ScaleNet: Scale Invariance Learning in Directed Graphs

Chengjia Wang; Michael Lones; Qin Jiang; Wei Pang; Yingfang Yuan

REVIEW 4 major objections 4 minor 44 references

ScaleNet: Scale Invariance Learning in Directed Graphs

T0 review · 4 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read Node classification in directed graphs is scale-invariant across adjacency-matrix products, the paper claims, and a model built on this invariance, ScaleNet, tops five of seven benchmarks.

desk verdict Solid empirical architecture with an unsupported central invariance claim; worth reviewing but needs major reframing. read the letter →

arxiv 2411.08758 v3 pith:V5IIXQHQ submitted 2024-11-13 cs.LG

Qin Jiang , Chengjia Wang , Michael Lones , Yingfang Yuan , Wei Pang This is my paper · ORCID

classification cs.LG

keywords scaleinvariancedirectedgraphsego-graphsnodeclassificationhomophilyheterophilygraphneuralnetworksmulti-scaleaggregation

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

Inside a GNN, the object being classified is the ego-graph of the center node. The paper extends image-style scale invariance to directed graphs by defining scaled ego-graphs, built from ordered sequences of directed edges encoded as products of the adjacency matrix $A$ and its transpose $A^T$. It claims and empirically supports that these scaled versions preserve node labels: for any $k\ge 1$, $f(G_v)=f(G_k(v))$. If this scale invariance holds, the products of adjacency matrices can be fused without destroying label information, which explains why DiGCN and SymDiGCN succeed on homophilic graphs and why they fail on heterophilic ones. The paper then introduces ScaleNet, which flexibly combines scaled graphs with a directional weight parameter and achieves state-of-the-art accuracy on five of seven datasets while matching the top model on the other two.

What carries the argument

The central object is the scaled adjacency matrix set $\mathcal{A}^k = \{a_1 a_2 \cdots a_k \mid a_i \in \{A, A^T\}\}$, whose matrices encode $k$th-scale edges (ordered sequences of directed in- and out-hops); a scaled ego-graph is the ego-graph reachable through one such scaled-edge type. Definition 7, $f(G_v) = f(G_k(v))$, is the load-bearing identity: it is what licenses fusing multiple scaled graphs without destroying label information. The architecture that exploits it is the bidirectional aggregation block $AGG\text{-}B_\alpha(M,N,X) = (1+\alpha)\alpha\, AGG(M,X) + (1+\alpha)(1-\alpha)\, AGG(N,X)$, whose parameter $\alpha$ selects, balances, unions, or intersects opposite-direction scaled matrices, combined layer-wise by a COMB function (Jumping Knowledge or addition) and optionally with self-loops, batch norm, and nonlinearities.

What would settle it

Train a single GNN on the original adjacency matrix $A$ for Chameleon or Squirrel, then apply it without any retraining to the scaled matrices $AAT$, $AA$, $A^T A$, and $A^T A^T$; if accuracy on any scale drops toward the no-input baseline (roughly 22%–38%), the invariance $f(G_v)=f(G_k(v))$ is false for that dataset, regardless of the comparable separately trained numbers in Table 1.

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Extended reading notes

Core claim

The paper's central claim is Definition 7: for any $k \ge 1$, $f(G_v) = f(G_k(v))$, meaning the classification of a node does not change when its ego-graph is replaced by a $k$th-scale ego-graph formed from ordered sequences of in- and out-edge hops. Table 1 shows, on seven benchmark datasets, that higher-scale graphs reach accuracies comparable to their lower-scale counterparts, even after removing edges shared with the lower scale; the paper reads this as confirming scale invariance. It further finds that on homophilic graphs both directions $A$ and $A^T$ carry signal, while on heterophilic graphs $A$-based scales preserve performance but $A^T$-based scales degrade because many nodes have no in-neighbors. From this it concludes that the success of digraph inception models rests on scale-invariant multi-scale fusion, that their expensive edge-weight computation is unnecessary (constant weight 1 or even random weights match or beat it), and that a direction-aware fusion of scaled graphs, ScaleNet, unifies homophilic and heterophilic node classification.

Load-bearing premise

The paper's evidence for Definition 7 is that separately trained GNNs reach similar accuracy on different scaled graphs; it assumes this implies one and the same classifier would label original and scaled ego-graphs identically, but Table 1 never shows that.

Editorial extensions

If this is right

Scale invariance makes multi-scale adjacency products safe to combine, explaining the empirical success of DiGCN and SymDiGCN on homophilic graphs.
The expensive eigenvalue- and normalization-based edge weights in those methods are unnecessary; constant unit weights give equal or better accuracy, and random weights can also outperform the learned ones.
On heterophilic graphs scale invariance is directional: $A$, $AA$, and $AAT$ preserve information, while $A^T$, $A^T A$, and $A^T A^T$ inject noise and zeros; excluding the bad direction recovers performance.
A single tunable architecture, ScaleNet, can thereby match or beat specialized models on both graph types, reaching state-of-the-art accuracy on five of seven datasets and matching on the remaining two.
Constant-weight scaled edges avoid the out-of-memory failures of SymDiGCN and DiGCN on larger datasets, improving scalability.

Reading between the lines

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

The paper's Table 1 trains a separate classifier per scale, so it does not directly test whether one fixed classifier labels original and scaled ego-graphs identically; verifying Definition 7 would require training once on $A$ and evaluating on the scaled matrices without retraining.
The consistent $A$-versus-$A^T$ asymmetry on heterophilic graphs suggests the grid-searched direction parameters $\alpha, \beta, \gamma$ could be replaced by a simple data statistic, such as the share of nodes with zero in-neighbors.
If constant-weight scaled edges keep beating random-walk-weighted ones, the label signal in higher-order scales may be purely topological; other proximity-based GNNs could similarly drop their spectral-weight machinery.
If scale invariance is confirmed for a fixed classifier, scaled ego-graphs become a principled, non-random data-augmentation scheme for node classification, with potential benefits for robustness to distribution shift.

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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

Dear [Colleague],

The central conceptual claim in this paper does not survive scrutiny, but the empirical work underneath is real and worth a referee's time. Scale invariance as stated in Definition 7—f(G_v) = f(G^k(v)) for the same classifier f—is asserted, not established. Table 1 compares separately trained models on single splits; that only shows that some classifier can do well at each scale, not that a single classifier is invariant. The stress-test note is on target: Section 3.2's 'confirming scale invariance' is a non sequitur. The paper's own reproducibility checklist says proofs of all novel claims are included, yet no proof of Definition 7 appears. This matters because the motivation for fusing scaled graphs rests on that equality.

What is genuinely new: the weight-1 ablation. Replacing DiGCN's expensive edge weights with a constant weight of 1 improves accuracy on several datasets, including Telegram, Chameleon, and Squirrel, and is simpler and faster. That is a clean, reproducible empirical finding, and the code is available. The flexible bidirectional aggregation (per-dataset alpha, beta, gamma) is also new, and the tuning patterns are sensible—homophilic graphs benefit from self-loops and bidirectional aggregation, heterophilic graphs prefer removing self-loops and dropping opposite-direction scaled edges. The Wilcoxon signed-rank tests on 30 splits are more careful than typical in this subfield.

Soft spots, in proportion: the abstract says seven datasets while the experiments use six—a simple but embarrassing inconsistency. The grid search over many per-dataset hyperparameters risks overfitting, though the significance tests mitigate that. And the invariance framing, as a formal property, is unsupported. It could be reframed as 'classification performance is roughly preserved across scales,' which is what Table 1 actually shows.

Who is this for? Anyone working on directed GNNs or heterophilic graphs will get value from the architecture and the weight-1 result. The paper deserves peer review, but the referee should focus on the invariance claim and demand either a direct test of Definition 7 or an explicit weakening of the claim. I would not desk-reject it.

Referee Report

4 major / 4 minor

Summary. The paper introduces the concept of "scaled ego-graphs" for directed graphs, obtained by ordered products of the adjacency matrix A and its transpose A^T. It claims that node classification is scale-invariant (Definition 7, Eq. 5), argues that this invariance explains the success of digraph inception models such as DiGCN and SymDiGCN, and proposes ScaleNet, a multi-scale architecture that combines features from these scaled graphs. The paper reports experiments on seven datasets (four homophilic, two heterophilic, plus an imbalanced setting), claiming state-of-the-art or tied-best accuracy on all datasets, and also proposes simplified inception variants with unit edge weights that outperform the original weighted versions. The code and detailed hyperparameter settings are provided.

Significance. If the scale-invariance claim were rigorously established, it would provide a principled explanation for the empirical success of multi-scale digraph models and motivate a unified treatment of homophilic and heterophilic graphs. The practical findings—that replacing DiGCN's expensive edge weights with constant weights improves or matches performance, and that ScaleNet achieves strong results across both graph types—are valuable and potentially publishable. The paper includes a reproducibility-conscious appendix with code, hyperparameter grids, and Wilcoxon signed-rank tests for the main comparisons. However, the central conceptual claim of scale invariance is not adequately supported by the evidence, which limits the significance of the theoretical framing.

major comments (4)

[Definition 7, Eq. (5) and Section 3.2] The invariance equality f(G_v) = f(G^k(v)) is not tested. Table 1 reports accuracies of separately trained GNNs on different scaled adjacency matrices, each on a single split; this does not show that the same classifier f labels the original and scaled ego-graphs identically. As written, Eq. (5) is not a well-defined property of the graph unless f is specified (e.g., a fixed trained model, an optimal classifier, or an explicit architecture family). The statement in Section 3.2 that "higher-scale graphs consistently achieve performance comparable to their lower-scale counterparts, confirming scale invariance" is not a valid inference from those comparisons. Please either specify f and test same-classifier invariance (for example, train on one scale and evaluate on another, or compare the predictions of a single model on original versus scaled ego-graphs), or reformulate the claim as "classification performance is roughly preserved across scales" and adjust the motivation accordingly.
[Table 1, heterophilic rows] The heterophilic results in Table 1 show strong scale dependence that contradicts the invariance claim. For Chameleon, A achieves 78 while AT achieves 30 (near the no-input baseline of 22), and AAT/AA achieve 68/70 while AT A/AT AT achieve 29/30. This is the opposite of f(G_v) = f(G^k(v)) for all k and all scaled variants; at best it shows that some scaled graphs preserve the discriminative information of A while others discard it. Please address this directly, either by testing Eq. (5) at the node level or by explicitly restricting the claimed invariance to the scales that preserve information, with a rationale for why those scales are the relevant ones.
[Section 4.2, Eq. (6)] The formula AGG-Bα(M,N,X) = (1+α)α AGG(M,X) + (1+α)(1−α) AGG(N,X) does not match the following sentence. For α=0, the formula gives weight 0 on M and weight 1 on N (i.e., it uses only N), whereas the text states "α=0 uses only M"; for α=1, the formula gives weight 2 on M and 0 on N (i.e., only M), whereas the text states "α=1 uses only N." Please correct either the formula or the text, and clarify how the reported α values in Table 4 (including the special modes α=2 and α=3 defined in Eqs. (7)-(8)) map to the formula, since the current inconsistency makes the method ambiguous.
[Appendix D, Reproducibility Checklist] The checklist states that "Proofs of all novel claims are included" and that "All theoretical claims are demonstrated empirically to hold," but no proof or direct verification of Definition 7 appears anywhere in the manuscript. The only evidence is the Table 1 accuracy comparison, which, as noted above, does not establish Eq. (5). Please either supply a proof for a clearly specified f, or correct the checklist to reflect that the scale-invariance claim is a hypothesis supported only indirectly by accuracy comparisons rather than a proven theoretical claim.

minor comments (4)

[Figures 2 and 4] Figure 2 and Figure 4 appear to be the same figure with slightly different captions; please unify them or differentiate their content to avoid redundancy.
[Appendix D] The word "satisficing" in the dataset checklist should be "satisfying" (or "satisfactory").
[Table 3 footnote] The footnote "When accuracy is below 45%, only one split is used" means that some entries (e.g., Dir-GNN on CiteSeer) are single-split results; please state this in the main text and clearly mark single-split entries so that the table does not mix single-split and multi-split statistics without indication.
[Table 1] The final column labeled "None" is described as "all zero input"; consider renaming it to "Zero input" for clarity, and consider reporting multiple splits or confidence intervals for the single-split accuracy columns, since the invariance claim rests entirely on these numbers.

Circularity Check

1 steps flagged · score 4.0 of 10

Scale invariance is asserted from the same per-scale GNN accuracy comparisons that the architecture is built to exploit, so the explanation of inception models is partly a renaming of that observed pattern; ScaleNet's empirical performance itself remains an independent contribution.

renaming known result [Section 4.1 (Digraph Inception Networks), building on Section 3.2/Table 1]
"The essence of their success is the equivalent edge augmentation, as shown in Figure 2. ... Given the observed scale invariance across all tested datasets, combining these scaled graphs might further enhance the performance of GNNs."

The 'scale invariance' invoked to explain DiGCN/SymDiGCN is the paper's own Table 1 observation: each scaled adjacency matrix (A, AT, AA, AAT, ATA, ATAT) is fed to a separately trained GNN, and comparable accuracies are read as 'confirming scale invariance.' DiGCN's success is then attributed to the same edge augmentation built from those scaled matrices. The explanation thus reduces to the known per-scale accuracy pattern of DiGCN-like products relabeled as 'scale invariance'; no independent fixed classifier f from Eq. 5 or first-principles argument is supplied.

full rationale

The paper has no load-bearing self-citation chain: the references are external, and ScaleNet is evaluated against external baselines with released code. The main circularity concern is explanatory, not experimental. Definition 7 states scale invariance as f(G_v)=f(G^k(v)) for an unspecified f, and Section 3.2 claims Table 1 'confirms' it by training separate GNNs on each scaled adjacency matrix and observing similar accuracies. Section 4.1 then explains the success of DiGCN and SymDiGCN by 'the equivalent edge augmentation' and by the 'observed scale invariance.' Because DiGCN's k-th order proximity matrices are the same products used in Table 1, the explanation restates the per-scale accuracy pattern under the new label 'scale invariance' rather than deriving it from an independent fixed classifier or from Eq. 5. This is a moderate circularity in the conceptual narrative: the phenomenon and its explanation are the same empirical observation. However, ScaleNet's actual performance numbers and the comparisons to Dir-GNN and MagNet are independent empirical contributions and do not reduce to this loop, so the score is 4 rather than higher.

Assumptions & free parameters 3 free parameters · 3 assumptions · 1 invented entities

The central claim rests on a re-labeling of known higher-order adjacency products plus per-dataset hyperparameter choices. The free parameters are the aggregation weights and architecture options selected by grid search. The main axiom, that accuracy preservation demonstrates classifier invariance, is asserted rather than proved, and the invented 'scaled ego-graph' construct has no independent evidence outside this paper.

free parameters (3)

directional aggregation parameters (α, β, γ) = dataset-specific values: e.g., (0.5,-1,-1), (2,-1,-1), (1,1,1)
Chosen by grid search per dataset in Section 5.3 and Table 4. These parameters control which scaled graphs are combined, so the reported accuracy depends on fitting them to each dataset.
self-loop handling mode = add, remove, or none depending on dataset
Grid-searched in Appendix A.2. The paper states that adding self-loops helps homophilic graphs and removing them helps heterophilic graphs, so this choice materially affects results.
architecture option flags (batch norm, ReLU, jumping knowledge) = enabled or disabled per dataset
Grid-searched options listed in Appendix A.2; for example, CiteSeer performs better when non-linear activations are removed. These flags are part of the per-dataset tuning that produces the reported SOTA numbers.

assumptions (3)

domain assumption Node classification of a node v is classification of its ego-graph G_v.
Stated in the Introduction: "it is actually the ego-graph of the center node that is classified." This assumption grounds the entire scale-invariance framing.
domain assumption Products of A and A^T represent scaled edges and preserve enough label information for classification.
Used in Definition 6 and Table 1. The paper verifies this empirically rather than proving it, and the preservation of label information is the central claim.
ad hoc to paper Scale invariance holds for the evaluated graph datasets.
The architecture is justified by the claimed empirical invariance in Section 3.2. This is the paper's own assertion, inferred from accuracy comparisons, not independently established.

invented entities (1)

scaled-edge and scaled ego-graph
purpose: Formal device to reinterpret multi-hop directed paths as different scales of the same graph.
This is a new conceptual label for products of A and A^T already used by DiGCN(ib). No independent falsifiable prediction is offered beyond the paper's own benchmark results.

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Pith. "Pith review of ScaleNet: Scale Invariance Learning in Directed Graphs." pith.science (2026). https://pith.science/paper/V5IIXQHQ

@misc{pith2026241108758,
  author       = {Pith},
  title        = {Pith review of: ScaleNet: Scale Invariance Learning in Directed Graphs},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/V5IIXQHQ}},
  note         = {Machine review of arXiv:2411.08758}
}

read the original abstract

Graph Neural Networks (GNNs) have advanced relational data analysis but lack invariance learning techniques common in image classification. In node classification with GNNs, it is actually the ego-graph of the center node that is classified. This research extends the scale invariance concept to node classification by drawing an analogy to image processing: just as scale invariance being used in image classification to capture multi-scale features, we propose the concept of ``scaled ego-graphs''. Scaled ego-graphs generalize traditional ego-graphs by replacing undirected single-edges with ``scaled-edges'', which are ordered sequences of multiple directed edges. We empirically assess the performance of the proposed scale invariance in graphs on seven benchmark datasets, across both homophilic and heterophilic structures. Our scale-invariance-based graph learning outperforms inception models derived from random walks by being simpler, faster, and more accurate. The scale invariance explains inception models' success on homophilic graphs and limitations on heterophilic graphs. To ensure applicability of inception model to heterophilic graphs as well, we further present ScaleNet, an architecture that leverages multi-scaled features. ScaleNet achieves state-of-the-art results on five out of seven datasets (four homophilic and one heterophilic) and matches top performance on the remaining two, demonstrating its excellent applicability. This represents a significant advance in graph learning, offering a unified framework that enhances node classification across various graph types. Our code is available at https://github.com/Qin87/ScaleNet/tree/July25.

Figures

Figures reproduced from arXiv: 2411.08758 by the authors.

**Figure 2.** Edge augmentation by stacking multi-scale graphs [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 4.** Edge augmentation by stacking multi-scale graphs. [PITH_FULL_IMAGE:figures/full_fig_p015_4.png] view at source ↗

Figures from the paper (3 more)

**Figure 6.** Figure 6: These sets included two-peak and three-peak dis [PITH_FULL_IMAGE:figures/full_fig_p015_6.png]

**Figure 5.** Figure 5: DiG VS. RiG (a) Accuracy with randomly generated edge weights (2 peaks): 36.5 ± 4.0 (b) Accuracy with randomly generated edge weights (3 peaks): 72.6±4.9 [PITH_FULL_IMAGE:figures/full_fig_p016_5.png]

**Figure 6.** Figure 6: Random edge weights D Reproducibility Checklist This paper: • Includes a conceptual outline and/or pseudocode description of AI methods introduced (yes/partial/no/NA) yes • Clearly delineates statements that are opinions, hypothesis, and speculation from objective fa…

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, " * write output.state after.block = add.period write newline

ENTRY address archivePrefix author booktitle chapter edition editor eid eprint howpublished institution isbn journal key month note number organization pages publisher school series title type volume year label extra.label sort.label short.list INTEGERS output.state before.all...
[44]

write newline

" write newline "" before.all 'output.state := FUNCTION n.dashify 't := "" t empty not t #1 #1 substring "-" = t #1 #2 substring "--" = not "--" * t #2 global.max substring 't := t #1 #1 substring "-" = "-" * t #2 global.max substring 't := while if t #1 #1 substring * t #2 gl...

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

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V.; and Galstyan, A

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N.; and Giannakis, G

Berberidis, D.; Nikolakopoulos, A. N.; and Giannakis, G. B. 2019. Adaptive Diffusions for Scalable Learning Over Graphs . IEEE Transactions on Signal Processing, 67(5): 1307--1321

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arXiv 2020

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S.; and Welling, M

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