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REVIEW 3 major objections 5 minor 27 references

Edge-Splitting MLP: Node Classification on Homophilic and Heterophilic Graphs without Message Passing

T0 review · 3 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read ES-MLP replaces message passing with an edge-splitting neighborhood loss, letting a pure MLP match or beat GNNs on homophilic and heterophilic graphs without ever using the adjacency matrix at inference.

desk verdict A solid, honestly-limited MLP alternative for heterophilic graphs; the abstract overclaims on Minesweeper but the core empirical work and ablations stand up. read the letter →

arxiv 2412.08310 v1 pith:ZSNTN2JR submitted 2024-12-11 cs.LG stat.ML

Matthias Kohn , Marcel Hoffmann , Ansgar Scherp This is my paper · ORCID
classification cs.LGstat.ML
keywords nodeclassificationheterophilicgraphshomophilyMLPwithoutmessagepassingedgesplittingneighborhoodcontrastivelossinference-timerobustnessgraphneuralnetworks
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

The paper proposes ES-MLP, a node-classification model that never touches the graph's edges at inference time. It combines Graph-MLP's neighborhood contrastive loss with ES-GNN's edge-splitting mechanism: node features are projected into task-relevant and task-irrelevant subspaces, and each edge receives a learned weight deciding how much of it belongs to each channel. The authors claim that this two-channel loss lets a plain MLP outperform all six baselines on the heterophilic datasets Actor, Roman, and Amazon, while staying within about two accuracy points of the best message-passing models on the homophilic citation graphs Cora, CiteSeer, and PubMed. If true, message passing can be dropped entirely whenever a node's class is readable from its own features, which brings robustness to edge noise and two-to-five times faster inference.

What carries the argument

The load-bearing object is the pair of splitting adjacency matrices A_R(i,j)=(1+alpha(i,j))/2 and A_IR(i,j)=(1-alpha(i,j))/2, where alpha(i,j) is a scalar in (-1,1) produced by a tanh-activated linear layer over the concatenated relevant and irrelevant embeddings of nodes i and j. Because A_R+A_IR equals the original adjacency matrix, every edge is partitioned into a task-relevant and a task-irrelevant channel. The r-th powers of these matrices feed a neighborhood contrastive loss computed separately on the Z_R and Z_IR embedding spaces, which is how the model learns from r-hop neighborhoods without message passing; an irrelevant consistency regularization term then minimizes the Z_IR distance of adjacent nodes whose predicted labels differ. At inference only the MLP on Z_R remains, so the adjacency matrix is never needed.

What would settle it

Run ES-MLP on a heterophilic graph whose node features carry no class information while neighbor features do (the Minesweeper setting); its accuracy will remain at chance because edges are absent at inference—and indeed the paper reports roughly 50% AUROC for ES-MLP on Minesweeper versus 88.9% for GraphSAGE.

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Extended reading notes

Core claim

ES-MLP's central claim is that the neighborhood structure of a graph can be digested during training, through a loss, and then discarded at inference. The model learns two exclusive splitting matrices A_R and A_IR, with A_R(i,j)+A_IR(i,j)=1 for every edge, by feeding the concatenated task-relevant and task-irrelevant embeddings of both endpoints into a tanh-activated linear layer. The neighborhood contrastive loss of Graph-MLP is then applied separately to the two embedding spaces, using powers of A_R and A_IR, so that task-relevant edges pull relevant representations together while irrelevant edges are handled in the other channel; an irrelevant consistency regularization further suppresses the irrelevant channel on edges whose endpoints are predicted to have different labels. Predictions are made solely from the task-relevant embedding Z_R. The paper demonstrates on seven real-world datasets and one CSBM synthetic family that this scheme yields accuracy on par with or better than GCN, GraphSAGE, ES-GNN, LINKX, and the two MLP baselines, and that the learned matrices separate homophilic from heterophilic edges as intended.

Load-bearing premise

At inference time the node's class must be recoverable from its own feature vector, because edges are used only during training; on any task where labels are determined by neighbor features, as in Minesweeper, the model cannot classify correctly.

Editorial extensions

If this is right

  • On the heterophilic datasets Actor, Roman, and Amazon, ES-MLP outperforms all six baselines, including ES-GNN and GraphSAGE.
  • On the homophilic citation graphs Cora, CiteSeer, and PubMed, ES-MLP remains within about two accuracy points of the best MPNN, and it beats Graph-MLP on Cora and PubMed.
  • Injecting uniform or categorical edge noise into the test graph leaves ES-MLP's accuracy unchanged, while GCN, GraphSAGE, and ES-GNN drop by up to 12.7 points on Cora and 2–3 points on Amazon.
  • ES-MLP runs two to five times faster than the fastest MPNN baseline at inference and can be fed the test nodes alone without losing accuracy, because edges are absent by design.
  • The learned adjacency matrices separate homophilic from heterophilic edges, with the relevant channel carrying a higher share of same-class edges on Cora and most edges pushed into the irrelevant channel on Actor.

Reading between the lines

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

  • The Minesweeper result marks a hard boundary for all edge-free models, not just ES-MLP: whenever labels are determined by neighbor features rather than the node's own features, dropping edges at inference caps accuracy at chance; homophily level is irrelevant to this boundary.
  • Because the splitting coefficients are computed from the endpoints' embeddings, they could serve as a reusable edge-relevance score, for example to prune graphs or to flag distribution shift between training and deployment graphs—uses the paper does not explore.
  • The robustness to test-time edge noise follows directly from discarding edges at inference, so a fair comparison against other edge-free models such as LINKX on noisy graphs would be needed to isolate how much of that robustness is due to the edge-splitting loss rather than the absence of edges.
  • Extending the splitting to directed edges, which the paper lists as future work, could give the model a way to encode asymmetric neighborhood patterns; directed splitting coefficients would be a natural next experiment on heterophilic benchmarks.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper proposes ES-MLP, a node-classification model that combines the edge-splitting mechanism of ES-GNN with the neighborhood-contrastive training of Graph-MLP. During training, node features are projected into task-relevant and task-irrelevant subspaces, the adjacency matrix is split into two exclusive weighted matrices, and the model is trained with a combined cross-entropy, neighborhood-contrastive, and irrelevant-consistency loss. At inference only the MLP on the task-relevant representation is used, so no adjacency matrix is required. The authors evaluate ES-MLP on seven real-world datasets and CSBM synthetic graphs, compare with six baselines, and report competitive accuracy on the homophilic citation graphs, best results on Actor, Roman, and Amazon, robustness to edge noise, and faster inference than MPNNs.

Significance. ES-MLP is a sensible combination of two existing ideas, and the empirical study includes useful elements: ablations of the two auxiliary losses, hyperparameter sensitivity analysis, robustness experiments, and an inference-time comparison on full-graph versus test-only inputs. If the claims are properly scoped, the paper gives evidence that an MLP trained with edge-splitting can match or exceed message-passing baselines on several heterophilic benchmarks while keeping the inference-time advantages of a pure MLP. The release of source code is a further strength. The main weaknesses are that the headline 'all datasets' claim is false as stated and one baseline appears to be configured in a way that weakens the comparison; both are fixable.

major comments (3)
  1. [Abstract, Table 2, Section 6] The abstract's statement that ES-MLP 'is on par with homophilic and heterophilic models on all datasets' is contradicted by Table 2 on Minesweeper: ES-MLP obtains 50.87 ± 2.03 AUROC, essentially chance, while GraphSAGE obtains 88.90 ± 2.37. Section 6 itself explains that this is because 'the node features are independent of the class of a node' and MLP-based models have no access to neighbor features at inference. This is a hard boundary of the model class, not a minor benchmark miss, and it should be stated explicitly in the title and abstract, e.g., by scoping the claim to settings where node features carry label information.
  2. [Appendix A Table 10 vs Table 2 and Section 6] For Actor and Amazon, the reported Graph-MLP configuration sets α_NC = 0, i.e., the neighborhood-contrastive term, which is the component that distinguishes Graph-MLP from an MLP, is disabled on exactly the heterophilic datasets used to argue that Graph-MLP underperforms. The discussion in Section 6 that Graph-MLP 'cannot compete on the heterophilic dataset, since the original neighborhood contrastive loss is based on the homophily assumption' is therefore not supported by the Actor/Amazon experiments. The authors should either report Graph-MLP with a nonzero α_NC on these datasets or explain why α_NC = 0 was chosen and how the Amazon result (Graph-MLP 37.07 vs MLP 44.05) arises under the stated reuse of MLP hyperparameters.
  3. [Section A.2 and Tables 9, 11] Section A.2 states that the adjacency power r is searched over {1, 2, 3}, but Table 9 reports r = 4 for Cora and Minesweeper. Table 11 also contains a malformed entry ('0.5.7') in the Amazon row. Because the reported results depend on these hyperparameters, the tables must be corrected and the search range reconciled; otherwise the experiments cannot be reproduced from the description.
minor comments (5)
  1. [References] References [21] and [23] are the same paper (Lim et al., 'Large scale learning on non-homophilous graphs'), which should be merged or cited consistently.
  2. [Section 3] The word 'seperately' should be 'separately' in the paragraph describing the per-channel forward pass.
  3. [Appendix C] The text says 'reporter in Table 12' and 'best results are marked in bolt'; these should read 'reported' and 'bold'.
  4. [Section 6] The sentence 'The difference to the best-performing models GCN and Graph-MLP is between 0.31 and 2.07 points' understates the CiteSeer gap: Graph-MLP achieves 77.79 and ES-MLP 75.67, a difference of 2.12 points.
  5. [Figure 3] The x-axis labels give homophily values but not the corresponding amount of edge noise; please add the noise levels or describe the mapping in the caption.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: empirical architecture paper with externally cited components and an honestly stated scope limitation, not a circular derivation.

full rationale

The paper makes no pretense of deriving a prediction from first principles; it is an empirical architecture proposal. The model combines the neighborhood contrastive loss of Graph-MLP (Hu et al.) with the edge-splitting channel and ICR loss of ES-GNN (Guo et al.), both cited as external prior work. None of the authors' own prior work is invoked as load-bearing, and there is no uniqueness theorem or ansatz smuggled in via self-citation. The key empirical claims (test accuracy on seven datasets, inference time, robustness to test-time edge noise) are evaluated against held-out splits and baselines, not generated by the model's defining equations. The only self-referential mechanism is the ICR loss weighting by the model's own predicted label vectors, but that is a training regularizer and does not define the final prediction, so it is not a derivation cycle. The paper also explicitly scopes its own validity: in Section 6 it states that on Minesweeper 'the node features are independent of the class of a node... Since MLP-based models do not have access to these features during inference time, they are unable to classify these nodes correctly,' and the Limitations paragraph repeats that ES-MLP 'cannot be applied to tasks where the class depends on the neighborhood structure during inference time.' That is an honest boundary statement rather than a circular justification. The abstract's 'on par ... on all datasets' is overstated given Table 2 (ES-MLP 50.87 AUROC vs. GraphSAGE 88.90 on Minesweeper), but overclaiming is a correctness and scope issue, not circularity. Accordingly, no circular step can be quoted and the score is 0.

Assumptions & free parameters 5 free parameters · 4 assumptions · 0 invented entities

The paper's central claims are empirical and depend on per-dataset hyperparameters, benchmark data, and the domain assumption that labels are recoverable from node features alone at inference. No new physical or mathematical entities are introduced; edge-splitting and neighborhood contrastive loss are adopted from prior work. The strongest limitation, that MLP-based models cannot classify nodes whose labels depend on neighbor features, is stated by the authors in the Limitations section.

free parameters (5)
  • adjacency_power_r = Cora:4, CiteSeer:2, PubMed:2, Actor:3, Amazon:3, Roman:1, Minesweeper:4
    Grid-searched per dataset (Table 9). Controls the r-hop neighborhood used in the neighborhood contrastive loss and in computing gamma_s(i,j).
  • neighborhood_contrastive_weight_alpha_NC = 1 for most datasets; 100 for Minesweeper
    Loss weight in L_final; tuned per dataset (Table 9).
  • irrelevant_consistency_weight_beta_ICR = varies: Cora 0.01, CiteSeer 0.0, PubMed 0.0, Actor 0.0001, Amazon 0.0001, Roman 0.00001, Minesweeper 0.01
    Tuned per dataset (Table 9); ablation shows it matters only on heterophilic datasets.
  • skip_connection_weights_epsilon_R_and_epsilon_IR = varies per dataset, e.g. Cora 0.5/0.5, Actor 0.0/0.7
    Tuned per dataset (Table 9).
  • base_hyperparameters = learning rate, hidden dim, dropout, weight decay per dataset (Tables 5-8)
    Standard model hyperparameters, tuned per dataset; the reported accuracies depend on these choices.
assumptions (4)
  • standard math Powers of the adjacency matrix encode r-hop neighborhood reachability.
    Used to define gamma_s(i,j) in the neighborhood contrastive loss (Section 3).
  • domain assumption Transductive setting, with test nodes present during training.
    Stated at the start of Section 3.
  • domain assumption Node labels are recoverable from node features alone at inference time.
    The design of ES-MLP uses no edges at inference; Section 6 Limitations acknowledges this fails on Minesweeper.
  • domain assumption Soft labels from the model can be used to weight the irrelevant consistency loss.
    The ICR loss uses predicted probability vectors y_hat to weight Z_IR distances (Section 3).

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Cite this review

Pith. "Pith review of Edge-Splitting MLP: Node Classification on Homophilic and Heterophilic Graphs without Message Passing." pith.science (2026). https://pith.science/paper/ZSNTN2JR

@misc{pith2026241208310,
  author       = {Pith},
  title        = {Pith review of: Edge-Splitting MLP: Node Classification on Homophilic and Heterophilic Graphs without Message Passing},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZSNTN2JR}},
  note         = {Machine review of arXiv:2412.08310}
}
read the original abstract

Message Passing Neural Networks (MPNNs) have demonstrated remarkable success in node classification on homophilic graphs. It has been shown that they do not solely rely on homophily but on neighborhood distributions of nodes, i.e., consistency of the neighborhood label distribution within the same class. MLP-based models do not use message passing, \eg Graph-MLP incorporates the neighborhood in a separate loss function. These models are faster and more robust to edge noise. Graph-MLP maps adjacent nodes closer in the embedding space but is unaware of the neighborhood pattern of the labels, i.e., relies solely on homophily. Edge Splitting GNN (ES-GNN) is a model specialized for heterophilic graphs and splits the edges into task-relevant and task-irrelevant, respectively. To mitigate the limitations of Graph-MLP on heterophilic graphs, we propose ES-MLP that combines Graph-MLP with an edge-splitting mechanism from ES-GNN. It incorporates the edge splitting into the loss of Graph-MLP to learn two separate adjacency matrices based on relevant and irrelevant feature pairs. Our experiments on seven datasets with six baselines show that ES-MLP is on par with homophilic and heterophilic models on all datasets without using edges during inference. We show that ES-MLP is robust to multiple types of edge noise during inference and that its inference time is two to five times faster than that of commonly used MPNNs. The source code is available at https://github.com/MatthiasKohn/ES-MLP.

Figures

Figures reproduced from arXiv: 2412.08310 by the authors.

Figure 1
Figure 1. The pipeline of ES-MLP. Node features are first projected into two different subspaces R and IR. A separate forward pass is performed to obtain embeddings ZR and ZIR. Both embeddings are then used for edge splitting to divide the original graph edges into two sets and for the neighbor￾hood contrastive loss. The task-relevant representation ZR is then used for the prediction task and task-irrelevant representation ZI… view at source ↗
Figure 2
Figure 2. Test accuracies for the CSBM datasets (left). The scores are averaged over 5 runs. The probability for intra-class edges is on the x-axis and the probability for inter-class edges is on the y-axis. The adjusted homophily ratios Hadj for the CSBM datasets (right). Inference Time. The results for the inference times are presented in [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Test accuracies of the robustness analysis on the Cora and Amazon datasets. Left to right: Results on Cora with uniform edge-noise, Cora with categorical edge-noise, Amazon with Uniform edge-noise, and Amazon with categorical edge-noise. Our ES-MLP, as it does not require edges at test time, remains constant on all datasets. Graph-MLP, not shown for brevity, also remains constant. 6 Discussion The results of our exp… view at source ↗
Figures from the paper (7 more)
Figure 4
Figure 4. Figure 4: Results for the edge-noise experiment on Cora with uniform (left) and categorical (right) noise. 17 [PITH_FULL_IMAGE:figures/full_fig_p017_4.png]
Figure 5
Figure 5. Figure 5: Hyperparameter sensitivity analysis on the neighborhood contrastive loss weight αNC hyperparameter 0.0 1e-05 0.0001 0.01 ICR on Cora 0.0 0.2 0.4 0.6 0.8 1.0 Test Accuracy 0.0 1e-05 0.0001 0.01 ICR on CiteSeer 0.0 0.2 0.4 0.6 0.8 1.0 Test Accuracy 0.0 1e-05 0.0001 0.01 …
Figure 6
Figure 6. Figure 6: Hyperparameter sensitivity analysis on the hyperparameter ICR loss weight βICR. 18 [PITH_FULL_IMAGE:figures/full_fig_p018_6.png]
Figure 7
Figure 7. Figure 7: Performance for ES-MLP and baseline models on Cora with generated edge noise on the test nodes with global homophily ratios {0.35, 0.43, 0.58, 0.76}. Results are reported for different local homophily ranges. J Adjacency Matrix Visualization We aggregated the r-th powe…
Figure 8
Figure 8. Figure 8: Adjacency Matrices Ar and Air on Cora. The ratio of homophilic edges and all edges in Ar is 77.53%, and in Air 66.82% We observe that Ar has a higher ratio of edges on the diagonal than AIR, i. e., the learned adjacency matrix is more homophilic. For Actor, the model l…
Figure 9
Figure 9. Figure 9: Adjacency Matrices Ar and Air on Actor. The ratio of homophilic edges and all edges in Ar is 32.33%5, and in Air 21.91% K Extended CSBM Results We provide additional homophily measures for the CSBM datasets measured with the edge homophily, node homophily, and class-in…
Figure 10
Figure 10. Figure 10: The edge homophily (top-left), node homophily,(top-right), and class-insensitive ho￾mophily (bottom) ratios of the CSBM dataset. 20 [PITH_FULL_IMAGE:figures/full_fig_p020_10.png]

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Reviewed August 11, 2026 · model on record in the stance chip above.