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arxiv: 2511.06443 · v1 · submitted 2025-11-09 · 💻 cs.LG

How Wide and How Deep? Mitigating Over-Squashing of GNNs via Channel Capacity Constrained Estimation

Zinuo You , Jin Zheng , John Cartlidge This is my paper

Pith reviewed 2026-05-17 23:19 UTC · model grok-4.3

classification 💻 cs.LG
keywords graph neural networksover-squashingchannel capacityinformation theoryhidden dimensionspropagation depthspectral GNNrepresentation learning
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The pith

Channel capacity estimation selects hidden dimensions and depths to mitigate over-squashing in GNNs.

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

The paper establishes a framework called C3E that selects hidden dimensions and propagation depths for spectral graph neural networks by modeling them as communication channels and solving a nonlinear program to optimize channel capacity. A sympathetic reader would care because typical heuristic choices for these settings often produce severe loss of information when signals propagate across graphs. The work shows that over-squashing arises specifically from cumulative compression inside the evolving representation matrices. Experiments across nine datasets indicate that wider hidden layers reliably lessen this compression while depth requires balancing compression against the growth in representational power.

Core claim

Spectral graph neural networks can be treated as communication channels whose capacity depends on hidden dimensions, propagation depth, the chosen propagation mechanism, and the structure of the input graph. Formulating the choice of dimensions and depth as a nonlinear programming problem that maximizes this capacity yields architectural settings that reduce the cumulative compression of information in representation matrices. Larger hidden dimensions directly counteract compression, whereas depth exhibits a more nuanced effect that reflects a trade-off between added compression and increased representation complexity.

What carries the argument

Channel Capacity Constrained Estimation (C3E), a nonlinear programming procedure that treats GNN propagation steps as a communication channel to compute suitable hidden dimensions and depth.

If this is right

  • Over-squashing occurs through the progressive compression of information inside successive representation matrices.
  • Larger hidden dimensions reduce the amount of information lost to compression at each step.
  • Propagation depth must be chosen to balance further compression against gains in representational complexity.
  • Settings produced by the capacity-constrained optimization improve representation learning on the evaluated datasets.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same capacity-modeling idea could be applied to message-passing rules in non-spectral GNNs if an analogous channel description can be written down.
  • For graphs with long diameters the depth trade-off might favor shallower networks paired with wider layers.
  • Running the nonlinear program on graphs with controlled changes in diameter or density would test how sensitive the resulting choices are to structure.

Load-bearing premise

Modeling spectral graph neural networks as communication channels whose capacity is determined by hidden dimensions, depth, and graph structure accurately reflects actual information flow during propagation.

What would settle it

Finding that architectures chosen by the nonlinear program produce no measurable drop in over-squashing or no gain in representation quality on a fresh collection of graphs would show the capacity model does not predict real behavior.

Figures

Figures reproduced from arXiv: 2511.06443 by Jin Zheng, John Cartlidge, Zinuo You.

Figure 1
Figure 1. Figure 1: The green-axis (left) denotes performance, and blue-axis (right) denotes parameters counts (in millions). C3E￾estimated solutions (red points, starred for optimal) consistently land in high-performance regions within dashed intervals defined in Eq. (12), outperforming baselines with heuristic dimensions (green points, e.g., 16, 32, 64, 128, 256, 512, 1024). Statistics Cora Citeseer Pubmed AmazonPhoto Amazo… view at source ↗
Figure 2
Figure 2. Figure 2: C3E-estimated models (red lines) avoid information loss by maintaining high H(Hl) across layers. In contrast, naively stacked baselines (green lines) suffer from over-squashing, with entropy collapsing to near-zero long before the final layer. Dotted lines indicate the maximum entropy for initial feature dimensions (brown) and fixed hidden dimensions (blue). 5.3 Entropy Transitions in Representation Matrix… view at source ↗
Figure 3
Figure 3. Figure 3: Average performance of C3E-estimated models versus η across nine benchmark datasets. Each curve shows the averaged C3E-estimated model performance per dataset. Page 19 of 29 [PITH_FULL_IMAGE:figures/full_fig_p019_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Examples of C3E solutions (hidden dimensions) on five benchmark datasets. Page 21 of 29 [PITH_FULL_IMAGE:figures/full_fig_p021_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The relationship between propagation depth ( [PITH_FULL_IMAGE:figures/full_fig_p022_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Learnable weight matrix entry distribution on Cora, [PITH_FULL_IMAGE:figures/full_fig_p024_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Representation matrix entry distribution on Cora, [PITH_FULL_IMAGE:figures/full_fig_p025_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Learnable weight matrix entry distribution on AmazonPhoto, [PITH_FULL_IMAGE:figures/full_fig_p026_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Representation matrix entry distribution on AmazonPhoto, [PITH_FULL_IMAGE:figures/full_fig_p027_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Learnable weight matrix entry distribution on AmazonComputers, [PITH_FULL_IMAGE:figures/full_fig_p028_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Representation matrix entry distribution on AmazonComputers, [PITH_FULL_IMAGE:figures/full_fig_p029_11.png] view at source ↗
read the original abstract

Existing graph neural networks typically rely on heuristic choices for hidden dimensions and propagation depths, which often lead to severe information loss during propagation, known as over-squashing. To address this issue, we propose Channel Capacity Constrained Estimation (C3E), a novel framework that formulates the selection of hidden dimensions and depth as a nonlinear programming problem grounded in information theory. Through modeling spectral graph neural networks as communication channels, our approach directly connects channel capacity to hidden dimensions, propagation depth, propagation mechanism, and graph structure. Extensive experiments on nine public datasets demonstrate that hidden dimensions and depths estimated by C3E can mitigate over-squashing and consistently improve representation learning. Experimental results show that over-squashing occurs due to the cumulative compression of information in representation matrices. Furthermore, our findings show that increasing hidden dimensions indeed mitigate information compression, while the role of propagation depth is more nuanced, uncovering a fundamental balance between information compression and representation complexity.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The manuscript proposes Channel Capacity Constrained Estimation (C3E), a nonlinear programming framework that selects hidden dimensions and propagation depths for spectral GNNs by modeling them as communication channels whose capacity depends on graph structure, depth, dimensions, and propagation mechanism. It attributes over-squashing to cumulative compression of information in representation matrices and reports that C3E-derived choices mitigate this effect while improving representation learning on nine public datasets; depth is characterized as playing a nuanced role that balances compression against representation complexity.

Significance. If the channel-capacity formulation accurately predicts information flow and the resulting parameters generalize, the work would supply a principled, information-theoretic alternative to heuristic choices of width and depth in GNNs, together with empirical evidence linking architectural decisions to measurable compression in representation matrices. The multi-dataset evaluation is a positive feature, though its interpretive weight hinges on the validity of the underlying capacity model.

major comments (3)
  1. [Abstract and §3] Abstract and §3: the nonlinear program is presented as directly connecting channel capacity to hidden dimension, depth, and graph structure, yet the manuscript supplies no explicit capacity formula, no description of how the program is solved, and no statement of the precise constraints or scaling constants employed; without these details the central claim that the selected parameters reduce observed compression cannot be verified.
  2. [§4] §4 (experimental results): consistent gains are reported across nine datasets, but the text does not indicate whether error bars, multiple random seeds, or statistical significance tests accompany the improvements; this omission weakens the assertion that C3E “consistently” mitigates over-squashing.
  3. [§3.2] §3.2 (capacity model): the formulation appears to treat each layer as an additive or linear channel whose mutual information scales directly with dimension and depth, yet the paper itself describes over-squashing as arising from nonlinear activations and repeated Laplacian applications; the mapping from empirical compression in representation matrices to the capacity objective is therefore not secured.
minor comments (2)
  1. [§3] Notation for the channel-capacity threshold and any scaling constants should be introduced with explicit symbols and distinguished from free parameters.
  2. [§3] A brief discussion of how graph-specific quantities (e.g., eigenvalues or effective diameter) are estimated without introducing circularity relative to the downstream task would improve clarity.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive feedback on our manuscript. We address each major comment below and have made revisions to strengthen the presentation of the C3E framework and its empirical validation.

read point-by-point responses

  1. Referee: [Abstract and §3] Abstract and §3: the nonlinear program is presented as directly connecting channel capacity to hidden dimension, depth, and graph structure, yet the manuscript supplies no explicit capacity formula, no description of how the program is solved, and no statement of the precise constraints or scaling constants employed; without these details the central claim that the selected parameters reduce observed compression cannot be verified.

    Authors: We appreciate this observation and agree that additional mathematical detail is necessary for verifiability. The original submission presented the nonlinear program at a high level without the full capacity expression or solver specifics. In the revised manuscript we have inserted the explicit channel-capacity formula (derived from the mutual information between input features and propagated representations under the spectral filter), the complete nonlinear program with all equality and inequality constraints, the chosen scaling constants, and a description of the interior-point solver together with convergence criteria. These additions directly support the claim that the optimized parameters reduce measured compression in the representation matrices. revision: yes

  2. Referee: [§4] §4 (experimental results): consistent gains are reported across nine datasets, but the text does not indicate whether error bars, multiple random seeds, or statistical significance tests accompany the improvements; this omission weakens the assertion that C3E “consistently” mitigates over-squashing.

    Authors: We concur that statistical reporting is essential to substantiate consistency claims. We have re-executed all experiments using ten independent random seeds, added standard-deviation error bars to every table and figure in §4, and included paired t-test p-values comparing C3E-derived configurations against baselines. The revised results confirm that the observed improvements remain statistically significant (p < 0.05) across the nine datasets, thereby reinforcing the assertion that C3E mitigates over-squashing. revision: yes

  3. Referee: [§3.2] §3.2 (capacity model): the formulation appears to treat each layer as an additive or linear channel whose mutual information scales directly with dimension and depth, yet the paper itself describes over-squashing as arising from nonlinear activations and repeated Laplacian applications; the mapping from empirical compression in representation matrices to the capacity objective is therefore not secured.

    Authors: We thank the referee for this precise critique of the modeling assumptions. Although over-squashing originates from nonlinear activations and iterated Laplacian multiplications, the C3E capacity model employs an effective information-theoretic upper bound that is calibrated against the empirically observed compression in the final representation matrices. In the revision we have augmented §3.2 with a short derivation showing how the spectral capacity expression bounds the end-to-end mutual information even in the presence of nonlinearities, together with additional plots that correlate the capacity objective directly with measured compression metrics. We believe these clarifications secure the mapping while preserving the abstraction’s utility. revision: partial

Circularity Check

0 steps flagged

No significant circularity: derivation grounded in external information-theoretic model

full rationale

The paper formulates hidden dimension and depth selection as a nonlinear program that maximizes channel capacity under a communication-channel model of spectral GNN propagation. This objective is defined from graph structure, propagation mechanism, and information-theoretic capacity expressions rather than from the downstream task loss or representation quality metrics used in evaluation. No equations reduce the chosen parameters to fitted values of the target performance; the optimization is solved once per graph and then applied to GNN training. No self-citation chains, self-definitional loops, or renaming of known empirical patterns appear in the derivation. The approach is therefore self-contained against external benchmarks and receives the default non-circularity finding.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on treating spectral GNNs as communication channels whose capacity can be computed from dimensions, depth, and graph structure; this modeling step is an unproven domain assumption rather than a derived result.

free parameters (1)
  • channel capacity threshold or scaling constants
    The nonlinear program necessarily introduces at least one tunable threshold or scaling factor to balance compression against representation complexity; its value is not derived from first principles.
axioms (1)
  • domain assumption Spectral graph neural networks can be modeled as communication channels whose capacity is a direct function of hidden dimension, propagation depth, propagation mechanism, and graph structure.
    This modeling choice is invoked to turn architecture selection into a capacity-constrained optimization problem.

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