LEED: Local Embedding Evolution Distance for over-smoothing estimation and virtual node selection in GNN

Killian Cressant; Pedro B. Velloso

REVIEW 4 major objections 4 minor 35 references

LEED: Local Embedding Evolution Distance for over-smoothing estimation and virtual node selection in GNN

T0 review · 4 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read LEED, a node-level embedding-evolution distance, claims to diagnose over-smoothing and to select virtual nodes better than classical centrality measures.

desk verdict Neat local metric idea, but the critical-node selection used in experiments is a static proxy that on featureless graphs reduces to degree centrality, so the headline claim of replacing centrality is not supported. read the letter →

arxiv 2608.09596 v1 pith:UVN2P32I submitted 2026-08-10 cs.LG cs.AI

Killian Cressant , Pedro B. Velloso This is my paper · ORCID

classification cs.LGcs.AI

keywords over-smoothingover-squashingvirtualnodesgraphrewiringnodecentralityembeddingevolutionmessage-passingGNNslocalmetric

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 LEED, a local metric that measures over-smoothing by tracking how each node's embedding changes as message passing proceeds through a GNN. The intended payoff is that this node-level view reveals heterogeneous collapse patterns—cliques and bridges—that the global Dirichlet energy hides, and that the same local scores can be used as a single criterion for choosing which nodes should receive virtual nodes or extra embedding width. The authors claim that using LEED alone, with no centrality heuristics, improves or matches GNN accuracy on six graph classification benchmarks, and that it is about as cheap as Dirichlet energy on sparse graphs. If these claims hold, one parameter-free score would jointly address over-smoothing and over-squashing, which are usually handled separately.

What carries the argument

The load-bearing object is the two-hop mean evolution operator $T_2$, a degree-weighted average of a node's neighbors' one-hop averages with an added self-loop weight, applied to node embeddings. LEED defines each node's over-smoothing score through $T_2$: the minimum squared distance from a node's evolved embedding to any neighbor's evolved embedding, Eq. (4), while the critical-node score $D_i = \|T_2(x_i^{(0)}) - x_i^{(0)}\|_2$, Eq. (12), applies the same operator once to the input embeddings. The proof that $L(X)$ is bounded by a scaled Dirichlet energy, Eq. (11), connects LEED to the global over-smoothing measure. For featureless graphs, the constant-feature reduction makes $D_i$ equal to $|d_i-2|/(d_i+2)$, so selection then coincides with degree centrality.

What would settle it

Train a GCN with several random initializations on a featureful dataset like MUTAG or PROTEINS, compute the top-K nodes from the static proxy $D_i$ (Eq. 12) and from the trajectory-based LEED scores $\ell_i$ (Eq. 4), and measure their overlap; if the intersections are at chance level across most runs, the one-shot proxy is not capturing the embedding dynamics the paper says drive criticality.

Watch

Extended reading notes

Core claim

On its own terms, the paper discovers that a per-node distance computed from a two-hop mean evolution of embeddings behaves like a local Dirichlet energy, and that ranking nodes by the one-shot distance between their initial embedding and its evolved counterpart identifies the nodes most worth rewiring. LEED's local score for node $i$ is the minimum squared distance between the evolved embedding of $i$ and that of any neighbor, $\ell_i = \min_{j\in\mathcal{N}_i} \|T_2(x_i) - T_2(x_j)\|_2^2$, and its graph-level sum $L(X)$ is bounded above by a constant times the Dirichlet energy, $L(X) \le (\max_i 2/d_i) C_{\hat{T}} E(X)$. When input features are uninformative, the score reduces to a monotone function of degree, $|d_i-2|/(d_i+2)$, so the claimed advantage must come from datasets where features carry structure. In experiments, LVN with LEED-selected critical nodes achieves the best average rank across six benchmarks, and PANDA-LEED improves three of them.

Load-bearing premise

The virtual-node selection rests on the assumption that a single static application of the two-hop evolution to the initial embeddings (Eq. 12) ranks critical nodes the same way the training-trajectory LEED metric does, and that this ranking remains informative on datasets whose node features are uninformative or absent.

Editorial extensions

If this is right

Over-smoothing can be monitored per node during training, so a practitioner can see which cliques or bridge nodes collapse first, not just the graph-wide average.
Graph-rewiring methods such as LVN and PANDA can drop the empirical search over PageRank, betweenness, closeness, and degree; LEED alone selects the top-K critical nodes.
Because the graph-level LEED sum is bounded by a Dirichlet-energy term, LEED could replace Dirichlet energy in regularization-based training while preserving the same global guarantees.
On sparse graphs LEED costs $O(l\cdot n\cdot \bar{d})$, comparable to Dirichlet energy, so the added node-level resolution does not require an asymptotic complexity increase.
The same LEED ranking improves a directed virtual-node variant, with the largest gain reported on ENZYMES.

Reading between the lines

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

The reduction of LEED to degree centrality on featureless datasets means that on datasets without node features the method is, in effect, a degree-hub strategy; an explicit comparison against naive degree top-K selection would show whether the embedding-evolution machinery adds anything in those cases.
The static proxy $D_i$ (Eq. 12) is applied once to input embeddings, whereas the LEED metric itself is defined over training trajectories; whether the one-shot ranking matches the trajectory-based ranking on deeper or randomly initialized GNNs is a testable assumption the paper does not directly evaluate.
Because the bound constant $C_{\hat{T}}$ can become large on big graphs, the authors rely on relative trajectory similarity rather than absolute values; normalizing LEED differently might tighten the link to Dirichlet energy and change the comparability on large datasets.
The study is confined to GCN message passing; if the same critical-node ranking transfers to attention-based or higher-order architectures, LEED could serve as an architecture-agnostic rewiring criterion, but that transfer is not demonstrated.

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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

The short version: LEED is a genuinely new local construction, but the experiments hinge on a static proxy that for featureless graphs collapses to degree centrality. That undermines the central claim that LEED replaces classical centrality measures.

What's new and good: the min-over-neighbors two-hop distance (Eq. 4) is a simple, parameter-free local metric, and the diagnostic plots (Figs. 4–6) make a plausible case that its graph-level aggregate tracks Dirichlet energy. The authors are also honest that the constant in their bound can be very large, so they don't oversell the theory. The complexity table is a nice addition, and the idea of using embedding evolution for node importance is worth exploring.

Where it gets shaky: the critical node score D_i (Eq. 12) is not the LEED metric of Eq. 4. It is a static one-step diffusion applied to the input, with no min over neighbors and no tracking across layers. The paper never checks whether the top-K of D_i agrees with the top-K of layer-wise LEED scores. On featureless graphs, D_i reduces to a monotonically increasing function of degree (Eq. 15), so on REDDIT-BINARY, IMDB-BINARY, and COLLAB the method is effectively degree centrality under a new name. That means the reported improvements on those datasets cannot be attributed to the embedding-evolution mechanism the paper claims to exploit. The concern is not minor; it is load-bearing for the virtual node selection story.

The experimental protocol also has integrity issues: some baselines are taken from other papers, PANDA uses a best-of between the authors' runs and the original paper, LEED's own hyperparameters are not disclosed, and no code is released. The reported gains are a few accuracy points on small benchmarks, so they are plausible but not established.

Bottom line: the diagnostic part of LEED could be a useful addition to the toolbox. The critical-node selection part needs direct validation of the static proxy against the true LEED ranking, and a clean uniform experimental setup. With code and honest baselines, this could become a reasonable contribution. In its current form, I would not cite it, and I would expect major revision before acceptance. Still, the core idea is interesting enough that I would send it to reviewers rather than desk reject.

Referee Report

4 major / 4 minor

Summary. The paper introduces LEED (Local Embedding Evolution Distance), a node-level metric defined in Eq. (4) as the minimum squared distance between a node's two-hop mean-transformed embedding and those of its neighbors, intended to quantify local over-smoothing. A graph-level aggregation L(X) is proposed and compared with Dirichlet energy. The paper then defines a critical-node score D_i in Eq. (12), applies it to select nodes for virtual-node insertion in LVN and width expansion in PANDA, and reports classification accuracies on six TUDataset benchmarks, concluding that LEED-based selection improves GNN performance.

Significance. If fully supported, a parameter-free local metric that both diagnoses over-smoothing and improves virtual-node placement would be a useful contribution to graph rewiring. The paper has some strengths: the LEED definition is interpretable, the complexity comparison in Table II is helpful, and experiments span six datasets with multiple runs. However, the theoretical link between L and E is not established (Eq. 11 involves an unbounded constant), the selection score D_i in Eq. (12) is not the LEED metric of Eq. (4), and on featureless graphs it reduces to a degree-based function (Eq. 15). As a result, the main empirical claims are not clearly attributable to the proposed metric. The reported gains are also modest and inconsistently significant across datasets, undercutting the 'globally better' conclusion. The core idea is interesting, but the current evidence is insufficient to support the central claims.

major comments (4)

[Section IV-B (Eq. 11)] The claimed bound L(X^(k)) ≤ (max_i 2/d_i) C_bT E(X^(k)) does not provide a meaningful theoretical link between LEED and Dirichlet energy. C_bT is defined as a supremum of a Rayleigh quotient, and the authors themselves state that it 'can become very large for large-scale datasets' (text after Eq. 11). The proof is therefore vacuous as a bound, and the subsequent assertion that 'in practice ∃ C s.t. C·E(X) ≤ L(X)' is an empirical assumption, not a derived property. The paper should either derive a controlled bound with a graph-dependent constant that is explicitly computed, or present the similarity to Dirichlet energy as purely empirical and remove the inequality from the theoretical claims.
[Section V (Eq. 12), Section IV-B (Eq. 4)] The critical-node score used in all experiments is D_i = ||T2(x_i^(0)) − x_i^(0)||^2, a static one-step application of T2 to the input embeddings. This is not the LEED metric of Eq. (4), which is l(x_i^(k)) = min_{j∈N_i} ||T2(x_i^(k)) − T2(x_j^(k))||^2 and is evaluated layer-by-layer during training. The score D_i contains no min over neighbors and no tracking across layers. No experiment checks whether the top-K of D_i agrees with the top-K of the layer-wise LEED scores. Without such validation (or direct use of Eq. (4) for node selection), the reported improvements from virtual-node construction cannot be attributed to LEED.
[Section V-A (Eq. 15)] For the three datasets without node features (REDDIT-BINARY, IMDB-BINARY, COLLAB), the paper assigns a constant feature vector and derives D_i = |d_i − 2|/(d_i + 2). This depends only on the degree d_i, so the resulting ranking is a degree-based heuristic, not an embedding-evolution measure. The text claims the formula 'increases monotonically with d_i', but it does not: f(1)=1/3 > f(2)=0 and f(3)=0.2. More fundamentally, since the rank is a deterministic function of degree, the gains on featureless datasets do not demonstrate that LEED replaces classical centrality measures; they demonstrate a particular degree-based reweighting. If the authors intend to claim otherwise, they must show that the full LEED (Eq. 4) produces a different ranking on these datasets.
[Section VII (Table I)] The conclusion 'It is clear that LVN with LEED performs globally better than any other model' is not supported by the reported results. On MUTAG, LVN-LEED (83.333±2.325) is lower than PANDA (86.068±2.198) and PANDA-LEED (86.838±1.653); on ENZYMES, LVN-LEED (31.094±1.748) is comparable to LVN (31.367±1.376) and below PANDA-LEED (33.182±1.921). The average rank of 2.3 does not reflect these pairwise differences, and no statistical significance testing (e.g., paired tests over the runs) is reported. The findings support at most 'LEED-based selection is competitive and sometimes better than existing centrality-based selection on these datasets'.

minor comments (4)

[General / Figure 1] There are several typos: 'gragh' in the Figure 1 caption should be 'graph'; 'Another import aspect' in Section IV-A should be 'important'; 'ArXive' appears in references [26] and [29]; reference [20] uses 'ICKM' instead of 'CIKM'.
[Section IV-B] The notation for the local score is inconsistent: Eq. (4) uses ℓ(x_i^(k)), while Eq. (8) sums over i as L(X^(k)) = Σ_i ℓ(x_i^(k)), but later text refers to a 'local score l_i'. The self-loop coefficient 2 in Eq. (6) is not justified; it is effectively a hyperparameter of T2, which is not acknowledged in the parameter-free claims.
[Section V-A] The condition 'except when d_p << d_i, ∀p ∈ N_i' in the discussion of Eq. (15) is undefined and unexplained; it should be made precise or removed.
[Section VI] The experimental section says 'For LVN, we extensively evaluated ... conducting 20 independent runs ... Given the lower variance reported in the original results, we retain their reported scores.' This mixing of reproduced and borrowed numbers should be clarified: for which cells in Table I are the reported means and errors taken from prior papers rather than from the authors' own runs? This matters for the statistical comparison.

Circularity Check

1 steps flagged · score 6.0 of 10

On featureless datasets, the 'LEED' selection rule of Eq. (12) collapses to degree centrality by Eq. (15), so the claimed replacement of centrality heuristics is partly a renaming of degree centrality; the dynamic LEED metric of Eq. (4) is not the criterion used.

renaming known result [Section V, Eq. (12) and Eq. (15); experiments in Section VII, Table I]
"Let Di = ||T2(x_i^(0)) - x_i^(0)||^2, i ∈ [1,n] (12) ... For undirected graphs with a symmetric binary adjacency matrix, this simplification allows the proposed measure to be expressed in closed form, yielding the following output: Di = |d_i - 2|/(d_i + 2) (15) ... This simple formula yields a value in [0,1], that increases monotonically with d_i, the degree of node i."

The criterion used in the LVN-LEED and PANDA-LEED experiments is D_i (Eq. 12), not the layer-wise local metric ℓ(x_i^(k)) of Eq. (4), which includes a minimum over neighbors and tracks evolution across layers. For the three datasets without node features (REDDIT-BINARY, IMDB-BINARY, COLLAB), the paper assigns a constant feature vector to every node, and Eq. (15) makes D_i a strictly increasing function of degree. Consequently, the top-K set S_k of Eq. (14) is exactly the top-K by degree centrality: the proposed 'unique criterion' reduces, by the paper's own closed form, to the classical degree heuristic it claims to replace. The reported gains on those datasets are therefore not evidence for a new embedding-evolution distance.

full rationale

The over-smoothing diagnostic itself is not circular: LEED is a parameter-free local distance, Eq. (11) is an inequality (an upper bound, not an identity) linking L to Dirichlet energy, and Figure 4 is a consistency check rather than a derivation from E. There is no load-bearing self-citation and no fitted parameter renamed as prediction. The circularity is confined to the virtual-node-selection claim: because the operational selection score in featureless graphs equals degree centrality by Eq. (15), and because it is not the metric of Eq. (4), the paper's central claim that LEED replaces multiple centrality heuristics is partly a renaming of degree centrality. This affects three of the six benchmark datasets, so a moderate score, not a high one, is appropriate.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The metric introduces no new physical or architectural entities; it reuses graph structure, node embeddings, and existing virtual node machinery. The main implicit inputs are design choices (self-loop weights, top-K fraction) and an assumed empirical link between LEED and Dirichlet energy that is not rigorously established.

free parameters (2)

self-loop weight in T2 (coefficient 2 in Eq. 6) = 2
Hand-chosen multiplier applied to the self-neighbor term in the two-hop evolution, intended to emphasize self-loops; no derivation or data fitting is provided.
top-K fraction k (Eq. 13) = not reported
Controls how many critical nodes are selected for virtual node insertion; the value used in experiments is not disclosed, yet it directly affects all reported accuracy numbers.

assumptions (4)

domain assumption GCN message-passing model with symmetric binary adjacency matrix and self-loops (Section II, Eq. 1)
The entire LEED construction and the bound in Eq. 11 assume this standard but specific message-passing setup.
domain assumption Analogy between GCN layers and a Markov diffusion kernel (Section IV-B, from [34])
The paper treats the layer index as analogous to a diffusion time parameter, which justifies defining T1 and T2 as averaging operations, but the analogy is not formalized.
ad hoc to paper Existence of a constant C such that C*E(X) <= L(X) in practice (Section IV-B, after Eq. 11)
The authors assert that simulations show a lower bound relating LEED to Dirichlet energy, but they do not prove it or characterize C; this empirical claim carries the diagnostic equivalence argument.
domain assumption One-hot embeddings in MUTAG and ENZYMES behave like an unbiased random distribution for the expectation formula (Section V-A)
The closed-form expressions for E[D_i^2] rely on a 'normal random distribution' assumption over one-hot features, which is not verified for the actual datasets.

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Cite this review

Pith. "Pith review of LEED: Local Embedding Evolution Distance for over-smoothing estimation and virtual node selection in GNN." pith.science (2026). https://pith.science/paper/UVN2P32I

@misc{pith2026260809596,
  author       = {Pith},
  title        = {Pith review of: LEED: Local Embedding Evolution Distance for over-smoothing estimation and virtual node selection in GNN},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UVN2P32I}},
  note         = {Machine review of arXiv:2608.09596}
}

read the original abstract

Graph Neural Networks (GNNs) suffer from two fundamental limitations: over-smoothing, where node representations become indistinguishable with depth, and over-squashing, where long-range information is compressed through limited message-passing channels. Existing metrics such as Dirichlet energy provide global characterizations of over-smoothing but lack the resolution to analyze node-level behavior and guide architectural improvements. In this paper, we propose LEED (Local Embedding Evolution Distance), a novel local metric that quantifies over-smoothing by tracking the evolution of individual node embeddings across layers. By operating at the node level, LEED enables fine-grained analysis of representation dynamics during training, revealing heterogeneous over-smoothing patterns that are invisible to global energy-based measures. This locality induces informative node importance scores, interpreted as embedding-driven centrality measures. We leverage LEED to design a more efficient strategy for virtual node selection. Unlike existing approaches that depend on multiple heuristic centrality measures, our method uses LEED as a unique criterion to guide the construction of Local Virtual Nodes to mitigate over-squashing. Experiments show that LEED provides more informative diagnostics than Dirichlet energy while preserving global evaluation, and enables more effective virtual node integration, improving GNN performance across datasets.

Figures

Figures reproduced from arXiv: 2608.09596 by the authors.

**Figure 3.** LEED score and the Heatmap highlighting critical nodes [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

**Figure 2.** One dimension of node embedding around a clique [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

Figures from the paper (4 more)

**Figure 4.** Figure 4: Comparison with 8 configurations between LEED and Dirichlet energy [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]

**Figure 5.** Figure 5: Cora distance for each layer between 2 and 10 [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]

**Figure 6.** Figure 6: Cora distance for every 10 epoch of training [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]

**Figure 7.** Figure 7: Dirichlet energy over sets of hyperparameters for LEED and other [PITH_FULL_IMAGE:figures/full_fig_p009_7.png]

Discussion (0). Continue with ORCID to comment.

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Simple spectral graph convolution,

H. Zhu and P. Koniusz, “Simple spectral graph convolution,” inICLR’21, 2021

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The effectiveness of curvature- based rewiring and the role of hyperparameters in GNNs revisited,

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Pith tools

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

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Fault detection in telecom networks using bi- level federated graph neural networks,

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Semi-supervised classification with graph convolutional networks,

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Graph neural networks in computer vision - architectures, datasets and common approaches,

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Improving expressivity of GNNs with subgraph-specific factor embedded normalization,

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A survey on over- smoothing in graph neural networks,

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Over-squashing in graph neural networks: A comprehen- sive survey,

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Simplifying graph convolutional networks,

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work page 2019