Beltrami Flow and Neural Diffusion on Graphs

B. Chamberlain, J. Rowbottom, Davide Eynard, Francesco Di Giovanni, Xiaowen Dong, M. Bronstein

Abstract

We propose a novel class of graph neural networks based on the discretised Beltrami flow, a non-Euclidean diffusion PDE. In our model, node features are supplemented with positional encodings derived from the graph topology and jointly evolved by the Beltrami flow, producing simultaneously continuous feature learning and topology evolution. The resulting model generalises many popular graph neural networks and achieves state-of-the-art results on several benchmarks.

Benchmark Results

Tested on 5 datasets 4 baseline values reported too low

Tables show presented results in global context (best 3 papers + GCN + MLP).

Classic

CiteSeer

Global leaderboard →

#	Model	Accuracy ↑	Paper
1	CNA code ↗ GNN global	0.9575 ±0.0058	Graph Neural Networks Need Cluster-Normalize-Activate Modules code ↗
2	IS-GIB code ↗ Hyb global	0.9390 ±0.0187	Individual and Structural Graph Information Bottlenecks for Out-of-Distribution Generalization code ↗
3	EERM code ↗ GNN global	0.9112 ±0.0145	Individual and Structural Graph Information Bottlenecks for Out-of-Distribution Generalization code ↗
44	GCN GNN global	0.8033 ±0.0088	NTFormer: A Composite Node Tokenized Graph Transformer for Node Classification
350	MLP code ↗ Trad global	0.7289 ±0.0010	Gumbel-MPNN: Graph Rewiring with Gumbel-Softmax code ↗
391	GDE GNN baseline	0.7250 ±0.0050	This paper
411	GODE GNN baseline	0.7240 ±0.0060	This paper
590	CGNN GNN baseline actual #590 vs paper #680 90 places worse	0.7050 ±0.0123 best reported	actual #590 vs paper #680 90 places worse paper 0.6810 ±0.0120 best at pub: GRAND: Graph Neural Diffusion (ICML 2021) best today: Gauge-Equivariant Graph Networks via Self-Interference Cancellation (2025) paper 0.6810 ±0.0120 Δ today 0.0240
665	AdaLanczos GNN baseline	0.6870 ±0.0100	This paper
745	Lanczos GNN baseline	0.6620 ±0.0190	This paper

PubMed

Global leaderboard →

#	Model	Accuracy ↑	Paper
1	LGTL GT global	0.9518 ±0.0011	Adaptive Tokenization: On the Hop-Overpriority Problem in Tokenized Graph Learning Models
2	MixHop (LO) code ↗ GNN global	0.9513 ±0.0038	LOGIN: A Large Language Model Consulted Graph Neural Network Training Framework code ↗
3	GraphSAGE (LO) code ↗ GNN global	0.9511 ±0.0036	LOGIN: A Large Language Model Consulted Graph Neural Network Training Framework code ↗
18	GCN GNN baseline actual #18 vs paper #585 567 places worse	0.9296 ±0.0015 best reported	actual #18 vs paper #585 567 places worse paper 0.7900 ±0.0050 best at pub: Tackling Over-Smoothing for General Graph Convolutional Networks (2020) best today: Adaptive Tokenization: On the Hop-Overpriority Problem in Tokenized Graph Learning Models (2025) paper 0.7900 ±0.0050 Δ at pub 0.1070 Δ today 0.1396
116	SAGE GNN global	0.8881 ±0.0015	Efficient Text-Attributed Graph Learning through Selective Annotation and Graph Alignment
210	MLP code ↗ Trad global	0.8690 ±0.0074	On the Equivalence of Graph Convolution and Mixup code ↗
345	BLEND-kNN Hyb ★ proposed	0.8150 ±0.0050	This paper
394	BLEND GNN ★ proposed	0.8070 ±0.0070	This paper
430	CGNN GNN baseline	0.8020 ±0.0030	This paper
440	GODE GNN baseline	0.8010 ±0.0030	This paper
459	GDE GNN baseline	0.7990 ±0.0030	This paper
654	Lanczos GNN baseline	0.7830 ±0.0030	This paper
670	AdaLanczos GNN baseline	0.7810 ±0.0040	This paper
754	GS-maxpool GNN baseline	0.7660 ±0.0080	This paper

Amazon-Photo

Global leaderboard →

#	Model	Accuracy ↑	Paper
1	GraphTARIF code ↗ Hyb global	0.9703 ±0.0019	Graph Transductive Sharpening: Leveraging Unlabeled Predictions in Node Classification code ↗
2	GAT GNN baseline actual #2 vs paper #503 501 places worse	0.9669 ±0.0014 best reported	actual #2 vs paper #503 501 places worse paper 0.8570 ±0.2000 best at pub: Graph Contrastive Learning with Adaptive Augmentation (WWW 2020) best today: Graph Transductive Sharpening: Leveraging Unlabeled Predictions in Node Classification (2026) Difference within reported uncertainty; not counted as underperformance. paper 0.8570 ±0.2000 Δ at pub 0.0686 Δ today 0.1099
3	GAT+T code ↗ Hyb global	0.9660 ±0.0007	Graph Transductive Sharpening: Leveraging Unlabeled Predictions in Node Classification code ↗
7	SAGE code ↗ GNN global	0.9643 ±0.0027	Graph Transductive Sharpening: Leveraging Unlabeled Predictions in Node Classification code ↗
8	GCN GNN baseline actual #8 vs paper #395 387 places worse	0.9632 ±0.0008 best reported	actual #8 vs paper #395 387 places worse paper 0.9120 ±0.0120 best at pub: Graph Contrastive Learning with Adaptive Augmentation (WWW 2020) best today: Graph Transductive Sharpening: Leveraging Unlabeled Predictions in Node Classification (2026) paper 0.9120 ±0.0120 Δ at pub 0.0122 Δ today 0.0512
82	CGNN GNN baseline actual #82 vs paper #378 296 places worse	0.9435 ±0.0056 best reported	actual #82 vs paper #378 296 places worse paper 0.9140 ±0.0150 best today: Graph Wave Networks (WWW 2025) paper 0.9140 ±0.0150 Δ today 0.0295
230	BLEND GNN ★ proposed	0.9290 ±0.0060	This paper
296	GDE GNN baseline	0.9240 ±0.0200	This paper
380	GS-mean GNN baseline	0.9140 ±0.0130	This paper
390	MLP code ↗ Trad global	0.9128 ±0.0063	Clarify Confused Nodes via Separated Learning code ↗
393	MoNet GNN baseline actual #393 vs paper #395 2 places worse	0.9124 ±0.0130 best reported	actual #393 vs paper #395 2 places worse paper 0.9120 ±0.0230 best today: Building Shortcuts between Distant Nodes with Biaffine Mapping for Graph Convolutional Networks (2023) Difference within reported uncertainty; not counted as underperformance. paper 0.9120 ±0.0230 Δ today 0.0004
403	GAT-ppr GNN baseline	0.9090 ±0.0030	This paper
435	GS-maxpool GNN baseline	0.9040 ±0.0130	This paper

Amazon-Computers

Global leaderboard →

#	Model	Accuracy ↑	Paper
1	GraphTARIF code ↗ Hyb global	0.9461 ±0.0017	Graph Transductive Sharpening: Leveraging Unlabeled Predictions in Node Classification code ↗
2	GCN GNN baseline actual #2 vs paper #389 387 places worse	0.9412 ±0.0008 best reported	actual #2 vs paper #389 387 places worse paper 0.8260 ±0.0240 best at pub: Learning on Attribute-Missing Graphs (2020) best today: Graph Transductive Sharpening: Leveraging Unlabeled Predictions in Node Classification (2026) paper 0.8260 ±0.0240 Δ at pub 0.0525 Δ today 0.1152
3	GAT GNN baseline actual #3 vs paper #442 439 places worse	0.9398 ±0.0022 best reported	actual #3 vs paper #442 439 places worse paper 0.7800 ±0.1900 best at pub: Graph Contrastive Learning with Adaptive Augmentation (WWW 2020) best today: Graph Transductive Sharpening: Leveraging Unlabeled Predictions in Node Classification (2026) Difference within reported uncertainty; not counted as underperformance. paper 0.7800 ±0.1900 Δ at pub 0.0893 Δ today 0.1598
8	SAGE code ↗ GNN global	0.9351 ±0.0006	Graph Transductive Sharpening: Leveraging Unlabeled Predictions in Node Classification code ↗
195	CGNN GNN baseline actual #195 vs paper #428 233 places worse	0.8831 ±0.0065 best reported	actual #195 vs paper #428 233 places worse paper 0.8030 ±0.0200 best today: Graph Wave Networks (WWW 2025) paper 0.8030 ±0.0200 Δ today 0.0801
257	BLEND GNN ★ proposed	0.8690 ±0.0060	This paper
300	GAT-ppr GNN baseline	0.8540 ±0.0030	This paper
320	MLP code ↗ Trad global	0.8463	GraphFM: A generalist graph transformer that learns transferable representations across diverse … code ↗
367	MoNet GNN baseline	0.8350 ±0.0220	This paper
385	GDE GNN baseline	0.8290 ±0.0060	This paper
391	GS-mean GNN baseline	0.8240 ±0.0180	This paper

Coauthor-CS

Global leaderboard →

#	Model	Accuracy ↑	Paper
1	DoG code ↗ GNN global	0.9690 ±0.0030	Diffusion on Graph: Augmentation of Graph Structure for Node Classification code ↗
2	GWN-fa code ↗ Hyb global	0.9667 ±0.0026	Graph Wave Networks code ↗
3	GWN-sym code ↗ GNN global	0.9666 ±0.0026	Graph Wave Networks code ↗
4	CGNN GNN baseline actual #4 vs paper #107 103 places worse	0.9621 ±0.0032 best reported	actual #4 vs paper #107 103 places worse paper 0.9230 ±0.0020 best today: Graph Wave Networks (WWW 2025) paper 0.9230 ±0.0020 Δ today 0.0391
6	GraphSAGE code ↗ GNN global	0.9593 ±0.0025	Graph Wave Networks code ↗
12	GCN GNN baseline actual #12 vs paper #148 136 places worse	0.9514 ±0.0025 best reported	actual #12 vs paper #148 136 places worse paper 0.9110 ±0.0050 best at pub: Unifying Graph Convolutional Neural Networks and Label Propagation (2020) best today: Graph Wave Networks (WWW 2025) paper 0.9110 ±0.0050 Δ at pub 0.0250 Δ today 0.0404
21	GAT GNN baseline actual #21 vs paper #156 135 places worse	0.9437 ±0.0028 best reported	actual #21 vs paper #156 135 places worse paper 0.9050 ±0.0060 best at pub: Unifying Graph Convolutional Neural Networks and Label Propagation (2020) best today: Graph Wave Networks (WWW 2025) paper 0.9050 ±0.0060 Δ at pub 0.0330 Δ today 0.0387
84	BLEND GNN ★ proposed	0.9290 ±0.0020	This paper
88	MLP Trad global	0.9280 ±0.0020	Revisiting the Necessity of Graph Learning and Common Graph Benchmarks
135	GDE GNN baseline	0.9160 ±0.0010	This paper
140	GAT-ppr GNN baseline	0.9130 ±0.0010	This paper
141	GS-mean GNN baseline	0.9130 ±0.0280	This paper
155	MoNet GNN baseline actual #155 vs paper #155	0.9082 ±0.0160 best reported	actual #155 vs paper #155 paper 0.9080 ±0.0060 best at pub: GraphMix: Improved Training of GNNs for Semi-Supervised Learning (2019) best today: Building Shortcuts between Distant Nodes with Biaffine Mapping for Graph Convolutional Networks (2023) Difference within reported uncertainty; not counted as underperformance. paper 0.9080 ±0.0060 Δ today 0.0002
186	GS-maxpool GNN baseline	0.8500 ±0.0110	This paper

SOTA performance (not significantly different from the best) Proposed method (this paper) Baseline / comparison (this paper) Global context (other papers) Best at pub — reported result worse than the best known when published (best reported in green) Best today — OK at publication; today's leaderboard best is elsewhere