Heterogeneous Sheaf Neural Networks

Luke Braithwaite; Alessio Borgi; Gabriele Onorato; Kristjan Tarantelli; Francesco Restuccia; Fabrizio Silvestri; Pietro Liò

cs.LGSep 12, 2024

Heterogeneous Sheaf Neural Networks

Authors: Luke Braithwaite, Alessio Borgi, Gabriele Onorato, Kristjan Tarantelli, Francesco Restuccia, Fabrizio Silvestri, Pietro Liò

Organizations: Department of Computer Science and Technology, University of Cambridge · Department of Computer, Control and Management Engineering, Sapienza University of Rome · Department of Electrical and Computer Engineering, Northeastern University

Abstract

Heterogeneous graphs, whose nodes and edges can belong to different types and feature spaces, arise in many real-world domains, including biology, recommendation, social networks, and computer systems. Existing heterogeneous graph neural networks typically handle this heterogeneity at the architectural level through relation-specific modules, meta-path machinery or type-aware attention, which often leads to increasingly specialised parameter-heavy designs. In this work, we propose HetSheaf, a framework for learning heterogeneous graphs through cellular sheaves. Instead of encoding heterogeneity solely in the architecture, HetSheaf represents it directly in the underlying data structure by assigning type-aware local feature spaces and learning restriction maps conditioned on node features, node types, and edge types. To support graph-level prediction, we further introduce SheafPool, a universal stalk-space readout that aggregates node representations while being invariant to local changes of basis, thereby making graph classification with sheaf networks well-defined and achieving an F1 Score up to 42 percentage points higher than mean pooling. Across a diverse suite of benchmarks (node classification, link prediction and graph classification). HetSheaf consistently achieves up to 2 percentage points higher performance (up to 94.97% Macro F1 Score on node classification and up to 99.62% on link prediction) on the Heterogeneous Graph Benchmark (HGB) framework against homogeneous (GCN, GAT, GIN, GraphSAGE), heterogeneous (R-GCN, HAT, HGT) and type-agnostic sheaf baselines, while reducing the number of parameters by up to 10 $\times$ .

Figures & tables

Panel of Figure 1 — Figure 1: Heterogeneous Sheaf Predictors. For an edge $e=(u,v)$ , $\phi(u)$ and $\phi(v)$ denote node types, $\psi(e)$ denotes the edge type, $x_{u},x_{v}$ are the raw node features, projected in the shared channel via ${\mathbf{z}}_{*}=\mathcal{P}_{\phi(*)}({\mathbf{x}}_{*})$ . $\|$ denotes concatenation, and $\mathrm{MLP}_{\psi(e)}$ denotes a separate MLP for each edge type.

Panel of Figure 2 — Figure 2: The SheafPool Architecture. Our readout processes node stalk embeddings in five steps. (1) Per-node Whitening removes GL(d) scaling and shearing. (2) Anchor Alignment resolves residual O(d) rotations. (3) Attention Weights are computed via stalk energies. (4) Stalk Pooling aggregates into a Late Receive-only Token without altering message passing. (5) Feature Extraction collapses the stalk dimension, yielding a permutation- and gauge-invariant graph embedding.

	ACM		DBLP		IMDB
	Macro F1	Micro F1	Macro F1	Micro F1	Macro F1	Micro F1
Homogeneous
GCN	92.25 $\pm$ 0.82	91.25 $\pm$ 0.84	90.87 $\pm$ 1.77	91.68 $\pm$ 1.83	58.26 $\pm$ 0.36	60.89 $\pm$ 0.26
GAT	92.16 $\pm$ 1.55	91.95 $\pm$ 0.87	92.33 $\pm$ 1.13	92.81 $\pm$ 0.99	60.23 $\pm$ 0.67	64.06 $\pm$ 0.80
GIN	88.70 $\pm$ 2.20	88.78 $\pm$ 1.98	94.83 $\pm$ 0.43	95.24 $\pm$ 0.43	60.30 $\pm$ 0.75	63.96 $\pm$ 0.52
SAGE	92.18 $\pm$ 1.61	92.07 $\pm$ 1.51	90.23 $\pm$ 1.31	90.69 $\pm$ 1.26	60.60 $\pm$ 1.75	63.73 $\pm$ 0.63

Table 2: Heterogeneous node classification. Macro/Micro F1, mean

\pm

std over five seeds. Top three by mean: first , second , third .

	LastFM		MovieLens		PubMed		Amazon		YouTube
	AUPR	AUROC	AUPR	AUROC	AUPR	AUROC	AUPR	AUROC	AUPR	AUROC
Homogeneous
GCN	97.05 $\pm$ 0.25	96.95 $\pm$ 0.17	99.67 $\pm$ 0.04	99.62 $\pm$ 0.04	96.03 $\pm$ 0.11	92.88 $\pm$ 0.28	92.44 $\pm$ 0.28	89.61 $\pm$ 0.27	89.44 $\pm$ 0.18	83.25 $\pm$ 0.17
GAT	63.03 $\pm$ 0.22	51.11 $\pm$ 0.50	97.50 $\pm$ 0.22	97.83 $\pm$ 0.18	96.06 $\pm$ 0.14	93.23 $\pm$ 0.27	88.59 $\pm$ 0.19	86.11 $\pm$ 0.25	66.79 $\pm$ 0.39	58.31 $\pm$ 0.28
GIN	97.98 $\pm$ 0.19	97.73 $\pm$ 0.30	99.58 $\pm$ 0.14	99.54 $\pm$ 0.04	96.11 $\pm$ 0.07	92.84 $\pm$ 0.12	92.04 $\pm$ 0.28	89.12 $\pm$ 0.29	88.62 $\pm$ 0.58	82.92 $\pm$ 0.49
SAGE	95.07 $\pm$ 1.76	92.70 $\pm$ 2.89	99.21 $\pm$ 0.09	99.12 $\pm$ 0.08	95.53 $\pm$ 0.52	88.66 $\pm$ 0.97	90.82 $\pm$ 0.39	87.82 $\pm$ 0.32	84.52 $\pm$ 0.09	80.22 $\pm$ 0.32

Table 3: Heterogeneous link prediction. Binary AUPR/AUROC; mean

\pm

std over five seeds. Top three by mean: first , second , third .

	MUTAG		PROTEINS		ENZYMES		NCI1
	Macro F1	Micro F1	Macro F1	Micro F1	Macro F1	Micro F1	Macro F1	Micro F1
Homogeneous
GCN	89.56 $\pm$ 1.04	90.00 $\pm$ 1.35	76.82 $\pm$ 2.45	80.34 $\pm$ 2.47	12.71 $\pm$ 1.92	28.30 $\pm$ 2.71	64.13 $\pm$ 2.12	64.94 $\pm$ 2.31
GAT	89.56 $\pm$ 1.02	90.00 $\pm$ 1.32	75.76 $\pm$ 1.51	78.54 $\pm$ 1.75	14.12 $\pm$ 1.29	16.60 $\pm$ 2.27	64.19 $\pm$ 2.44	64.90 $\pm$ 2.07
GIN	94.88 $\pm$ 0.72	95.00 $\pm$ 0.78	70.64 $\pm$ 1.38	72.32 $\pm$ 1.32	20.10 $\pm$ 2.46	28.33 $\pm$ 2.43	64.17 $\pm$ 2.18	64.91 $\pm$ 2.86
SAGE	89.56 $\pm$ 1.04	90.00 $\pm$ 1.35	74.59 $\pm$ 1.56	78.55 $\pm$ 1.35	8.55 $\pm$ 1.80	18.33 $\pm$ 2.26	63.64 $\pm$ 0.85	65.41 $\pm$ 1.09

Table 4: Heterogeneous graph classification. Macro/Micro F1; mean

\pm

std over five seeds. Top three by mean: first , second , third .

Appendix figures & tables12 assets

Supplementary material from the paper’s appendix.

Appendix

Hyperparameter	Search space
Sheaf type	$\left\{\text{general},O(d),\text{diagonal},\text{SheafAN}\right\}$
Sheaf learner	$\{\text{NSD},\text{TE},\text{EE},\text{NE},\text{types},\text{NT},\text{ET}\}$
Attention Learner †	$\{\text{Yes},\text{No}\}$
$d$	$\left\{$ 2345 $\right\}$
# Layers	$\{$ 2345678 $\}$
input_dropout	$\left[$ 00.9 $\right]$

Appendix

Table 5: Hyperparameter search space.

Node classification	# Nodes	# Node types	# Edges	# Edge types	Target	# Classes
DBLP	26,128	4	239,566	6	author	4
IMDB	21,420	4	86,642	6	movie	5
ACM	10,942	4	547,806	7	paper	3
Link prediction					Target
LastFM	20,612	3	141,521	3	user-artist
MovieLens	10,352	2	201,672	2	user-movie

Appendix

Table 6: Statistics of benchmark datasets.

Biomedical	# Nodes	# Node types	# Edges	# Edge types	Target
CTD-DDA	12,765	2	43,492	1	drug-disease
DrugBank-DDI	2,191	1	242,027	1	drug-drug
NDFRT-DDA	13,545	2	4,709	1	drug-disease
String-PPI	15,131	1	359,776	1	protein-protein

Appendix

Table 7: Statistics of biomedical link prediction datasets.

Networking	# Nodes	# Node types	# Edges	# Edge types	Task	Target
AS-733	6,474	1	13,895	3	LP	customer-provider
RouteNet Traffic Models	$\sim$ 674	2	$\sim$ 1,200	2	NC	link congestion

Appendix

Table 9: Statistics of networking datasets.

ACM

DBLP

IMDB

Family

d\!=\!2

d\!=\!3

d\!=\!4

d\!=\!5

d\!=\!2

d\!=\!3

d\!=\!4

d\!=\!5

d\!=\!2

d\!=\!3

d\!=\!4

d\!=\!5

Diag

93.7/93.6

93.0/92.9

92.6/92.5

93.0/92.9

94.0/94.5

94.1/94.5

93.5/94.0

62.4/65.4

63.6/66.6

62.8/65.9

63.2/65.8

Bundle

92.6/92.5

91.2/91.1

91.0/90.8

92.1/92.1

93.8/94.3

93.7/94.2

93.2/93.7

93.4/93.9

64.4/ 67.2

64.5 /66.6

63.9/66.5

64.2/66.4

General

91.8/91.7

91.7/91.7

91.9/91.8

92.1/92.0

93.8/94.3

94.0/94.4

94.7/95.1

94.0/94.4

63.5/65.6

62.5/65.5

62.4/65.4

63.0/65.8

Diag-P

93.3/93.2

93.1/93.0

94.2/94.1

93.5/93.4

94.9/95.3

94.7/95.1

94.9/95.3

94.6/95.0

61.5/64.7

61.6/64.6

61.1/64.1

60.2/63.6

Bundle-P

90.3/90.2

92.1/92.0

89.3/89.1

90.3/90.2

94.7/95.1

93.3/93.8

94.6/95.0

93.6/94.0

61.5/64.3

61.8/64.6

61.0/64.3

61.9/65.2

Appendix

Table 17: Node classification: Macro-F1 / Micro-F1 (%) by restriction map family and stalk dimension

d

. Cyan = best in family, bold = global best per dataset.

MUTAG

PROTEINS

ENZYMES

NCI1

Family

d\!=\!2

d\!=\!3

d\!=\!4

d\!=\!5

d\!=\!2

d\!=\!3

d\!=\!4

d\!=\!5

d\!=\!2

d\!=\!3

d\!=\!4

d\!=\!5

d\!=\!2

d\!=\!3

d\!=\!4

d\!=\!5

Diag

80.0/80.0

83.3/83.3

76.9/76.9

74.6/75.9

67.3/67.9

73.6/75.9

61.5/67.9

53.0/53.3

46.1/46.7

34.2/35.0

39.1/40.0

72.1/72.1

68.5/68.5

73.7/73.7

61.8/61.8

Bundle

78.0/80.0

84.7/85.0

94.9/95.0

84.7/85.0

74.8/76.8

75.8/77.7

78.8/80.4

54.4/54.5

50.8/50.0

51.2/53.3

62.3 / 63.3

51.0/51.7

62.6/62.6

57.6/57.6

68.0/68.0

60.4/60.4

General

89.0/90.0

94.9/95.0

89.9/90.0

74.9/75.0

71.3/72.3

64.2/74.1

78.7/81.2

64.2/67.0

49.9/51.7

47.7/50.0

50.7/51.7

46.9/46.7

63.2/63.2

64.4/64.4

67.5/67.5

62.1/62.1

Diag-P

95.7/95.7

96.0/96.0

100.0 / 100.0

51.8/67.9

78.9/ 82.1

71.6/74.1

63.2/74.1

22.7/28.3

25.6/30.0

14.0/21.7

14.9/23.3

60.2/60.2

62.1/62.1

61.4/61.4

Bundle-P

100.0 / 100.0

75.6/77.7

78.8/80.4

78.4/80.4

79.8/ 82.1

52.6/53.3

46.5/48.3

51.9/51.7

56.7/56.7

69.2/69.2

54.4/54.4

74.7/74.7

76.3/76.3

Appendix

Table 18: Graph classification: Macro-F1 / Micro-F1 (%) by restriction map family and stalk dimension

d

. Cyan = best in family, bold = global best per dataset.

LastFM

MovieLens

Amazon

YouTube

Family

d\!=\!2

d\!=\!3

d\!=\!4

d\!=\!5

d\!=\!2

d\!=\!3

d\!=\!4

d\!=\!5

d\!=\!2

d\!=\!3

d\!=\!4

d\!=\!5

d\!=\!2

d\!=\!3

d\!=\!4

d\!=\!5

Diag

98.4/98.2

98.5/98.2

98.4/98.2

98.5/98.2

99.7 /99.6

99.7/99.6

99.7 /99.6

93.8/91.5

93.8/91.6

93.8/91.5

90.5/85.4

90.7/86.0

90.6/85.7

90.6/85.9

Bundle

98.5/98.2

98.4/98.3

98.5/98.2

99.7/ 99.6

99.7/99.6

93.7/91.3

93.8/91.6

93.7/91.5

93.8/91.6

90.1/84.4

89.3/83.2

90.4/85.4

90.7/85.9

General

98.5/98.3

99.7/99.6

99.7/ 99.6

93.8/91.6

93.8/91.7

90.7/86.0

90.1/84.9

91.3 / 87.3

Diag-P

98.3/97.9

98.4/98.0

98.3/98.0

98.4/98.1

99.6/99.5

99.7/99.6

99.6/99.6

99.7/99.6

93.2/90.6

93.3/90.5

93.5/90.9

93.3/90.8

90.1/85.3

90.5/85.8

90.7/86.5

90.6/85.4

Bundle-P

98.3/98.0

98.3/98.1

98.4/98.0

99.7/99.6

93.5/91.1

93.6/91.1

93.6/91.2

89.4/83.4

89.4/83.3

Appendix

Table 19: Link prediction: (AUPR / AUROC, %). by restriction map family and stalk dimension

d

. Cyan = best in family, bold = global best per dataset.

ACM

DBLP

IMDB

L\!=\!2

L\!=\!3

L\!=\!4

L\!=\!5

L\!=\!6

L\!=\!7

L\!=\!8

L\!=\!2

L\!=\!3

L\!=\!4

L\!=\!5

L\!=\!6

L\!=\!7

L\!=\!8

L\!=\!2

L\!=\!3

L\!=\!4

L\!=\!5

L\!=\!6

L\!=\!7

L\!=\!8

Homogeneous baselines

GCN

92.3/92.2

92.1/92.0

65.2/67.3

53.6/57.0

38.6/42.3

30.8/39.1

32.8/40.9

82.2/83.0

88.4/89.2

89.8/90.6

90.4/91.1

91.0/91.7

87.0/88.2

15.5/25.8

58.3/60.9

15.4/39.4

57.1/60.1

23.3/45.2

58.2/62.1

16.6/28.2

GAT

92.1/92.0

92.2/92.1

83.1/84.1

28.1/38.2

29.2/37.6

28.8/38.3

17.0/34.3

84.4/85.1

90.3/91.0

89.4/89.9

92.3/92.8

90.0/90.7

91.1/91.7

91.4/92.0

24.7/32.9

60.2/64.0

14.1/40.0

59.0/61.3

24.7/32.9

55.7/61.3

23.3/45.2

GIN

88.7/88.8

86.8/86.5

35.3/45.2

49.0/49.4

42.2/43.3

41.6/45.4

41.3/43.7

79.9/80.5

93.6/93.9

93.2/93.7

92.9/93.4

94.8/95.2

92.9/93.3

53.1/56.9

60.2/63.9

55.7/60.0

57.3/60.8

52.2/58.5

59.1/62.2

52.1/56.7

SAGE

91.4/91.3

91.8/91.7

92.0/91.9

89.1/89.0

90.6/90.5

92.2/92.1

85.1/85.0

83.6/84.6

88.0/89.0

90.1/90.7

88.4/88.9

89.8/90.5

88.9/89.7

87.7/88.5

58.2/62.4

58.6/61.2

59.5/61.8

60.6/63.8

55.6/57.6

59.2/63.4

58.1/63.6

Appendix

Table 20: Node classification: Macro-F1 / Micro-F1 (%) by model and number of layers

L

. Cyan = best in family, bold = global best per dataset.

MUTAG

PROTEINS

L\!=\!2

L\!=\!3

L\!=\!4

L\!=\!5

L\!=\!6

L\!=\!7

L\!=\!8

L\!=\!2

L\!=\!3

L\!=\!4

L\!=\!5

L\!=\!6

L\!=\!7

L\!=\!8

Homogeneous baselines

GCN

89.6/90.0

76.4/80.4

71.8/77.7

69.5/71.4

75.8/78.6

74.1/76.8

69.0/70.5

75.8/78.6

GAT

89.6/90.0

58.3/60.0

84.0/85.0

56.2/56.2

75.8/78.6

71.8/77.7

48.8/70.5

71.8/77.7

74.6/77.7

52.2/52.7

GIN

89.6/90.0

94.9/95.0

89.9/90.0

85.0/85.0

89.6/90.0

84.7/85.0

70.3/71.4

51.3/71.4

64.1/69.6

70.6/72.3

–

66.8/67.9

62.7/68.8

SAGE

60.0/70.0

89.6/90.0

85.0/85.0

89.6/90.0

69.9/78.6

71.8/77.7

64.7/75.9

71.8/77.7

74.7/78.6

Appendix

Table 21: Graph classification: Macro-F1 / Micro-F1 (%) by model and number of layers

L

. Cyan = best in family, bold = global best per dataset.

LastFM

MovieLens

L\!=\!2

L\!=\!3

L\!=\!4

L\!=\!5

L\!=\!6

L\!=\!7

L\!=\!8

L\!=\!2

L\!=\!3

L\!=\!4

L\!=\!5

L\!=\!6

L\!=\!7

L\!=\!8

Homogeneous baselines

GCN

96.7/96.5

96.5/96.0

96.9/96.8

96.8/96.8

97.1/97.0

97.1/96.9

97.0/97.0

99.5/99.5

99.6/99.6

GAT

62.7/50.4

63.1/51.1

62.7/50.4

62.9/50.7

97.0/97.4

97.6/97.9

96.8/97.3

96.8/97.2

GIN

97.4/96.3

98.1/97.6

98.0/97.7

97.9/97.7

96.7/96.6

97.0/96.9

96.9/96.5

99.6/99.5

99.6/99.6

99.6/99.5

99.6/99.6

99.3/99.3

99.5/99.5

SAGE

92.7/89.2

88.0/86.4

83.5/80.1

95.1/92.7

90.0/86.4

88.5/86.5

88.0/86.6

98.8/98.8

99.1/99.0

99.2/99.1

99.1/99.0

99.2/99.0

99.2/99.1

Appendix

Table 22: Link prediction: AUPR / AUROC (%) by model and number of layers

L

. Cyan = best in family, bold = global best per dataset.

Predictor Name	Acronym Meaning	Inputs Used
Sheaf-NSD	N eural S heaf D iffusion	${\mathbf{z}}$
HetSheaf-TE	T ype E mbeddings	${\mathbf{z}},\phi,\psi$
HetSheaf-NE	N ode E mbeddings	${\mathbf{z}},\phi$
HetSheaf-EE	E dge E mbeddings	${\mathbf{z}},\psi$
HetSheaf-types	All types only	$\phi,\psi$
HetSheaf-NT	N ode T ypes only	$\phi$

Appendix

Table 26: Summary of Heterogeneous Sheaf Predictors and their inputs.

Model	Computational complexity
Non-sheaf baselines
GCN	$\mathcal{O}(nc^{2}+mc)$
GAT	$\mathcal{O}(nc^{2}h+mch)$
HAN	$\mathcal{O}(tnc^{2}h+tmch)$
HetSheaf predictors (diag)
NSD	$\mathcal{O}(nc^{2}+mdc)$

Appendix

Table 27: Per-layer computational complexities of heterogeneous sheaf predictors and baselines. Notation:

n

nodes,

m

edges,

s

node types,

t

edge types, stalk dimension

d

, channels

f

c=df

(or

c=f

for non-sheaf models), attention heads

h

. Diagonal restriction maps are denoted (diag); general or orthogonal maps are denoted (non-diag).

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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.

Abstract

Heterogeneous GNNs

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