm no_dropout no_residual no_ffn no_pe all_off

ogbg-molhiv (AUROC, ↑ better)

better

worse better

worse

Diffusion x Map Parameterization x Stalk dimension d Figure 5: Complete screening grid, OGB. Rows are GNN+ ablations, columns are the 18 architecture cells, color is withindataset goodness (brighter is better, direction-aware), and the printed value is test-at-best-val. ogbg-ppa is the one dataset that systematically deviates from the trends seen elsewhere. The cause is the evaluation-time behavior of BatchNorm, discussed in the main paper.

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