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REVIEW 4 major objections 5 minor 53 references

Local Virtual Nodes for Alleviating Over-Squashing in Graph Neural Networks

T0 review · 4 major / 5 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read Local virtual nodes relieve GNN over-squashing without rewiring

desk verdict Useful new LVN construction with honest empirical work, but the centrality-placement mechanism is unablated and the causal claim remains unsupported. read the letter →

arxiv 2508.20597 v1 pith:JMAMC72D submitted 2025-08-28 cs.LG

Tuğrul Hasan Karabulut , İnci M. Baytaş This is my paper · ORCID
classification cs.LG
keywords graphneuralnetworksover-squashinglocalvirtualnodesmessagepassingrewiringnodecentralityeffectiveresistancelong-rangedependencies
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 claims that a lightweight preprocessing step—replacing a handful of high-centrality nodes with small groups of virtual nodes, each with a trainable embedding—reduces over-squashing in message-passing GNNs. The virtual nodes inherit the edges of the central node they replace, adding extra pathways exactly where bottlenecks are likely; shared trainable embeddings let distant central regions exchange information without adding layers. On graph and node classification benchmarks, the authors report that this approach outperforms graph-rewiring baselines and a width-expansion baseline while leaving original edges and global topology intact. If correct, it offers a plug-in remedy for long-range tasks that works with any GNN architecture.

What carries the argument

The central object is the LVN group: for each of ns centrality-selected nodes, the original node is replaced by nc virtual nodes, each connected to the original node's neighbors (undirected strategy) or receiving all neighbor messages while sending to only a subset (directed strategy). The graph fed to the GNN is the induced subgraph after removing the central nodes, so no original edges are deleted. Each LVN is initialized with one row of a shared trainable embedding matrix P^{nc × D}; sharing across groups is what enables cross-region communication. The load-bearing formal identity is the augmented edge set, which merges the original edge set with copied central-node edges and, when two ce

What would settle it

Run the same LVN pipeline on graphs where true bottleneck regions are identified by curvature or effective resistance, with three placements: centrality, oracle bottlenecks, and random nodes. If random or oracle placements match or beat centrality, the centrality heuristic is not carrying the reported benefit; if centrality matches oracle, the method's placement assumption is confirmed.

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Extended reading notes

Core claim

The central claim is that augmenting a graph with Local Virtual Nodes (LVNs) at central regions, then training GNNs with shared embeddings for those virtual nodes, mitigates over-squashing both structurally and representationally. Structurally, each selected central node is replaced by a group of nc virtual nodes that copy its connections, increasing the number of paths through bottleneck regions and lowering total effective resistance among non-central nodes. Representationally, each LVN starts from one of nc trainable embeddings shared across all central regions; because those embeddings update from feedback in multiple graph regions, they carry long-range information independent of layer

Load-bearing premise

Centrality scores reliably mark the regions where bottlenecks actually form, so placing LVNs at the most central nodes is assumed to be the right placement.

Editorial extensions

If this is right

  • Any message-passing GNN can be augmented with LVNs as a preprocessing step plus an embedding table; no changes to the aggregation or update functions are required.
  • Graphs with community-like bottlenecks, such as social networks, stand to gain the most; datasets without node features still benefit because the trainable embeddings supply learnable signals.
  • The structural benefit is measurable before training: LVNs reduce total effective resistance and increase the number of walks between non-central nodes.
  • Because original edges are never removed, the method avoids the edge-level ambiguity that graph rewiring can create, making it safer for tasks where edge semantics matter.
  • Long-range information no longer has to travel layer by layer; shared LVN embeddings provide a shortcut that behaves like extra receptive field at constant depth.

Reading between the lines

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

  • The placement heuristic—degree, PageRank, or label-propagation centrality—is doing causal work the paper credits to bottlenecks. Comparing LVN placement against oracle bottleneck locations would separate the heuristic's contribution from the mechanism itself.
  • Shared trainable embeddings act as a small distributed memory at selected hubs; this suggests LVNs could be combined with rewiring methods that intervene on different resources, such as adding edges elsewhere.
  • The smaller gains on node classification hint that over-smoothing, not over-squashing, dominates those local tasks; a principled LVN variant might need to suppress redundant messages within a group to avoid adding smoothing pressure.
  • The 'add' mode is structurally similar to additive positional encodings in transformers; a testable extension is to condition the shared embeddings on node features or graph structure rather than keeping them globally shared.
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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

4 major / 5 minor

Summary. This manuscript proposes Local Virtual Nodes (LVNs), an augmentation that replaces a set of high-centrality nodes with groups of virtual nodes. Each LVN inherits the edges of the original central node, and LVNs receive trainable embeddings shared across different groups. The authors argue that this creates local pathways around bottlenecks and enables long-range communication via shared embeddings without global rewiring. They evaluate with a GCN backbone on six graph classification and six node classification datasets, compare against rewiring baselines (DIGL, SDRF, FoSR, BORF, LASER), PANDA, and Last Layer FA, and report total effective resistance and path-count improvements. The central empirical claim is that LVNs improve structural connectivity and downstream accuracy relative to these baselines.

Significance. The idea is simple and attractive: instead of rewiring globally or adding a complete-graph supernode, one can expand capacity at a small number of bottleneck candidate regions and keep the original topology otherwise intact. The paper includes useful structural analyses (effective resistance, path counts), reports 50-split confidence intervals, and releases code. If the mechanism were validated, this would be a practical, architecture-agnostic remedy for over-squashing. However, the two main mechanisms—centrality-based placement and shared trainable embeddings—are not isolated by ablations, so the current evidence does not establish that bottleneck targeting and trainable sharing are what drive the observed gains. The conclusion in Section VI itself identifies the centrality heuristic as crucial and unresolved, which strengthens the need for direct experimental control.

major comments (4)
  1. [IV.A, V, VI] The paper's central claim is that placing LVNs at high-centrality nodes targets bottlenecks. This is never tested directly. All experiments compare centrality-selected LVNs against non-augmented GCNs or rewiring baselines; there is no control with random node selection or low-centrality selection at the same ns, nc, embedding budget, and training setup. Because LVNs add trainable parameters, the observed gains could arise from added capacity alone rather than from bottleneck targeting. The paper itself states in Section VI that 'determining the set of nodes to replace with virtual node groups is a crucial part of our algorithm' and proposes future learnable selection. I request an ablation with random, low-centrality, and (if feasible) uniform node selection under identical resource budgets, showing that centrality placement outperforms these controls.
  2. [IV.C, V.G, VI] The second mechanism, shared trainable LVN embeddings, is also not isolated. No comparison is made against fixed random embeddings, per-group non-shared embeddings, or LVNs initialized with copied central-node features. The embedding analyses in Section V.G show only that embeddings change during training and that task-supervised embeddings improve an MLP on the same train/test splits; they do not demonstrate that sharing or trainability per se enables long-range communication. To support the abstract's claim that 'trainable LVN embeddings shared across selected central regions facilitate communication,' please add ablations: (a) fixed random embeddings, (b) non-shared trainable embeddings, and (c) shared trainable embeddings, all with the same ns and nc.
  3. [V.C, Tables III-IV] The reported hyperparameters are incomplete. The manuscript never states the selected ns, nc, centrality function, replace/add mode, or edge strategy for any dataset, although the captions of Figures 6-7 note that the number of LVNs is tuned. Without these values and the search procedure, the experiments are not reproducible, and it is impossible to judge whether the comparison is fair—for example, whether LVN's good results rely on more extensive tuning than the baselines. Please add a table with per-dataset configurations and the tuning protocol.
  4. [Tables III-IV] Some headline comparisons are not supported by the reported intervals. For example, on MUTAG, LVN (Directed) is 84.778 ± 2.902 versus PANDA 83.000 ± 1.876; on COLLAB, LVN (Undirected) is 71.520 ± 0.661 versus LASER 71.880 ± 0.567. These error bars overlap substantially. The text states that the method 'outperforms ... by a large margin' and 'significantly improve performance'; please either provide paired statistical tests across the 50 splits or temper the claims to the datasets where the intervals are separated. The current wording overstates the evidence.
minor comments (5)
  1. [III] The notation [N] is defined as {0, 1, 2, ..., N}, which is non-standard and inconsistent with the later use of [ns] and [nc] as index sets. Please use a consistent convention (e.g., {1, ..., N}).
  2. [IV.B, Eq. (8)] The second union in Eq. (8) connects all pairs of LVNs between adjacent central nodes, creating a complete bipartite subgraph between the two groups. Please clarify whether this is intended and how the directed edge distribution strategy modifies this formalization.
  3. [V.G.2] The mutual-similarity analysis concludes that embeddings are distinct 'not due to random initialization.' However, random vectors in high dimension also tend to have near-zero cosine similarity; this analysis does not distinguish learned diversity from initialization. A comparison against the initial embedding distribution would be more informative.
  4. [V.G.3, Table V] The discriminative-power experiment uses task-supervised LVN embeddings as MLP inputs and trains the MLP on the same labels. This makes the improvement over raw input features difficult to interpret as 'structural information.' Please include a baseline with random pre-trained embeddings and with embeddings from a non-augmented GCN trained under the same protocol.
  5. [References] Reference [40] has an incomplete author name ('U. B. and'); please correct the citation.

Circularity Check

1 steps flagged · score 2.0 of 10

Main performance and connectivity claims are independent; only the auxiliary LVN-embedding analysis in Section V-G.3 is partly circular.

  1. fitted input called prediction [Section V-G.3 ('Discriminative Power'), around Table V]
    "To assess the contribution of virtual node embeddings, we compare MLP performance using input features from the raw graph versus input features from graphs equipped with LVNs and their embeddings. We use pre-trained LVN embeddings as the MLP's input for the second case. ... Thus, we infer that LVN embeddings can capture valuable structural information during training."

    The 'pre-trained' embeddings are the LVN embedding matrix P optimized end-to-end on the same graph-classification labels and the same 50 data splits used later to train the MLP. Feeding P (or raw features + P) into an MLP trained on those labels and observing higher accuracy than without P is a direct consequence of fitting P to those labels; it does not isolate structural information from task-label information. The experiment reads a fitted parameter back as evidence of structure, so the inference that LVN embeddings 'capture valuable structural information' is partly self-definitional. This is auxiliary: the main performance comparison is genuine held-out test accuracy, and the structural metrics in Sections V-E and V-F involve deleted central nodes and fixed non-central node sets, so t

full rationale

The paper is primarily an empirical method paper. The central claim that adding LVNs with trainable embeddings improves connectivity and downstream task performance is evaluated by genuine held-out test accuracy against GCN, rewiring baselines, PANDA, and LASER (Tables III and IV); no fitted parameter is renamed as a test-set prediction. The centrality-based placement heuristic is load-bearing but is an external assumption, not a circular step, and the paper itself flags it in Section VI as a future replacement. The structural connectivity analyses (Sections V-E and V-F) are computed on the transformed graph with selected central nodes removed and restricted to a fixed non-central node set, so the reported reductions in effective resistance and increases in walk counts are not tautological edge-count effects. The only partially circular element is the auxiliary embedding analysis in Section V-G.3, where task-fitted LVN embeddings are fed into an MLP for the same task and then interpreted as evidence of 'structural information'; that conclusion is partly guaranteed by the training objective and does not affect the main performance claim. There is also a self-citation [10] in the related-work discussion of over-smoothing, but it is not load-bearing.

Assumptions & free parameters 6 free parameters · 5 assumptions · 1 invented entities

The central claim depends mainly on heuristic hyperparameters (ns, nc, centrality, mode, strategy) and two domain assumptions: centrality locates bottlenecks, and removing central nodes does not destroy useful structure. The LVN entity has independent evidence via benchmark results.

free parameters (6)
  • ns (number of central nodes) = not reported per dataset
    Hyperparameter selecting how many central nodes to augment; tuned per dataset (Figure 4 shows sweeps over ns values).
  • nc (LVNs per group) = not reported per dataset
    Group size hyperparameter; tuned per dataset.
  • centrality function = not reported per dataset
    Choice among degree, PageRank, label propagation; not specified which was used for each reported result.
  • embedding mode (replace/add) = not reported per dataset
    Whether LVN initial features use only trainable embeddings or add input features; tuned per dataset.
  • edge distribution (standard/directed) = both reported
    Two variants presented; best variant is dataset-dependent as shown in Tables III and IV.
  • Shared trainable LVN embeddings P (nc x D) = learned during training
    The central claim that trainable embeddings help relies on these learned vectors; they are fitted to the training data.
assumptions (5)
  • domain assumption High-centrality nodes coincide with bottleneck regions
    Section IV-A states this correlation is discussed in [16] and [24]; the method's node selection rests on this premise.
  • domain assumption Adding extra pathways through central regions reduces over-squashing
    Inherited from the over-squashing literature (e.g., [23]); LVNs create such pathways.
  • ad hoc to paper Replacing central nodes with LVN groups preserves enough of the input graph's structure and domain knowledge
    Section IV-B removes the original central nodes; the paper claims no significant global topology change, but this is not quantitatively established and is challenged by Eq. 8.
  • standard math Standard message-passing GNN framework (Eq. 1-2)
    Used throughout as the backbone; not proven in the paper.
  • standard math Effective resistance as a proxy for over-squashing (Eq. 11-12)
    Borrowed from [19]; used to measure connectivity improvement.
invented entities (1)
  • Local Virtual Nodes (LVNs) independent evidence
    purpose: Added around central nodes to create extra pathways and carry trainable embeddings for long-range communication
    The method itself is new; performance on standard benchmarks and connectivity metrics provide falsifiable handles outside the paper's own training data.

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Pith. "Pith review of Local Virtual Nodes for Alleviating Over-Squashing in Graph Neural Networks." pith.science (2026). https://pith.science/paper/JMAMC72D

@misc{pith2026250820597,
  author       = {Pith},
  title        = {Pith review of: Local Virtual Nodes for Alleviating Over-Squashing in Graph Neural Networks},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JMAMC72D}},
  note         = {Machine review of arXiv:2508.20597}
}
read the original abstract

Over-squashing is a challenge in training graph neural networks for tasks involving long-range dependencies. In such tasks, a GNN's receptive field should be large enough to enable communication between distant nodes. However, gathering information from a wide range of neighborhoods and squashing its content into fixed-size node representations makes message-passing vulnerable to bottlenecks. Graph rewiring and adding virtual nodes are commonly studied remedies that create additional pathways around bottlenecks to mitigate over-squashing. However, these techniques alter the input graph's global topology and disrupt the domain knowledge encoded in the original graph structure, both of which could be essential to specific tasks and domains. This study presents Local Virtual Nodes (LVN) with trainable embeddings to alleviate the effects of over-squashing without significantly corrupting the global structure of the input graph. The position of the LVNs is determined by the node centrality, which indicates the existence of potential bottlenecks. Thus, the proposed approach aims to improve the connectivity in the regions with likely bottlenecks. Furthermore, trainable LVN embeddings shared across selected central regions facilitate communication between distant nodes without adding more layers. Extensive experiments on benchmark datasets demonstrate that LVNs can enhance structural connectivity and significantly improve performance on graph and node classification tasks. The code can be found at https://github.com/ALLab-Boun/LVN/}{https://github.com/ALLab-Boun/LVN/.

Figures

Figures reproduced from arXiv: 2508.20597 by the authors.

Figure 1
Figure 1. Comparison between graph rewiring, global virtual node, and local [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Procedure for adding virtual nodes to a graph. In [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. A graph augmented with LVNs directed (Left) and undirected (Right) [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: Change in the total effective resistance of the non-virtual node set for TUDataset [ [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]
Figure 5
Figure 5. Figure 5: Average change in number of paths with increasing walk length [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]
Figure 6
Figure 6. Figure 6: The evolution of the embedding weights during training for TUDataset [ [PITH_FULL_IMAGE:figures/full_fig_p011_6.png]
Figure 7
Figure 7. Figure 7: LVN embedding heatmaps for graph classification datasets (top row) and node classification datasets (bottom row). The number of LVNs is tuned. [PITH_FULL_IMAGE:figures/full_fig_p011_7.png]

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Pith tools

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