{"id":"3826c4f4-53ff-4e1a-8914-e3097279cdd3","arxiv_id":"2607.28259","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"Sliding-window interlevel Betti sequences (Topo-Scan) plus Transformers match or beat strong GNN and TDA baselines on graph classification and molecular property tasks while avoiding full persistence diagrams.","lead":"TopoFormer turns graphs into short sequences of topological tokens via sliding-window filtrations (Topo-Scan), then runs a Transformer on those sequences. It is a practical way to inject multi-scale topology into attention models without full persistent-homology diagrams, with competitive accuracy and clearer compute.","discovery_kind":"new_method","skeptic_critique":{"model":"grok-4.5","headline":"Empirical gains may not isolate Topo-Scan sequential structure from Transformer capacity and same-family hyperparameter selection.","rationale":"The reader correctly identifies the weakest link: attribution of gains to the minimal Topo-Scan token and sequential representation, given m/filtration choices on the evaluation families, fingerprint fusion for MPP strength, and error bars that swallow many margins. Table 5 already shows that feeding ordinary Betti vectors as sequences (PH-TR) captures much of the benefit; Topo-Scan’s extra lift is real but modest and not stress-tested under held-out hyperparameter protocols or on larger non-molecular graphs beyond one OGB number. Stability theory is appropriate for what it claims (ℓ1 robustness of windowed Betti counts) and does not over-reach. No internal contradiction or proof gap is load-bearing. The contribution remains accept-shaped as a clean, parallelizable alternative to full PH pipelines, but the strongest claim should stay tempered exactly as the reader recommends. Verdict stays CONDITIONAL; no upgrade or downgrade is warranted.","tokens_in":30637,"tokens_out":612,"duration_ms":13078,"concrete_test":"Re-run the Table 5 protocol with a single fixed filtration and m held out (e.g., choose m on BZR/COX2 only, evaluate locked m=2 on MUTAG/PROTEINS/IMDB/REDDIT and OGBG-MOLHIV). Report TopoFormer vs PH-TR mean accuracy difference and whether it remains positive outside one standard error on at least 5/7 datasets; if the gap collapses or becomes insignificant, the attribution to Topo-Scan weakens.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central claim attributes SOTA/competitive results (Tables 2–5) to Topo-Scan’s short ordered interlevel tokens (β0, β1, |Vi|, |Ei|) rather than to Transformer capacity, multi-filtration fusion, or fingerprint fusion (TOPOFORMER*). The key control is Table 5 / §C.7: PH-TR (same filtrations + Betti curves as sequences into a Transformer) already closes most of the gap over PH-MLP; TopoFormer adds a further ~1–3 points. Window width m is chosen as m=2 after experiments on the same classification families (Table 6, §C.5), and the main multi-filtration menu (degree/ORC/HKS) is likewise selected on those families. On small graphs, SOTA margins often sit inside large standard deviations (e.g., MUTAG 94.68±4.30). Thus the load-bearing assumption—that the sliding-window sequential representation itself is what drives the reported gains—is only partially isolated. Stability (Thm 3.1 / Cor 3.2) supports robustness of the tokens, not that they are the source of the accuracy lift.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The paper proposes TopoFormer, a graph-level architecture whose core module, Topo-Scan, replaces nested sublevel persistent homology with overlapping interlevel slices of a fixed clique complex under scalar node/edge filtrations. Each slice is summarized by a four-dimensional token (β0, β1, |Vi|, |Ei|); the resulting short ordered sequences are fed to Transformer encoders (optionally fused across filtrations or with molecular fingerprints as TopoFormer*). The authors prove discrete ℓ1 stability of the Topo-Scan sequences with respect to the bottleneck distance of interlevel modules (Theorem 3.1, Corollary 3.2, Appendix B), report competitive or best-in-table accuracy on eight graph-classification benchmarks plus OGBG-MOLHIV, and strong scaffold-split ROC-AUC on seven MoleculeNet tasks, together with runtime comparisons against PH, PersLay, and TopoGCL and ablations versus PH-MLP/PH-TR, window width, and multi-filtration choices.","tokens_in":30856,"tokens_out":1711,"duration_ms":39964,"significance":"If the empirical gains are genuinely driven by the sequential interlevel representation, the work offers a practical, parallelizable bridge between TDA and attention models that avoids full persistence diagrams and vectorization design. Strengths that should be credited include: (i) a clean stability reduction to standard algebraic stability of interlevel modules with an explicit proof sketch in Appendix B; (ii) broad empirical coverage (classification, OGBG-MOLHIV, MoleculeNet) with multiple filtration families and runtime tables; (iii) public code; and (iv) a useful control (PH-TR) showing that treating Betti curves as sequences already helps. The contribution is incremental rather than foundational, but it is a credible and usable design pattern for topology-aware graph transformers.","major_comments":[{"comment":"§4.3 / Table 5 and the extended Table 15: the central attribution claim—that gains come from Topo-Scan’s sliding-window sequential tokens rather than from Transformer capacity or filtration choice—is only partially isolated. PH-TR (identical filtrations and Betti information, same Transformer backend) already closes most of the gap over PH-MLP; TopoFormer adds a further ~1–3 points on average. On several datasets the TopoFormer–PH-TR difference sits inside reported standard deviations (e.g., PROTEINS, IMDB-B under degree). A load-bearing control is missing: e.g., (a) shuffled or randomly ordered Topo-Scan tokens into the same Transformer, and/or (b) a fixed Transformer trained on non-topological fixed-length sequences of comparable dimension. Without such a control, the claim that the interlevel sequential structure itself is the driver remains under-supported.","section":"§4.3, Table 5, Table 15"},{"comment":"§C.5 / Table 6: window width m is selected as m=2 after experiments on the same graph-classification families later used for SOTA reporting. The multi-filtration menu (degree, Ollivier–Ricci, HKS) is likewise tuned on those families (§C.5, Table 13). This is ordinary hyperparameter practice, but it weakens the claim that the method is ‘lightweight’ and that gains transfer without dataset-specific topological engineering. Please either (i) fix m and the filtration menu a priori (or via a held-out meta-validation split) and re-report main tables, or (ii) clearly mark main results as post-selection and add a frozen-hyperparameter column.","section":"§C.5, Table 6, Table 13"},{"comment":"Tables 2–3 and Table 4: SOTA / ‘best or second-best on 7/8’ language should be tempered where margins are smaller than cross-fold standard deviations (MUTAG 94.68±4.30; BZR 92.36±4.11) and where TopoFormer* (not pure TopoFormer) is the reported model. On OGBG-MOLHIV, TopoFormer* reaches 78.19±0.19 versus Graphormer 80.51±0.53 and TopER 80.21±0.15; calling this ‘underscoring scalability and strength of topological signals’ overstates a ~2-point deficit under fingerprint fusion. For MPP (Table 3), pure topological performance is secondary to ECFP fusion (Table 7); the abstract and §4.2 should separate pure TopoFormer from TopoFormer* more sharply when claiming competitiveness with KANO/MV-Mol.","section":"Tables 2–4, Table 7, Abstract, §4.2"},{"comment":"§3.1 / Theorem 3.1 and Corollary 3.2: the stability result is correct as a discrete sampling bound on interlevel Betti numbers, but the constant C = C(bG, {αi}, m, s) is left existential and can grow with the number of windows and bar complexity. For the ‘predictable and efficient compute’ and ‘robust encodings’ claims in the abstract, either give a concrete bound in terms of T and the number of bars, or clarify that stability alone does not imply that the four-dimensional token retains task-relevant information (Remark B.3 already notes the expressivity gap). A short remark on information loss relative to full barcodes would keep the theory claim proportionate.","section":"§3.1, Theorem 3.1, Corollary 3.2, Remark B.3"}],"minor_comments":[{"comment":"Figure 3 caption and body: β0/β1 sequences are written with angle brackets in a way that can be confused with inner products; use parentheses or explicit tuples consistently.","section":"Figure 3"},{"comment":"§3: notation switches between bGi, bG, and script G; unify clique-complex notation early.","section":"§3"},{"comment":"Table 2 header marks 1st/2nd/3rd by color; ensure a non-color cue (bold/underline) for accessibility and print.","section":"Table 2"},{"comment":"§4.1 Computational Complexity: the claim ‘no cycle-rank identity due to clique complexes’ is slightly opaque; one sentence explaining why β1 needs edge–triangle operators would help non-TDA readers.","section":"§4.1"},{"comment":"Related work: Graph Filtration Learning (Hofer et al., 2020) is cited in the appendix remark but deserves a clearer main-text contrast (learn filtration vs. fix filtration and change representation).","section":"§2, Remark B.3"},{"comment":"Typos / style: ‘Topoformer’ vs ‘TOPOFORMER’ vs ‘Topo-Scan’ capitalization is inconsistent in the abstract and title block; ‘early-saturate’ hyphenation; arXiv id line appears inside the manuscript body.","section":"Abstract, title block"}],"recommendation":"major_revision","confidential_remarks":"The paper is a solid ICLR-style methods contribution and is within scope. The main risk is over-claiming isolation of Topo-Scan given that PH-TR already captures much of the lift and that m/filtrations are tuned on the reported families. I would not reject on novelty grounds: bypassing PDs with interlevel tokens for Transformers is a legitimate design point. If the authors add the shuffle/random-sequence control and temper SOTA language around TopoFormer*, this could move to accept/minor revision quickly. Code link is a plus for reproducibility."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"The useful core here is Topo-Scan: instead of nested sublevel PH that saturates early on graphs, they take a scalar filtration, cut overlapping value windows, and emit short ordered tokens (β0, β1, |V|, |E|) that a Transformer can eat directly. That design choice is clear, parallelizable, and cheaper than diagram pipelines. Code is linked; the stability argument (Thm 3.1 / Cor 3.2, App. B) is the standard interlevel bottleneck reduction done cleanly, not oversold as expressivity.\n\nWhat they do well is the empirical isolation against PH. Table 5 and the expanded Table 15 keep filtration fixed and show PH-TR (Betti curves into a Transformer) already beats PH-MLP, with Topo-Scan adding another consistent 1–3 points across seven filtrations. Runtime tables vs PH/PersLay/TopoGCL and the early-saturation plots (Figs. 2, 6, 7) make the practical case. Classification coverage is broad; multi-filtration fusion and the fingerprint hybrid for MPP are reported honestly rather than hidden.\n\nSoft spots are real but proportionate. On small sets the SOTA margins often sit inside large error bars (MUTAG etc.). Window width m=2 and the filtration menu were chosen on the same families (Table 6, C.5), so the load-bearing claim that the sliding-window representation itself drives the lift is only partly isolated from Transformer capacity and same-family tuning. MPP strength leans on ECFP fusion (TopoFormer*). Foundation-model language in the intro/conclusion is aspirational; the paper is a competent graph-level front-end, not a pretrained foundation model. None of that breaks the central engineering result.\n\nThis is for people who already care about TDA-for-graphs or fixed-size graph embeddings and want a drop-in sequential alternative to PersLay-style pipelines. Math, citations, and reproducibility look solid for an ICLR methods paper. I would send it to referees; I would also bring the Topo-Scan vs PH-TR ablation to reading group. Worth engaging and citing when you need a lightweight topological sequence encoder.","headline":"Solid methods paper: sliding-window interlevel Betti tokens for Transformers beat same-filtration PH baselines and look competitive, with real runtime wins; SOTA claims need tempering on small-data variance and hyperparameter selection.","tokens_in":31587,"tokens_out":548,"would_cite":true,"duration_ms":12321,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.5","headline":"Topo-Scan turns graphs into short topological token sequences that Transformers can read, matching or beating strong GNN and homology baselines without full persistence diagrams.","keywords":["graph representation learning","Topo-Scan","persistent homology","Transformers","topological data analysis","graph classification","molecular property prediction","interlevel Betti numbers"],"falsifier":"On the same filtration, thresholds, and clique 2-skeleton, replace Topo-Scan windows by ordinary sublevel Betti vectors (or randomize slice order) and check whether the Transformer’s accuracy and late-threshold Betti activity collapse to the weaker PH-MLP/PH-TR baselines reported in the ablations.","tokens_in":31418,"feed_emoji":"🔗","tokens_out":928,"duration_ms":19057,"temperature":0.7,"pith_summary":"TopoFormer claims that multi-scale graph structure can be fed to ordinary Transformers by first slicing a graph along a filtration into overlapping windows and recording a few simple numbers per window—connected components, cycles, node count, and edge count. That module, Topo-Scan, deliberately drops the nested filtration and expensive persistence-diagram step of classical topological data analysis, keeping only ordered, parallelizable slice tokens. The paper argues these tokens stay stable when the filtration signal is perturbed, capture late-emerging structure that ordinary sublevel filtrations often bury, and, once attended over, produce graph-level embeddings competitive with strong graph neural nets and topology baselines on classification and molecular property tasks. A sympathetic reader cares because the method offers a fixed-budget, attention-native way to inject topological inductive bias without the usual computational bottlenecks or bespoke vectorizations.","feed_headline":"Graphs become short topological tokens Transformers can read","feed_subtitle":"Topo-Scan skips persistence diagrams yet matches strong GNN and homology baselines on classification and molecules","key_machinery":"Topo-Scan: slice a graph into overlapping induced subgraphs along thresholds of a filtration function, compute (β0, β1, |V|, |E|) on each slice’s clique complex, and feed the resulting fixed-length token sequence to a Transformer; stability follows from bottleneck control of the underlying interlevel persistence modules.","core_discovery":"A short ordered sequence of interlevel Betti numbers and size counts, obtained by sliding a window over a node or edge filtration on the clique complex, is a stable, Transformer-ready summary of multi-scale graph topology that matches or exceeds strong GNN and persistent-homology baselines while remaining fully parallelizable and free of persistence-diagram computation.","pith_inferences":["If Topo-Scan tokens transfer across domains, pretraining a single Transformer on heterogeneous graph corpora of these sequences could yield a lightweight topology-only foundation model that never sees raw nodes.","Learnable or task-conditioned filtrations plugged into the same sliding-window backbone would test whether the method’s gains survive when the filtration is no longer hand-chosen.","Extending per-slice tokens beyond (β0, β1, sizes) to richer local invariants, while keeping sequence length fixed, is a direct route to higher expressivity without restoring full persistence diagrams."],"forward_implications":["Graph-level models can inject multi-scale topology as short token sequences without global boundary-matrix reductions or custom persistence vectorizations.","Late-emerging components and cycles that classical sublevel filtrations bury remain visible to attention because slices are range-localized rather than cumulative.","Stability of the discrete token sequences is controlled by the bottleneck distance of interlevel modules, hence by the sup-norm of the filtration signal.","The same backbone can fuse multiple filtrations by independent Transformers plus learnable attention, and can be hybridized with molecular fingerprints for property prediction.","Fixed-length topological sequences are a natural substrate for future topology-aware graph foundation-model pretraining."],"fun_headline_variants":["Topo-Scan turns graphs into short Transformer-ready tokens","Filtration slices yield stable topological sequences for attention","Graphs become ordered Betti tokens Transformers can process","Parallel topological encodings match GNNs without diagrams","Multi-scale graph topology as short sequences attention reads"],"cache_read_input_tokens":128,"weakest_assumption_plain":"The four simple numbers per slice, on a fixed clique skeleton with a small hand-chosen window width and a few fixed filtrations, keep enough task-relevant structure that the gains come from the sequential Topo-Scan idea itself rather than from fusion tricks or model capacity alone.","fun_headline_variants_meta":{"raw":{"variants":["Topo-Scan turns graphs into short Transformer-ready tokens","Filtration slices yield stable topological sequences for attention","Graphs become ordered Betti tokens Transformers can process","Parallel topological encodings match GNNs without diagrams","Multi-scale graph topology as short sequences attention reads"]},"model":"grok-4.5","effort":"low","cost_usd":0.003031,"raw_usage":{"total_tokens":1051,"prompt_tokens":710,"num_sources_used":0,"completion_tokens":57,"cost_in_usd_ticks":30308000,"prompt_tokens_details":{"text_tokens":710,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":284,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":710,"tokens_out":57,"duration_ms":6391,"temperature":1.0,"reasoning_tokens":284,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-31T13:02:04.449835+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"On the same filtration, thresholds, and clique 2-skeleton, replace Topo-Scan windows by ordinary sublevel Betti vectors (or randomize slice order) and check whether the Transformer’s accuracy and late-threshold Betti activity collapse to the weaker PH-MLP/PH-TR baselines reported in the ablations.","supporting_citations":[],"review_version":1}