Graph Neural Networks Are More Than Filters: Revisiting and Benchmarking from A Spectral Perspective

Jundong Li; Patrick Soga; Song Wang; Yinhan He; Yushun Dong

REVIEW 3 major objections 6 minor 12 references

Graph Neural Networks Are More Than Filters: Revisiting and Benchmarking from A Spectral Perspective

T0 review · 3 major / 6 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read Filters don't decide what GNNs can learn: a benchmark of 14 models shows graph neural networks can recreate frequency components missing from their input, so spectral analysis of the aggregation filter alone is incomplete.

desk verdict A useful empirical challenge to the filter-only view of GNNs, but the benchmark's claims need a no-graph control before they can be trusted. read the letter →

arxiv 2412.07188 v1 pith:YJUZC74M submitted 2024-12-10 cs.LG cs.AI

Yushun Dong , Patrick Soga , Yinhan He , Song Wang , Jundong Li This is my paper · ORCID

classification cs.LGcs.AI

keywords graphneuralnetworksspectralanalysisfrequencycomponentsLaplacianneighborhoodaggregationnodeclassificationbenchmarkfilteringnon-linearlayers

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper challenges the prevailing view that a GNN's spectral behavior is determined by the frequency response of its neighborhood aggregation filter. Through a regression experiment where the input signal and the prediction target live in disjoint frequency bands, the paper shows that GCN can still produce outputs aligned with the target band, meaning the network as a whole can regenerate frequency components that are absent from its input. To make this measurable, the paper designs a benchmark: labels are generated by binning graph-Laplacian eigenvectors by eigenvalue and discretizing them, and each of 14 GNNs is scored on its accuracy per frequency bin. The results show V-shaped accuracy curves across models and datasets (lowest and highest frequencies are easiest, the middle band hardest) and rankings that favor models rarely analyzed spectrally, such as SAGE, GCNII, and GATv2. The upshot is that spectral analyses treating the aggregator as the sole filter are incomplete; other modules, notably non-linear layers, materially reshape a GNN's output spectrum.

What carries the argument

The load-bearing object is a pair of spectral quantities built on the normalized graph Laplacian $L = I - D^{-1/2}AD^{-1/2}$. The first is per-bin signal energy: eigenvectors of $L$ are sorted by eigenvalue (frequency), binned, and the mean eigenvector per bin is used as input or target signal, with energy per bin defining the Energy Distribution Field of a label matrix. The second is the evaluation protocol: each bin's continuous values are discretized into class labels, a GNN is trained to predict those labels, and its accuracy per bin traces an accuracy curve across the spectrum, summarized by the Normalized Area Under the Accuracy Curve (AUAC). The theoretical support is an energy distribution function $e(v) = (Av) \odot (Av)/\|Av\|^2$ with orthonormal $A$, shown to be Lipschitz on the unit sphere, plus a bound (Theorem 4.4) that discretizing an eigenvector into $k$ equal intervals shifts its energy distribution by at most $2(4n/k^2)^{1/n}$. This justifies reading classification accuracy on binned-eigenvector labels as a measure of how well a GNN captures the information encoded in that frequency component.

What would settle it

Replace a GNN's aggregation step with a hard band-stop projection that sets the filter response to exactly zero on one bin of the Laplacian spectrum while keeping the non-linear layers, and run the paper's frequency-incentive protocol. If the output energy in the zeroed bin stays at zero even when the supervision targets that bin, the aggregation filter's frequency response does determine the GNN's spectral behavior in that regime, contradicting the central claim; conversely, if the GNN regenerates the band, the claim survives a strict test.

Watch

Extended reading notes

Core claim

The paper's central claim: the frequency response of the filter of a neighborhood aggregator does not necessarily determine the behavioral characteristics of its host GNN in the frequency domain. Concretely, the paper constructs node regression tasks in which the input features are the mean eigenvectors from one third of the spectral range (say low) while the prediction target is the mean of eigenvectors from another third (say high), so the target frequencies carry zero input energy. GCN trained under MSE loss still matches the target energy distribution in both directions (low-to-high and high-to-low), and does so on par with FA, a GNN built on a learnable spectral filter. The paper concludes that non-linear layers and other non-filter components let GNNs flexibly shift output energy across the spectrum, and that filter-based spectral analysis alone cannot explain a GNN's strengths and weaknesses. It then formalizes this into a benchmark problem, proposes an accuracy-curve protocol with a Normalized AUAC score, proves an upper bound on the spectral deviation introduced by discretizing continuous eigenvector targets into class labels, and reports the resulting rankings for 14 GNNs on six datasets.

Load-bearing premise

The load-bearing premise of the evaluation protocol is that predicting labels made from graph-Laplacian eigenvectors in a frequency bin measures how well a GNN captures the information in that frequency component; if real supervised tasks encode their label information differently, the benchmark's rankings and curves may not transfer.

Editorial extensions

If this is right

Spectral analyses that read a GNN's behavior off its aggregator's frequency response alone are incomplete; conclusions about low-pass or band-pass behavior must be re-checked on the whole network.
On the benchmark's protocol, all 14 models show V-shaped accuracy curves, so middle-frequency information appears to be an inherent weak spot of neighborhood aggregation, not a fixable filter-design issue.
The measured rankings give practitioners a model-choice rule: pick a GNN by the dominant frequency band of the task labels (e.g., SAGE and Cheb for low bands, GATv2 and GCN for middle bands, Transformer and GPS for high bands), and the six-dataset case study shows these rankings track actual task rankings better than random rankings or original-task rankings.
The theoretical bound guarantees that discretizing continuous eigenvector targets into class labels changes the target's energy distribution by at most $2(4n/k^2)^{1/n}$, so node-classification results from the benchmark are consistent with the node-regression exploratory findings.
Deepening GNNs from two to four layers does not change the shape of the accuracy curves, so conclusions drawn from analyzing stacked aggregation filters across depths need revision.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If GNNs can regenerate frequencies that the input lacks, then feature design and supervision signal matter as much as filter design; a natural test is whether GNNs trained with middle-band targets on graphs engineered for homophily can be pushed above the observed V-shaped floor, which the paper does not attempt.
The band-specific rankings suggest a falsifiable design rule: a GNN with an explicit mid-frequency bias (e.g., aggregation on a band-passed neighborhood) should beat all 14 baselines on middle-band targets, and the benchmark provides the exact protocol to check this.
The benchmark's validity rests on synthetic eigenvector-derived labels; extending the Kendall-tau case study to a broader suite of real tasks whose labels' dominant band is measured from the Laplacian spectrum would tell whether the rankings transfer beyond the six datasets tested.
The finding that plain GCN matches FA, a model with a learnable filter, on regeneration tasks implies that expensive filter designs may yield their gains not from the filter itself but from the non-linear readout; comparing FA against GCN with matched non-linear layers would separate these contributions.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Here's the short version: the exploratory experiment is the real contribution; the benchmark is a useful first attempt but its central interpretation is currently unvalidated because there is no graph-agnostic control. The theory is window dressing and should be cut down.

What's new and good: the paper demonstrates with energy plots that a standard GCN, trained on low-frequency-only input, can produce high-frequency output, and vice versa. That's a clean, reproducible check on the claim that a GNN isn't just its aggregation filter. The benchmark, with 14 GNNs and six datasets, gives a concrete way to compare models across the spectral axis, and the Normalized AUAC metric is simple enough to adopt. The Kendall-tau case study is a reasonable start at external validation.

Where it falls down: the V-shaped accuracy curves are interpreted as an 'inherent limitation of neighborhood aggregation,' but the protocol never includes a no-graph classifier. An MLP could show the same V-shape if mid-band labels are simply harder to learn from the raw node features. Until that control is run, the benchmark is measuring something about the synthetic label-generation process, not necessarily about GNN spectral capability. The case study doesn't resolve this because it only checks whether frequency-range rankings transfer to real tasks; it doesn't tell us what drives the curve shape.

The theoretical section is also not up to the claim. Theorem 4.4 gives a bound that is about 2 for any graph of realistic size, so it says nothing useful about energy-distribution preservation under discretization. The appendix even contains a few 'we consider these outliers' exclusions, which contradict the main text's 'all cases' claim and are not justified.

Who reads this: people working on spectral analysis of GNNs or building GNN benchmarks. It's a serious attempt at an important problem. Send it to reviewers, but they should demand a no-graph baseline, a major rewrite of the theory, and honest handling of outliers. With those revisions, the benchmark could become useful.

Referee Report

3 major / 6 minor

Summary. This paper revisits the widely held view that the neighborhood-aggregation filter dominates GNN behavior in the spectral domain. It first presents an exploratory study in which a GCN is trained to regress target signals concentrated in frequency bands that are absent from the input; the outputs align with the target energy distribution, suggesting that nonlinear layers can create frequency components that the linear filter would suppress. The paper then proposes a benchmark protocol: eigenvectors of the normalized graph Laplacian are binned by eigenvalue, the entries of the binned vectors are discretized into classes, and the accuracy of 14 GNNs in predicting these synthetic labels is plotted against frequency to form 'accuracy curves' and a Normalized AUAC ranking. A theoretical analysis (Theorems 4.2–4.4) is offered to justify that discretization does not significantly perturb the energy distribution. The empirical results show V-shaped accuracy curves and model rankings that challenge filter-based analyses, plus a case study relating benchmark rankings to rankings on six real node-classification datasets.

Significance. If the central claim holds, the paper's exploratory result would be a valuable caution against attributing GNNs' spectral behavior solely to the aggregation filter, and the proposed benchmark would provide a common spectral yardstick for comparing spatial and spectral GNNs. The paper also has concrete strengths: it ships code, uses six real datasets and 14 architectures, reports standard deviations, and includes a case study with Kendall distances. However, the theoretical foundation is currently not solid, and the main empirical claims lack a no-graph control; both are needed to support the benchmark's stated purpose.

major comments (3)

[Section 4.2 / Appendix B, Theorem 4.4] The bound max ||e_u - e(v)||_2 ≤ 2(4n/k^2)^(1/n) is vacuous for the graph sizes and class counts used in the benchmark: for n = 1,000 and k = 10 the right-hand side is about 2.0, which exceeds the maximum possible L2 distance (√2) between two nonnegative unit-norm energy distributions. The proof also contains a dimensional error: the diagonal of a hypercube with side length 2/k is 2√n/k, not (4n/k^2)^(1/n), and the step replacing (1/n)^(1/n) by e^(1/e) in Eq. (2) goes in the wrong direction. Therefore the stated theoretical support for the discretization protocol is not established; the bound should be recomputed or the theoretical claim should be withdrawn.
[Sections 4.1 and 5.2] The benchmark protocol lacks a no-graph baseline. The V-shaped accuracy curves are interpreted as evidence about the frequency-capturing capability of GNNs and specifically as revealing 'an inherent limitation of neighborhood aggregation mechanism' (Section 5.2). However, the labels are quantized entries of graph-Laplacian eigenvectors and the input features are fixed node attributes; a plain MLP or logistic-regression model that ignores the adjacency matrix could in principle exhibit the same V-shape if mid-frequency labels are intrinsically harder to predict from the node features. Without reporting accuracy curves and AUAC rankings for such a control, the attribution of the V-shape to neighborhood aggregation is unsupported. Adding an MLP baseline is necessary to validate RQ1 and RQ2.
[Section 4.1 / Appendix A.2] The theoretical parameter k is not connected to the experimental protocol. Theorems 4.3 and 4.4 concern the number k of equal-length intervals in [-1,1] used to discretize an eigenvector's entries, but the main benchmark uses frequency bins of width 0.1 and the number of classes is only illustrated with an example ('5') in Appendix A.2. The actual k per dataset is not reported, and the theorems do not directly address whether a classifier trained on the discretized labels preserves the frequency-specific incentives of the continuous target. Even a sharp bound on energy deviation of some unit vector with the same NCL matrix would not, by itself, imply that the node classification accuracy curves measure the intended frequency-capturing capability. Please spell out the mapping or provide a direct argument.

minor comments (6)

[Section 2] The notation paragraph says 'we employ Xi and Xi to indicate its i-th row and column' but the two symbols are identical; this is presumably a typo and should be fixed.
[Abstract and Section 1] There are several grammatical slips, e.g., 'may also significantly affecting' in the abstract; the manuscript should be carefully proofread.
[Theorem 4.4 / Appendix B] The theorem statement has 'Laplacian amtrix' instead of 'Laplacian matrix'; this typo also appears in the proof heading.
[Section 5.4] The text defines Kendall's τ as 'total number of inversions' and then says smaller values indicate larger similarity; this is actually a Kendall distance, not the normalized τ in [-1,1]. Please clarify the definition and report the normalization used.
[Figure 4 and text] The figure caption and x-axis include 'Texas' but the text introduces only Airport-Brazil, Wisconsin, Cornell, Squirrel, and Chameleon; please reconcile the dataset list.
[Appendix E, Tables 3–5] The table headers say 'multiplied by 10' but the surrounding text says 'All metrics are multiplied by 100'; these should be made consistent.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the benchmark and exploratory study measure model behavior directly, and the theoretical analysis does not reduce to the paper's conclusions.

full rationale

The paper's central claim—that GNNs can flexibly shift output energy across frequency components despite the frequency response of their aggregation filter—is supported by direct experiments in which synthetic targets are constructed from graph-Laplacian eigenvectors and models are trained against those targets. The observed output energy distributions and accuracy curves are measurements, not quantities derived from fitted parameters that are then called predictions. No parameter is fitted to a subset of the benchmark data and then used to predict a closely related quantity; the Normalized AUAC scores are descriptive aggregations of measured accuracies. The theoretical analysis in Section 4.2 proves an upper bound on the energy-distribution deviation caused by discretizing eigenvector entries into class labels; it does not assume the benchmark's ranking conclusions, and it is a self-contained mathematical result about the constructed labels rather than a reduction of the target claim to its inputs. The exploratory argument that graph filters cannot create frequencies absent from the input is a valid logical premise, and the empirical demonstration that GCN outputs energy in target frequencies is evidence rather than circularity. The paper contains only a minor background self-citation (Dong et al., 2021) that is not load-bearing, and the external Kendall-tau case study validates benchmark rankings on new datasets without fitting the benchmark to those datasets. Potential concerns about the lack of a no-graph baseline or about whether synthetic eigenvector-derived labels reflect real task-relevant information are experimental-validity issues, not circularity. No load-bearing step reduces by construction to its own inputs.

Assumptions & free parameters 4 free parameters · 5 assumptions · 2 invented entities

The benchmark depends on several hand-chosen discretization parameters (bin width, class count, frequency-range split) and on the assumption that synthetic eigenvector labels measure spectral capability. The paper introduces two formal constructs, the EDF and NCL matrix, but they are theoretical scaffolding rather than independently evidenced entities. No physical or ontological entity is postulated.

free parameters (4)

Frequency bin width / number of bins = 20 bins in exploratory study; bin width 0.1 in main benchmark
Protocol granularity chosen by hand; accuracy curves and AUAC scores depend on this discretization.
Discretization class count k for NCL labels = Not specified precisely; appendix example uses 5 classes
Thresholds convert continuous eigenvector entries into class labels, and the theoretical bound depends on k.
Low/mid/high frequency split thresholds = Bottom, middle, and top thirds
Frequency-range-specific rankings in Section 5.3 rely on this arbitrary partition.
GNN architecture hyperparameters = 2 layers, hidden dimension 64, learning rate 0.001, 500 epochs
Fixed across models for fairness; the parameter study in Appendix F checks sensitivity to width and depth.

assumptions (5)

standard math Normalized graph Laplacian eigenvectors form an orthonormal frequency basis, and energy is the squared projection coefficient.
Used throughout to define frequency bins and energy distributions (Section 2, Definition 4.1).
standard math The energy distribution function e(v) is Lipschitz with constant 2 under the paper's definitions.
Theorem B.2 depends on orthonormal A and unit-norm v; the proof is straightforward but relies on these structural assumptions.
ad hoc to paper Each section of the unit sphere by the k^n hypercubes corresponds to the inverse image of some Energy Distribution Field.
Theorem B.3 asserts this correspondence with a proof sketch, but the mapping between sphere sections and NCL matrices is not fully formalized.
domain assumption Predicting labels derived from binned eigenvector entries is a valid operationalization of capturing frequency-specific information.
The entire benchmark interprets node classification accuracy on synthetic eigenvector labels as spectral capability (Section 4.1).
domain assumption Nonlinear layers are the explanation for the exploratory results because linear graph filters cannot create absent frequency components.
The paper infers this mechanism without a direct ablation that isolates nonlinearities from other learned components.

invented entities (2)

Energy Distribution Field (EDF)
purpose: Formal set of energy distributions consistent with a node class label matrix; used to prove discretization closeness.
A new mathematical object defined in Definition 4.1; it exists only inside the paper's theoretical framework.
Node Class Label (NCL) matrix
purpose: Discretized one-hot labels used to turn eigenvector entries into classification targets.
A construction specific to the proposed benchmark; no external evidence of its validity beyond the paper's Kendall-tau case study.

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Cite this review

Pith. "Pith review of Graph Neural Networks Are More Than Filters: Revisiting and Benchmarking from A Spectral Perspective." pith.science (2026). https://pith.science/paper/YJUZC74M

@misc{pith2026241207188,
  author       = {Pith},
  title        = {Pith review of: Graph Neural Networks Are More Than Filters: Revisiting and Benchmarking from A Spectral Perspective},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YJUZC74M}},
  note         = {Machine review of arXiv:2412.07188}
}

read the original abstract

Graph Neural Networks (GNNs) have achieved remarkable success in various graph-based learning tasks. While their performance is often attributed to the powerful neighborhood aggregation mechanism, recent studies suggest that other components such as non-linear layers may also significantly affecting how GNNs process the input graph data in the spectral domain. Such evidence challenges the prevalent opinion that neighborhood aggregation mechanisms dominate the behavioral characteristics of GNNs in the spectral domain. To demystify such a conflict, this paper introduces a comprehensive benchmark to measure and evaluate GNNs' capability in capturing and leveraging the information encoded in different frequency components of the input graph data. Specifically, we first conduct an exploratory study demonstrating that GNNs can flexibly yield outputs with diverse frequency components even when certain frequencies are absent or filtered out from the input graph data. We then formulate a novel research problem of measuring and benchmarking the performance of GNNs from a spectral perspective. To take an initial step towards a comprehensive benchmark, we design an evaluation protocol supported by comprehensive theoretical analysis. Finally, we introduce a comprehensive benchmark on real-world datasets, revealing insights that challenge prevalent opinions from a spectral perspective. We believe that our findings will open new avenues for future advancements in this area. Our implementations can be found at: https://github.com/yushundong/Spectral-benchmark.

Figures

Figures reproduced from arXiv: 2412.07188 by the authors.

**Figure 2.** We note that only partial results are presented here due to space limit, and see Appendix D [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

**Figure 3.** Performance comparison in the average ranking of 14 GNNs on six real-world datasets. [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗

Figures from the paper (12 more)

**Figure 4.** Figure 4: Kendall’s τ comparison on new datasets between a random ranking (orange), the rankings from the original graph learning task (blue), and the average rankings from our benchmark (green). In this subsection, we conduct a case study to explore RQ3. Specifically, we propos…

**Figure 5.** Figure 5: Accuracy curves in the spectral domain across different GNNs on the Coauthor-CS dataset. [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]

**Figure 6.** Figure 6: An illustration of key insights for the proof of Theorem B.4. (a) a unit sphere be [PITH_FULL_IMAGE:figures/full_fig_p016_6.png]

**Figure 7.** Figure 7: Energy distributions for CS. 0 5 10 15 20 Frequency bin 0.00 0.05 0.10 0.15 Energy 0 5 10 15 20 Frequency bin 0.0 0.1 0.2 0.3 Energy 0 5 10 15 20 Frequency bin 0.0 0.1 0.2 0.3 Energy (a) FA 0 5 10 15 20 Frequency bin 0.0 0.1 0.2 0.3 Energy (b) GCN [PITH_FULL_IMAGE:fig…

**Figure 8.** Figure 8: Energy distributions for Computers. C COMPLEMENTARY RESULTS OF EXPLORATORY STUDY Figures 7, 8, 9, 10, 11 illustrates a comparison between the energy distributions of input and output frequency components of FA and GCN on the CS, Computers, Cora Full, Photo and Physics …

**Figure 9.** Figure 9: Energy distributions for Cora-Full. 0 5 10 15 20 Frequency bin 0.00 0.05 0.10 0.15 0.20 Energy 0 5 10 15 20 Frequency bin 0.0 0.1 0.2 0.3 0.4 Energy 0 5 10 15 20 Frequency bin 0.0 0.1 0.2 Energy (a) FA 0 5 10 15 20 Frequency bin 0.0 0.1 0.2 0.3 Energy (b) GCN [PITH_FU…

**Figure 10.** Figure 10: Energy distributions for Amazon-Photo. trated in the lower frequency bins while the target signal is predominantly in the higher frequencies. Despite this discrepancy, the model’s output aligns closely with the target frequency, successfully capturing the desired high…

**Figure 11.** Figure 11: Energy distributions for Physics. flexibility suggests that GNNs are not constrained solely by their neighborhood aggregation mechanisms in the spectral domain. Instead, other components, such as the non-linear layers in GNNs, play a crucial role in shaping the outpu…

**Figure 13.** Figure 13: We have the observations as follows, which are consistent with the conclusion we draw [PITH_FULL_IMAGE:figures/full_fig_p019_13.png]

**Figure 12.** Figure 12: The accuracy curves of different GNNs (Transformer, SAGE, GNNII, GATv2, GAT, [PITH_FULL_IMAGE:figures/full_fig_p020_12.png]

**Figure 13.** Figure 13: The accuracy curves of different GNNs (Cheb, FA, GCN, GIN, 1-GNN, GPS and Gat [PITH_FULL_IMAGE:figures/full_fig_p020_13.png]

**Figure 14.** Figure 14: Parameter studies for Photo (top row), DBLP (moddle row) and Cora [PITH_FULL_IMAGE:figures/full_fig_p022_14.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

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Model Computers Cora CS DBLP Photo Physics Avg. Ranking FA 55.00 ± 0.07 44.14 ± 0.10 39.53 ± 0.05 46.36 ± 0.02 58.19 ± 0.20 46.85 ± 0.13 13.17 ± 1.21 APPNP 54.34 ± 0.05 46.59 ± 0.09 43.01 ± 0.05 48.53 ± 0.02 54.21 ± 0.20 46.87 ± 0.03 12.83 ± 0.69 GIN 53.92 ± 0.07 55.60 ± 0.68 48.87 ± 0.11 51.99 ± 0.50 59.39 ± 0.26 51.05 ± 0.13 11.67 ± 0.75 GAT 57.66 ± 0.2...

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The third equation can be verified easily with the definition of Hadamard multiplica- tion. The first “ ≤” is derived from the observation that any entry of Av1 and Av2 has an absolute value less or equal to one since those two vectors are unit vectors. The second “ ≤” follows by the Cauchy-Schwarz inequality. Finally, the last equality follows from the f...

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Model Computers Cora CS DBLP Photo Physics Avg. Ranking FA 55.00 ± 0.07 44.14 ± 0.10 39.53 ± 0.05 46.36 ± 0.02 58.19 ± 0.20 46.85 ± 0.13 13.17 ± 1.21 APPNP 54.34 ± 0.05 46.59 ± 0.09 43.01 ± 0.05 48.53 ± 0.02 54.21 ± 0.20 46.87 ± 0.03 12.83 ± 0.69 GIN 53.92 ± 0.07 55.60 ± 0.68 48.87 ± 0.11 51.99 ± 0.50 59.39 ± 0.26 51.05 ± 0.13 11.67 ± 0.75 GAT 57.66 ± 0.2...

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