Beyond Oversquashing: Understanding Signal Propagation in GNNs Via Observables

Eden Nagar, Yaowei Lin, Ron Levie

Abstract

Graph Neural Networks (GNNs) perform computations on graphs by routing the signal between graph regions using a graph shift operator or a message passing scheme. Often, the propagation of the signal leads to a loss of information, where the signal tends to diffuse across the graph instead of being deliberately routed between regions of interest. Two notions that depict this phenomenon are oversmoothing and oversquashing. In this paper, we propose an alternative approach for modeling signal propagation, inspired by quantum mechanics, using the notion of observables. Specifically, we model the place in the graph where the signal lies, how much the signal is concentrated there, and how much of the signal is propagated towards a location of interest when applying a GNN. Using these new concepts, we prove that standard spectral GNNs have poor signal propagation capabilities. We then propose a new type of spectral GNN, termed Schr\"odinger GNN, which we show has a superior capacity to route the signal across the graph.

Benchmark Results

Tested on 10 datasets 18 baseline values reported too low Non-standard metric on 1 dataset

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

TU Dortmund

MUTAG

Global leaderboard →

#	Model	Accuracy ↑	Paper
1	MSH-GNN GNN global	0.9910 ±0.0030	Multi-Scale Harmonic Encoding for Feature-Wise Graph Message Passing
2	ID-GNN+BEC Hyb global	0.9830 ±0.0360	Depth-Adaptive Graph Neural Networks via Learnable Bakry-Émery Curvature
3	SupCosine Hyb global	0.9830 ±0.0250	Supervised Contrastive Learning with Structure Inference for Graph Classification
39	GIN GNN baseline actual #39 vs paper #657 618 places worse	0.9330 ±0.0290 best reported	actual #39 vs paper #657 618 places worse paper 0.7819 ±0.0557 best at pub: ProtGNN: Towards Self-Explaining Graph Neural Networks (AAAI 2021) paper 0.7819 ±0.0557 Δ at pub 0.1511
63	GCN GNN baseline actual #63 vs paper #739 676 places worse	0.9220 ±0.0440 best reported	actual #63 vs paper #739 676 places worse paper 0.7324 ±0.0627 best at pub: Depth-Adaptive Graph Neural Networks via Learnable Bakry-Émery Curvature (KDD 2025) paper 0.7324 ±0.0627 Δ at pub 0.1896
158	GraphSAGE code ↗ GNN global	0.9040 ±0.0780	Asynchronous Neural Networks for Learning in Graphs code ↗
183	GAT GNN baseline actual #183 vs paper #709 526 places worse	0.9010 ±0.0580 best reported	actual #183 vs paper #709 526 places worse paper 0.7521 ±0.0641 best at pub: Breaking the Limits of Message Passing Graph Neural Networks (ICML 2021) best today: The Expressive Power of Graph Neural Networks: A Survey (2023) paper 0.7521 ±0.0641 Δ at pub 0.1489 Δ today 0.1489
403	MLP code ↗ Trad global	0.8660 ±0.0495	Breaking the Limits of Message Passing Graph Neural Networks code ↗
640	Schrödinger PMO GNN ★ proposed	0.7925 ±0.0619	This paper
700	UniGCN GNN baseline	0.7574 ±0.0667	This paper
704	Adaptive Unitary PMO GNN baseline	0.7562 ±0.0624	This paper
705	Adaptive Unitary GNN baseline	0.7553 ±0.0595	This paper
707	Schrödinger GNN ★ proposed	0.7542 ±0.0611	This paper

PROTEINS

Global leaderboard →

#	Model	Accuracy ↑	Paper
1	MSH-GNN GNN global	0.9410 ±0.0330	Multi-Scale Harmonic Encoding for Feature-Wise Graph Message Passing
2	GRDL GNN global	0.8260 ±0.0120	Graph Classification via Reference Distribution Learning: Theory and Practice
3	GMN code ↗ Hyb global	0.8225	Memory-Based Graph Networks code ↗
32	GIN GNN baseline actual #32 vs paper #414 382 places worse	0.7880 ±0.0410 best reported	actual #32 vs paper #414 382 places worse paper 0.7188 ±0.0308 best at pub: Permutation-Invariant graph partitioning: How graph neural networks capture structural interactions? (2023) paper 0.7188 ±0.0308 Δ at pub 0.0692
49	GAT GNN baseline actual #49 vs paper #397 348 places worse	0.7800 ±0.0300 best reported	actual #49 vs paper #397 348 places worse paper 0.7231 ±0.0328 best at pub: CI-GNN: A Granger Causality-Inspired Graph Neural Network for Interpretable Brain Network-Based Psychiatric Diagnosis (2023) paper 0.7231 ±0.0328 Δ at pub 0.0569
67	SAGE GNN global	0.7726 ±0.0228	Semi-Supervised Hierarchical Graph Classification
148	GCN GNN baseline actual #148 vs paper #429 281 places worse	0.7600 ±0.0320 best reported	actual #148 vs paper #429 281 places worse paper 0.7141 ±0.0304 best at pub: CI-GNN: A Granger Causality-Inspired Graph Neural Network for Interpretable Brain Network-Based Psychiatric Diagnosis (2023) best today: Simplifying Graph Convolutional Networks with Redundancy-Free Neighbors (2025) Difference within reported uncertainty; not counted as underperformance. paper 0.7141 ±0.0304 Δ at pub 0.0459 Δ today 0.0459
149	MLP Trad global	0.7600 ±0.0320	Simplifying Graph Convolutional Networks with Redundancy-Free Neighbors
379	Schrödinger PMO GNN ★ proposed	0.7268 ±0.0305	This paper
419	Adaptive Unitary GNN baseline	0.7179 ±0.0333	This paper
420	Adaptive Unitary PMO GNN baseline	0.7177 ±0.0284	This paper
429	Schrödinger GNN ★ proposed	0.7157 ±0.0256	This paper
465	UniGCN GNN baseline	0.6919 ±0.0301	This paper

ENZYMES

Global leaderboard →

#	Model	Accuracy ↑	Paper
1	GMN code ↗ Hyb global	0.7866	Memory-Based Graph Networks code ↗
2	MemGNN code ↗ Hyb global	0.7550	Memory-Based Graph Networks code ↗
3	TFGW SP Hyb global	0.7510 ±0.0500	Template based Graph Neural Network with Optimal Transport Distances
17	GatedGCN code ↗ GNN global	0.7000 ±0.0494	GNNBenchmark code ↗
20	GAT GNN baseline actual #20 vs paper #225 205 places worse	0.6850 ±0.0524 best reported	actual #20 vs paper #225 205 places worse paper 0.3113 ±0.0348 best at pub: GNNBenchmark (JMLR 2023) paper 0.3113 ±0.0348 Δ at pub 0.3737
26	GraphSAGE code ↗ GNN global	0.6817 ±0.0545	GNNBenchmark code ↗
32	GIN GNN baseline actual #32 vs paper #222 190 places worse	0.6767 ±0.0583 best reported	actual #32 vs paper #222 190 places worse paper 0.3193 ±0.0316 best at pub: GNNBenchmark (JMLR 2023) paper 0.3193 ±0.0316 Δ at pub 0.3574
36	GCN GNN baseline actual #36 vs paper #222 186 places worse	0.6650 ±0.0691 best reported	actual #36 vs paper #222 186 places worse paper 0.3166 ±0.0535 best at pub: Hierarchical Graph Capsule Network (AAAI 2020) paper 0.3166 ±0.0535 Δ at pub 0.3484
124	MLP code ↗ Trad global	0.5583 ±0.0352	GNNBenchmark code ↗
175	Schr"odinger PMO ★ proposed	0.4370 ±0.0337	This paper
176	Schr"odinger ★ proposed	0.4350 ±0.0489	This paper
183	Adaptive Unitary PMO GNN baseline	0.4183 ±0.0444	This paper
184	Adaptive Unitary GNN baseline	0.4160 ±0.0518	This paper
186	UniGCN GNN baseline	0.4030 ±0.0663	This paper

LRGB

PascalVOC-SP

Global leaderboard →

#	Model	F1 ↑	Paper
1	NeuralWalker code ↗ Walk global	0.4912 ±0.0042	Chimera: State Space Models Beyond Sequences code ↗
2	PPGT GT global	0.4641 ±0.0033	Plain Transformers Can be Powerful Graph Learners
3	CRaWl code ↗ Trad global ✎	0.4588 ±0.0079	Benchmarking Positional Encodings for GNNs and Graph Transformers code ↗
14	GPS Hyb baseline actual #14 vs paper #39 25 places worse	0.4440 ±0.0065 best reported	actual #14 vs paper #39 25 places worse paper 0.3884 ±0.0055 best at pub: Where Did the Gap Go? Reassessing the Long-Range Graph Benchmark (TMLR 2023) best today: Plain Transformers Can be Powerful Graph Learners (2025) paper 0.3884 ±0.0055 Δ at pub 0.0556 Δ today 0.0556
19	Exphormer GT baseline actual #19 vs paper #62 43 places worse	0.4242 ±0.0044 best reported	actual #19 vs paper #62 43 places worse paper 0.3430 ±0.0008 best at pub: Benchmarking Positional Encodings for GNNs and Graph Transformers (KDD 2024) paper 0.3430 ±0.0008 Δ at pub 0.0812
41	GatedGCN GNN baseline actual #41 vs paper #70 29 places worse	0.3880 ±0.0040 best reported	actual #41 vs paper #70 29 places worse paper 0.2922 ±0.0018 best at pub: Where Did the Gap Go? Reassessing the Long-Range Graph Benchmark (TMLR 2023) best today: Plain Transformers Can be Powerful Graph Learners (2025) paper 0.2922 ±0.0018 Δ at pub 0.0958 Δ today 0.0958
42	Gated-GCN code ↗ GNN global	0.3880 ±0.0040	Chimera: State Space Models Beyond Sequences code ↗
54	Schrödinger GNN ★ proposed	0.3507 ±0.0019	This paper
69	Lie UniGCN GNN baseline	0.3153 ±0.0035	This paper
71	UniGCN GNN baseline	0.2852 ±0.0016	This paper
75	GINE GNN baseline actual #75 vs paper #83 8 places worse	0.2718 ±0.0054 best reported	actual #75 vs paper #83 8 places worse paper 0.2125 ±0.0009 best at pub: Where Did the Gap Go? Reassessing the Long-Range Graph Benchmark (TMLR 2023) best today: Chimera: State Space Models Beyond Sequences (TMLR 2025) paper 0.2125 ±0.0009 Δ at pub 0.0593 Δ today 0.0593
81	GCN GNN baseline actual #81 vs paper #91 10 places worse	0.2380 ±0.0160 best reported	actual #81 vs paper #91 10 places worse paper 0.1338 ±0.0007 best at pub: When Graph Neural Networks Meet Dynamic Mode Decomposition (ICLR 2024) paper 0.1338 ±0.0007 Δ at pub 0.1042
93	MLP Trad global	0.1140 ±0.0230	When Graph Neural Networks Meet Dynamic Mode Decomposition

COCO-SP

Global leaderboard →

#	Model	F1 ↑	Paper
1	FloydNet code ↗ Hyb global	0.4901	FloydNet: A Learning Paradigm for Global Relational Reasoning code ↗
2	CRaWl Trad baseline	0.4588 ±0.0079	This paper
3	GPS Hyb baseline actual #3 vs paper #3	0.4440 ±0.0065	actual #3 vs paper #3 paper 0.4440 ±0.0065 prev best: Where Did the Gap Go? Reassessing the Long-Range Graph Benchmark (TMLR 2023) paper 0.4440 ±0.0065 Δ at pub 0.0556
5	Schrödinger GNN ★ proposed	0.4259 ±0.0034	This paper
6	Lie UniGCN GNN baseline	0.4005 ±0.0067	This paper
10	Exphormer GT baseline actual #10 vs paper #10	0.3960 ±0.0027	actual #10 vs paper #10 paper 0.3960 ±0.0027 prev best: A Scalable and Effective Alternative to Graph Transformers (AAAI 2024) paper 0.3960 ±0.0027 Δ at pub 0.0505
16	GatedGCN GNN baseline actual #16 vs paper #16	0.3880 ±0.0040	actual #16 vs paper #16 paper 0.3880 ±0.0040 prev best: Where Did the Gap Go? Reassessing the Long-Range Graph Benchmark (TMLR 2023) paper 0.3880 ±0.0040 Δ at pub 0.0958
23	UniGCN GNN baseline	0.3516 ±0.0070	This paper
31	DRew Trad baseline	0.3314 ±0.0024	This paper
33	Gated-GCN code ↗ GNN global	0.2922 ±0.0018	Chimera: State Space Models Beyond Sequences code ↗
34	GINE GNN baseline actual #34 vs paper #34	0.2718 ±0.0054	actual #34 vs paper #34 paper 0.2718 ±0.0054 prev best: Where Did the Gap Go? Reassessing the Long-Range Graph Benchmark (TMLR 2023) paper 0.2718 ±0.0054 Δ at pub 0.0593
45	GCN GNN baseline actual #45 vs paper #45	0.2078 ±0.0031	actual #45 vs paper #45 paper 0.2078 ±0.0031 prev best: Where Did the Gap Go? Reassessing the Long-Range Graph Benchmark (TMLR 2023) paper 0.2078 ±0.0031 Δ at pub 0.0740
74	MLP Trad global	0.0310 ±0.0160	When Graph Neural Networks Meet Dynamic Mode Decomposition

Heterophily Benchmark

Roman-empire

Global leaderboard →

Non-default metric. This paper reports AP; the dataset default is Accuracy. Values below use AP only.

Sorted by AP. Highlighted rows are from other papers.

#	Model	AP ↑	Paper
1	Schrödinger GNN ★ proposed	0.8856 ±0.0071	This paper
2	UniGCN GNN baseline	0.8721 ±0.0076	This paper
3	GT GNN baseline	0.8651 ±0.0073	This paper
4	SAGE GNN baseline	0.8574 ±0.0067	This paper
5	Lie UniGCN GNN baseline	0.8550 ±0.0022	This paper
6	GAT GNN baseline	0.8087 ±0.0030	This paper
7	GCN GNN baseline	0.7369 ±0.0074	This paper

Minesweeper

Global leaderboard →

#	Model	ROC-AUC ↑	Paper
1	BuNN code ↗ Hyb global	0.9899 ±0.0016	Bundle Neural Networks for message diffusion on graphs code ↗
2	DC-GNN GNN global	0.9850 ±0.0021	Differentiable Cluster Graph Neural Network
3	Tango -GPS global	0.9839 ±0.0054	TANGO: Graph Neural Dynamics via Learned Energy and Tangential Flows
12	Schrödinger GNN ★ proposed	0.9631 ±0.0049	This paper
14	Lie UniGCN GNN baseline	0.9611 ±0.0010	This paper
25	UniGCN GNN baseline	0.9427 ±0.0058	This paper
36	GAT GNN baseline actual #36 vs paper #60 24 places worse	0.9391 ±0.0035 best reported	actual #36 vs paper #60 24 places worse paper 0.9201 ±0.0068 best at pub: Relieving the Over-Aggregating Effect in Graph Transformers (NeurIPS 2025) paper 0.9201 ±0.0068 Δ at pub 0.0190
42	GraphSAGE code ↗ GNN global	0.9358 ±0.0005	Gumbel-MPNN: Graph Rewiring with Gumbel-Softmax code ↗
46	SAGE GNN baseline	0.9351 ±0.0057	This paper
63	GT GNN baseline	0.9185 ±0.0076	This paper
68	GCN GNN baseline actual #68 vs paper #88 20 places worse	0.9150 ±0.0006 best reported	actual #68 vs paper #88 20 places worse paper 0.8975 ±0.0052 best at pub: Gumbel-MPNN: Graph Rewiring with Gumbel-Softmax (2025) paper 0.8975 ±0.0052 Δ at pub 0.0175
156	MLP Trad global	0.5099 ±0.0147	Graph Homophily Booster: Rethinking the Role of Discrete Features on Heterophilic Graphs

Tolokers

Global leaderboard →

#	Model	ROC-AUC ↑	Paper
1	Polynormer-r code ↗ GT global	0.8591 ±0.0074	Polynormer: Polynomial-Expressive Graph Transformer in Linear Time code ↗
2	DC-GNN GNN global	0.8588 ±0.0081	Differentiable Cluster Graph Neural Network
3	TANGO-GPS Hyb global	0.8566 ±0.0101	TANGO: Graph Neural Dynamics via Learned Energy and Tangential Flows
10	Lie UniGCN GNN baseline	0.8518 ±0.0043	This paper
13	Schrödinger GNN ★ proposed	0.8509 ±0.0017	This paper
17	UniGCN GNN baseline	0.8483 ±0.0068	This paper
28	GAT GNN baseline actual #28 vs paper #47 19 places worse	0.8443 ±0.0100 best reported	actual #28 vs paper #47 19 places worse paper 0.8370 ±0.0047 best at pub: TopoBench: A Framework for Benchmarking Topological Deep Learning (2024) Difference within reported uncertainty; not counted as underperformance. paper 0.8370 ±0.0047 Δ at pub 0.0073
50	GCN GNN baseline	0.8364 ±0.0067	This paper
61	GT GNN baseline	0.8323 ±0.0064	This paper
76	SAGE GNN baseline	0.8243 ±0.0044	This paper
77	GraphSAGE code ↗ GNN global	0.8243 ±0.0044	Relieving the Over-Aggregating Effect in Graph Transformers code ↗
139	MLP code ↗ Trad global	0.7362 ±0.0020	Gumbel-MPNN: Graph Rewiring with Gumbel-Softmax code ↗

Questions

Global leaderboard →

#	Model	ROC-AUC ↑	Paper
1	NID_GAT code ↗ global	0.9703 ±0.0002	Node Identifiers: Compact, Discrete Representations for Efficient Graph Learning code ↗
2	NID_GCN code ↗ global	0.9685 ±0.0010	Node Identifiers: Compact, Discrete Representations for Efficient Graph Learning code ↗
3	UniGAP global	0.8421 ±0.0080	UniGAP: A Universal and Adaptive Graph Upsampling Approach to Mitigate Over-Smoothing in Node …
9	Schrödinger GNN ★ proposed	0.8014 ±0.0020	This paper
11	Lie UniGCN GNN baseline	0.8001 ±0.0043	This paper
15	UniGCN GNN baseline	0.7921 ±0.0079	This paper
16	GCN GNN baseline actual #16 vs paper #62 46 places worse	0.7902 ±0.0060 best reported	actual #16 vs paper #62 46 places worse paper 0.7609 ±0.0127 best at pub: UniGAP: A Universal and Adaptive Graph Upsampling Approach to Mitigate Over-Smoothing in Node Classification Tasks (2024) paper 0.7609 ±0.0127 Δ at pub 0.0293
26	GAT GNN baseline actual #26 vs paper #38 12 places worse	0.7835 ±0.0116 best reported	actual #26 vs paper #38 12 places worse paper 0.7743 ±0.0120 best at pub: Node Identifiers: Compact, Discrete Representations for Efficient Graph Learning (ICLR 2024) Difference within reported uncertainty; not counted as underperformance. paper 0.7743 ±0.0120 Δ at pub 0.0092
30	GT GNN baseline	0.7795 ±0.0068	This paper
46	GraphSAGE GNN global	0.7721 ±0.0130	UniGAP: A Universal and Adaptive Graph Upsampling Approach to Mitigate Over-Smoothing in Node …
58	SAGE GNN baseline	0.7644 ±0.0062	This paper
96	MLP code ↗ Trad global	0.7082 ±0.0010	Gumbel-MPNN: Graph Rewiring with Gumbel-Softmax code ↗

GNNBenchmark

MNIST

Global leaderboard →

#	Model	Accuracy ↑	Paper
1	Schrödinger GNN ★ proposed	0.9913 ±0.0004	This paper
2	CNN GNN baseline actual #2 vs paper #2	0.9907 ±0.0007	actual #2 vs paper #2 paper 0.9907 ±0.0007 prev best: Equivariance with Learned Canonicalization Functions (ICML 2022) paper 0.9907 ±0.0007 Δ at pub 0.0397
3	MPNN GNN baseline	0.9895 ±0.0006	This paper
11	GatedGCN GNN global	0.9855 ±0.0002	RandAlign: A Parameter-Free Method for Regularizing Graph Convolutional Networks
32	GIN GNN baseline actual #32 vs paper #32	0.9833 ±0.0011	actual #32 vs paper #32 paper 0.9833 ±0.0011 prev best: Graph-Coupled Oscillator Networks (ICML 2022) paper 0.9833 ±0.0011 Δ at pub 0.0110
86	GAT GNN baseline actual #86 vs paper #119 33 places worse	0.9755 ±0.0003 best reported	actual #86 vs paper #119 33 places worse paper 0.9594 ±0.0071 best at pub: RandAlign: A Parameter-Free Method for Regularizing Graph Convolutional Networks (2024) paper 0.9594 ±0.0071 Δ at pub 0.0161
95	GraphSAGE GNN global	0.9731 ±0.0010	Differential Encoding for Improved Representation Learning Over Graphs
121	ChebConv GNN baseline	0.9572 ±0.0074	This paper
126	MLP code ↗ Trad global	0.9534 ±0.0014	GNNBenchmark code ↗
135	GCN GNN baseline actual #135 vs paper #136 1 place worse	0.9405 ±0.0005 best reported	actual #135 vs paper #136 1 place worse paper 0.9209 ±0.0028 best at pub: RandAlign: A Parameter-Free Method for Regularizing Graph Convolutional Networks (2024) paper 0.9209 ±0.0028 Δ at pub 0.0196

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 at pub — reported value improved over the best known when published