{"id":"f726411c-fe56-42b3-a3f6-828a9b98d2fd","arxiv_id":"2608.09596","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"LEED is a new local distance for GNNs that detects per-node over-smoothing and selects critical nodes for virtual-node-based graph rewiring.","lead":"The authors introduce LEED, a node-level score that tracks how much each node's embedding changes across GNN layers and shows where representations collapse. The score also picks nodes for graph rewiring tricks (virtual nodes), replacing the usual menu of centrality heuristics with one number.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Critical-node selection uses static one-step proxy Eq. 12, not the LEED metric of Eq. 4; in featureless graphs it reduces to degree centrality, so LEED's claimed advantage over centrality is unsupported.","rationale":"The reader identified the same weak point, and I agree. The concern is sharper than a mere possibility: Eq. 12 is what was actually used to select critical nodes in the experiments, and it is not the LEED metric defined in Eq. 4. On featureless datasets the paper's own derivation shows the ranking degenerates to degree centrality, so the claimed replacement of classical centrality is not achieved in that regime. The theoretical LEED-Dirichlet bound in Eq. 11 has an unbounded constant C_bT, and the paper acknowledges this; that weakens the over-smoothing diagnosis claim but is not the sharpest issue. The metric's simplicity and reported O(l * n * dbar) complexity are genuine strengths, and the empirical trajectory comparison in Figure 4 is suggestive. However, the key architectural claim rests on a static proxy that has not been validated against the dynamic metric it is supposed to approximate. I therefore retain the CONDITIONAL verdict, with an explicit added condition: the authors must release code and run an Eq.12-versus-Eq.4 ablation, or soften the claim to 'degree-based centrality with a principled derivation'.","tokens_in":13886,"tokens_out":4427,"duration_ms":41693,"concrete_test":"Run an ablation on MUTAG, ENZYMES, and REDDIT-BINARY using identical LVN/PANDA hyperparameters, with three critical-node selectors: (i) the static proxy Eq. 12 used in the paper; (ii) the actual layer-wise LEED of Eq. 4, for example scores averaged over layers or taken at a mid-training checkpoint; (iii) plain degree centrality. Report the top-20% overlap and Spearman correlation between selectors, and final test accuracy for each selector. If (i) and (iii) are indistinguishable on featureless graphs, or if (ii) does not reproduce the reported gains, the central claim fails. A quick independent check is to compute both rankings on CORA and measure their overlap.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The selection rule used in all experiments is not the LEED metric. Eq. 4 defines a per-layer, neighbor-relative distance l(x_i^(k)) = min over j in N_i of ||T_2(x_i^(k)) - T_2(x_j^(k))||^2. Eq. 12 instead computes D_i = ||T_2(x_i^(0)) - x_i^(0)||^2, a single static diffusion step applied to the input, with no min over neighbors and no tracking across training layers. No experiment in the paper checks that the top-K of D_i agrees with the top-K of the layer-wise LEED scores. For featureless datasets, Section V.A derives D_i = |d_i - 2|/(d_i + 2), which is strictly increasing in degree, so the 'LEED' ranking is exactly degree centrality. Thus on REDDIT-BINARY, IMDB-BINARY and COLLAB, the method does not replace classical centrality measures; it is degree centrality under a new name. If the static proxy diverges from the embedding-evolution metric that the paper claims to exploit, the reported virtual-node improvements cannot be attributed to LEED.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":14234,"tokens_out":8450,"duration_ms":70671,"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":[{"comment":"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":"Section IV-B (Eq. 11)"},{"comment":"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":"Section V (Eq. 12), Section IV-B (Eq. 4)"},{"comment":"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":"Section V-A (Eq. 15)"},{"comment":"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'.","section":"Section VII (Table I)"}],"minor_comments":[{"comment":"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":"General / Figure 1"},{"comment":"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":"Section IV-B"},{"comment":"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":"Section V-A"},{"comment":"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.","section":"Section VI"}],"recommendation":"major_revision","confidential_remarks":"The paper fits the scope of a GNN representation-learning venue, but the gap between the stated LEED metric and the implemented selection procedure is substantial. In its current form, the experiments on three of six datasets effectively test a degree-based score, not the proposed embedding-evolution metric. A major revision with direct validation of Eq. (4) (or a clear redefinition of the proposed method) is needed before the claims can be evaluated fairly."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nWhat'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.\n\nWhere 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.\n\nThe 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.\n\nBottom 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.","headline":"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.","tokens_in":14667,"tokens_out":3979,"would_cite":false,"duration_ms":35694,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"LEED, a node-level embedding-evolution distance, claims to diagnose over-smoothing and to select virtual nodes better than classical centrality measures.","keywords":["over-smoothing","over-squashing","virtual nodes","graph rewiring","node centrality","embedding evolution","message-passing GNNs","local metric"],"falsifier":"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.","tokens_in":13612,"feed_emoji":"🎯","tokens_out":10134,"duration_ms":77461,"temperature":0.7,"pith_summary":"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.","feed_headline":"Local embedding score improves GNN rewiring on six benchmarks","feed_subtitle":"One parameter-free metric pinpoints over-smoothed and bottleneck nodes, replacing dataset-specific centrality tuning.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Establishes the GCN layer operation whose per-layer embedding evolution LEED tracks.","marker":"[3]"},{"why":"Defines the Dirichlet-energy baseline and the bounded-energy framework that LEED is compared and linked to.","marker":"[12]"},{"why":"Supplies the six graph classification benchmarks used in all accuracy comparisons.","marker":"[17]"},{"why":"Provides the over-smoothing and over-squashing trade-off argument that motivates a joint treatment.","marker":"[20]"},{"why":"Documents when virtual nodes help or worsen over-smoothing, framing the design targets of LEED.","marker":"[21]"},{"why":"Provides the expander-graph rewiring baseline and the constant-feature strategy used for featureless datasets.","marker":"[26]"},{"why":"Introduces the Local Virtual Node framework whose centrality selection LEED replaces.","marker":"[29]"},{"why":"Introduces the width-expansion model PANDA, whose central-node selection is also replaced by LEED.","marker":"[30]"},{"why":"Motivates the two-hop mean evolution via the Markov diffusion distance analogy.","marker":"[34]"}],"fun_headline_variants":["Local metric pinpoints over-smoothed nodes for GNN rewiring","LEED: one distance to rank nodes and boost GNN accuracy","Beyond Dirichlet energy: node-level measure guides virtual nodes","Embedding distance picks virtual nodes, improves GNN benchmarks","Per-node evolution score replaces heuristic centrality for GNN tuning"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Local metric pinpoints over-smoothed nodes for GNN rewiring","LEED: one distance to rank nodes and boost GNN accuracy","Beyond Dirichlet energy: node-level measure guides virtual nodes","Embedding distance picks virtual nodes, improves GNN benchmarks","Per-node evolution score replaces heuristic centrality for GNN tuning"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000602,"raw_usage":{"total_tokens":2840,"prompt_tokens":1001,"completion_tokens":1839,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":617,"completion_tokens_details":{"reasoning_tokens":1753}},"tokens_in":617,"tokens_out":1839,"duration_ms":13179,"temperature":1.0,"reasoning_tokens":1753,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T14:21:00.829069+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Dirichlet energy constrained learning for deep graph neural networks,","cited_arxiv_id":null,"evidence_quote":"Defines the Dirichlet-energy baseline and the bounded-energy framework that LEED is compared and linked to."},{"cited_title":"Tudataset: A collection of benchmark datasets for learning with graphs,","cited_arxiv_id":null,"evidence_quote":"Supplies the six graph classification benchmarks used in all accuracy comparisons."},{"cited_title":"On the trade- off between over-smoothing and over-squashing in deep graph neural networks,","cited_arxiv_id":null,"evidence_quote":"Provides the over-smoothing and over-squashing trade-off argument that motivates a joint treatment."},{"cited_title":"un- derstanding virtual nodes: oversquashing and node heterogeneity,","cited_arxiv_id":null,"evidence_quote":"Documents when virtual nodes help or worsen over-smoothing, framing the design targets of LEED."},{"cited_title":"Cayley graph propagation,","cited_arxiv_id":null,"evidence_quote":"Provides the expander-graph rewiring baseline and the constant-feature strategy used for featureless datasets."},{"cited_title":"Local virtual nodes for alleviating over-squashing in graph neural networks,","cited_arxiv_id":null,"evidence_quote":"Introduces the Local Virtual Node framework whose centrality selection LEED replaces."},{"cited_title":"PANDA: Expanded width-aware message passing beyond rewiring,","cited_arxiv_id":null,"evidence_quote":"Introduces the width-expansion model PANDA, whose central-node selection is also replaced by LEED."},{"cited_title":"Simple spectral graph convolution,","cited_arxiv_id":null,"evidence_quote":"Motivates the two-hop mean evolution via the Markov diffusion distance analogy."}],"review_version":1}