{"id":"44cf80ff-fe39-4f5c-b804-e060c269caca","arxiv_id":"2501.09178","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":7,"one_line_summary":"Localized persistent homology features can make graph neural networks more expressive and slightly more accurate, but the state-of-the-art claim is not uniformly supported.","lead":"This paper adds extra structural fingerprints, computed locally around each node or edge, to graph neural networks and tests them on node classification and link prediction. It matters because the fingerprints can sometimes tell apart graphs that standard powerful graph algorithms cannot, which may improve learning on complex networks.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The separation examples require a hand-crafted edge tie-break filter not used by the model, leaving the expressiveness theorems unproven for the deployed vectorized features.","rationale":"The reader's weakest assumption and my analysis converge: the paper proves expressiveness for exact EPDs under a tie-break filter, while the deployed system consumes persistence images (or PI+) under a different filter. Both observations are explicitly supported by the manuscript: the footnote after Equation (2) says the relaxed function is only for the proof and that experiments show the alternative does not hurt performance; Section 4.1 says PIs discard zero-persistence and makes level/edge counts hard to extract, motivating PI+. The discrepancy is not a fatal error. The theorems remain genuine results for their stated filter, and Appendix D provides empirical evidence that the filter change does not harm accuracy on the tested benchmarks. The empirical SOTA claim is also weakened by Table 2, where PEGN is below GAT on Cora/Citeseer and below GCN(+PI+) on Photo, and Table 3 shows PEGN(PI) beating PEGN on several link-prediction datasets, but these are secondary to the theory/implementation mismatch. The concrete test I propose directly closes the transfer gap: check whether the separation examples survive under the standard filter and the PI/PI+ vectorization. If they survive, the weakest assumption is resolved; if not, the central theoretical justification is narrower than claimed and the verdict should remain CONDITIONAL with a clear caveat. I therefore keep CONDITIONAL rather than REJECT, because the empirical contributions and the theoretical results for the tie-break filter still stand, but the load-bearing transfer from theorem to model remains unproven.","tokens_in":29189,"tokens_out":1647,"duration_ms":16569,"concrete_test":"Verify Theorem 1 without Equation (2): recompute the 1D EPD of the 1-hop vicinity of a vertex in the Shrikhande graph and the 4x4 Rook graph using the standard filter f(uv)=max(f(u),f(v)) and the deployed PI/PI+ vectorization (Eq. 4). If the resulting vectors coincide for the two graphs, the strict separation result does not transfer to the implemented setting, and the theoretical justification for the practical method is narrower than claimed.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Theorems 1 and 2 are proved using engineered filter functions: Equation (2) sets f(uv)=max(f(u),f(v)) when endpoint levels differ but f(u)+0.5 when they are equal, and the tie-break is essential because it lets the 1D EPD read off the Betti number of the layer-1 induced subgraph. The deployed PEGN instead uses ordinary shortest-path/Ollivier-Ricci distance filters with standard lower/upper-star extensions (Section 4.2), which the authors themselves flag as 'slightly different' from the proof filter in the footnote to Equation (2). Theorem 3 similarly assumes the tie-break filter. Appendix D shows empirically that swapping filters gives nearly identical node-classification results, but no proof is given that the standard filter preserves the separation power of the counterexamples, nor is the lossy persistence-image or PI+ vectorization shown to preserve it. The paper's own Section 4.1 concedes that level counts and intra/inter-layer edge counts are 'difficult to recover' from a plain persistence image, and PI+ adds these counts by hand. The theoretical upper bound also rests on an unformalized hash-simulation argument (Appendix B, Eqs. 13-14). The gap between the proved expressiveness of exact EPDs under tie-break filters and the features actually fed to the GNN is the load-bearing weak point.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a family of localized topological features for graph representation learning, based on extended persistent homology (EPH) computed on k-hop vicinity graphs of nodes (for node classification) and on intersections of k-hop neighborhoods (for link prediction). The diagrams are vectorized as persistence images (PI) and an augmented version (PI+) that concatenates level counts and intra/inter-layer edge counts. These features are combined with GNN backbones in the proposed Persistence Enhanced Graph Network (PEGN), with variants that either combine multiple precomputed filter functions or learn the filter function end-to-end. The paper reports accuracy and AUC improvements over several baselines on standard node classification and link prediction benchmarks. It also states four theorems (Theorems 1-4) about the expressiveness of exact extended persistence diagrams relative to k-WL, including examples where EPDs separate pairs that 3-WL or 4-WL cannot distinguish, an upper bound showing 1D EPD is less powerful than 4-WL, and a result on distinguishing most pairs of regular graphs.","tokens_in":29396,"tokens_out":7412,"duration_ms":74737,"significance":"If the expressiveness results were proved for the features actually fed into the GNN, this would be a meaningful contribution integrating topological data analysis with GNNs, with concrete guidance for practitioners. The empirical study is broad: it covers seven node-classification and five link-prediction datasets, compares against a wide range of strong baselines, reports mean and standard deviation over ten runs, and releases source code. The counterexample-based separation results for exact EPDs are interesting and directly relevant to the TDA-for-graphs literature. The main weakness is that the theoretical claims are established only for exact persistence diagrams under a hand-crafted tie-break filter, while the implemented model consumes persistence-image vectors computed with standard lower/upper-star filtrations; no theorem bridges this gap. The end-to-end learning variants are honestly evaluated, including their computational cost, which is a strength of the paper.","major_comments":[{"comment":"Theorems 1 and 2 are proved for exact extended persistence diagrams computed with the tie-break filter function in Eq. (2), namely f(uv)=max(f(u),f(v)) when endpoint levels differ and f(u)+0.5 when they are equal (and the analogous descending rule in Eq. (3)). The deployed PEGN model in Section 4.2 instead uses standard lower-star and upper-star extensions of shortest-path or Ollivier-Ricci distance filters, and then compresses the diagrams into persistence images (or PI+ vectors, Eq. (4)). The footnote to Eq. (2) and Appendix D report only that node-classification accuracies under the two filters are nearly identical on the benchmark datasets; they do not establish that the specific separating pairs from Theorems 1 and 2 (Shrikhande vs. 4x4 Rook, and the G(0)/G(1) pair) are still separated by the persistence-image vectorization used in the model, with the stated resolution and sigma parameters. Since the abstract and introduction use the expressiveness results to justify the proposed framework, this gap is load-bearing: the proven statements concern an object that the model does not consume. The authors should either prove that the deployed features preserve the separation for these pairs, or explicitly restrict the theoretical contribution to exact EPDs and adjust the corresponding claims about the implemented method.","section":"Section 4.1 (Eq. (2)) and Section 4.2"},{"comment":"The proof of Theorem 3 relies on the assertion that 'there exists a bijective hashing function HASH that can approximate the computation of the 1D EPDs' (Eqs. (13)-(14)). This is not a formal simulation argument: Algorithm 1 is a union-find procedure whose state includes the evolving disjoint-set partition and the active cycle representatives C_i, and the paper does not show that the subgraph-WL coloring sequence injectively encodes this state at every iteration. Moreover, the word 'approximate' is too weak for an upper-bound theorem, because an approximate simulation could in principle merge persistence points that the exact algorithm distinguishes. Consequently, the conclusion that 1D EPD is 'less powerful than 4-WL' does not follow from the argument as written. A precise reduction is needed, such as showing that each extended persistence point (b,d) is determined by a histogram computed by subgraph-WL, or a direct argument that any graph pair distinguished by the 1D EPD is distinguished by 4-WL.","section":"Appendix B (Theorem 3)"},{"comment":"The paragraph before Theorem 1 claims that adding EPDs as topological position embeddings makes the 'resulting enhanced GNN' more expressive than 2-WL, not less expressive than 3-WL, and less expressive than 4-WL. However, Theorems 1-3 are statements about the distinguishing power of the exact EPD itself, not about the GNN architecture in Eqs. (5)-(6) or about the vectorized PI/PI+ representation that the GNN actually receives. No theorem shows that the message-passing network with PI or PI+ features can recover the relevant persistence points, e.g., the (1.5, 0.5) points used in the proof of Theorem 1, even though the paper asserts these queries 'can be easily implemented using MLP.' Thus the theoretical results justify a property of an idealized feature extractor, not of the implemented architecture. The authors should either add a formal statement about the implemented model (for instance, showing that an MLP on PI+ can compute the layer counts and Betti numbers used in the separation proofs) or clearly delimit the expressiveness claims to exact EPDs.","section":"Section 4.1 bullet list and Theorem 4"}],"minor_comments":[{"comment":"The abstract and conclusion state that the method 'achieve[s] state-of-the-art performance,' but Table 2 shows that PEGN does not beat GAT on Cora (82.7 vs. 83.0) or Citeseer (71.9 vs. 72.5), and Table 3 shows HGCN outperforms PEGN on Citeseer link prediction (96.6 vs. 95.1). The claim should be qualified as state-of-the-art on most benchmarks.","section":"Abstract and Section 5.1"},{"comment":"The introduction says 'We propose the first method to adapt persistent homology for localized graph representation,' then states that the work builds on two preliminary conference papers that were 'the first to adapt persistent homology for localized node-level and edge-level representation learning.' This phrasing should be reconciled to avoid an apparent novelty contradiction.","section":"Section 1"},{"comment":"In Eq. (4) and the surrounding text, the counts n_j, n_{j,j}, and n_{j,(j+1)} are defined for the node vicinity graph G^k_u, but in the link prediction setting the vicinity graph is the intersection G^k_{uv}. The notation should be adapted or clarified so that PI+ is unambiguously defined for the pair-wise setting.","section":"Section 4.2.2"},{"comment":"The proof of Theorem 2 relies on the cited result that the constructed graphs G(0) and G(1) 'cannot be differentiated by 4-WL (Yan et al., 2024b).' Since this is the sole external load-bearing result for the 4-WL part of the theorem, the reference should include the specific theorem or proposition being invoked, so that the reader can verify the claim without recovering it from a broader paper.","section":"Appendix A"},{"comment":"The notation for persistence points is inconsistent: Section 3.2 states that for loops the death time is smaller than the birth time, while the proof of Theorem 1 refers to 'persistence points with persistence value (1.5, 0.5)' without specifying whether the pair is (birth, death) or (death, birth). Please standardize the order and define it once in Section 3.2.","section":"Section 3.2 and Theorem 1 proof"}],"recommendation":"major_revision","confidential_remarks":"This is an extended journal version of two conference papers by the same group (Zhao et al., 2020; Yan et al., 2021). The incremental contribution over those papers is the expressiveness analysis and the end-to-end learning discussion, which fits the scope of JMLR. The main risk is the unproven transfer from exact-EPD expressiveness to the deployed persistence-image pipeline; if the authors cannot close that gap, the theoretical section should be substantially narrowed. The self-citation pattern is heavy but mostly relevant to prior work in the same line. The empirical claims are mostly consistent with the tables, though the 'state-of-the-art' phrasing is too strong. I see no evidence of fabrication or circularity in the experiments."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: this is a real contribution, not a repackaging. The expressiveness results—EPDs separating the Shrikhande/Rook pair and a CFI-style pair that 4-WL cannot separate, plus the regular-graph distinguishment bound—are new and concrete. The PI+ encoding is a sensible response to the known lossiness of persistence images, and the end-to-end study is honest about cost and benefit. The empirical core is defensible: adding PI or PI+ to GCN consistently improves dense benchmarks, and the link prediction results hold up. The citation pattern is fine; the heavy self-citation points to the actual foundations.\n\nThe main soft spot is the one the stress-test note flags. Theorems 1-3 are proved for exact extended persistence diagrams using the tie-break filter in Equation (2), while the deployed PEGN uses ordinary shortest-path or Ollivier-Ricci filters with standard lower/upper-star extensions. The authors are transparent about this in the footnote, and Appendix D shows near-identical accuracy with the proof filter, which is decent evidence but not a proof that the separation power transfers. More importantly, no theorem carries the expressiveness from exact diagrams to the persistence-image or PI+ vectors the model actually consumes. The paper itself concedes that level counts and intra/inter-layer edge counts are hard to recover from a plain persistence image, which is why PI+ adds them by hand. That is an honest fix, but the separation results are not shown to survive vectorization.\n\nTwo smaller issues. Theorem 3 relies on an informal \"HASH can approximate the computation\" argument; it needs a formal simulation. And \"state-of-the-art\" in the abstract is looser than the tables: PEGN alone trails on Cora and Citeseer, though the PI+ variants do top those benchmarks, so the overstatement is minor.\n\nOverall this deserves serious refereeing. The gaps are fixable, and the paper would be stronger with a theorem or a careful statement that the practical claim is about the enriched features, not the exact-diagram expressiveness. For a TDA-for-GNN reader this is worth the time; I would bring it to a reading group to argue about the vectorization gap.","headline":"A solid extension of the authors' own localized-PH line, with genuinely new expressiveness bounds—but the theorems are proved for exact diagrams under a tie-break filter, while the deployed model uses lossy vectorized features, so the theoretical support for the implemented method is looser than the abstract suggests.","tokens_in":30053,"tokens_out":3998,"would_cite":true,"duration_ms":42460,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["55N31","05C60","68R10","68T07"],"pacs":[],"model":"deepseek-v4-flash","headline":"Localized loop-counting features from extended persistent homology make graph neural networks strictly more expressive than 3-WL and 4-WL on specific graph pairs, and lift node classification and link prediction accuracy.","keywords":["Persistent Homology","Topological Data Analysis","Graph Neural Network","Graph Representation Learning","Graph Isomorphism","Extended Persistent Homology","Node Classification","Link Prediction"],"falsifier":"Take the Shrikhande and 4×4 Rook graphs and the CFI pair, compute the implemented 25-pixel persistence image with the paper's default Gaussian smoothing plus the PI+ counts, and check whether the feature vectors, and the trained GNN embeddings, are identical up to symmetry; if they coincide, the separation proved for exact EPDs has been lost in vectorization and the implemented method's expressiveness claim fails.","tokens_in":28880,"feed_emoji":"🕸️","tokens_out":12216,"duration_ms":112827,"temperature":0.7,"pith_summary":"This paper argues that graph neural networks miss local loop and connectivity structure, and that this structure can be extracted in advance with the topological tool of extended persistent homology. The method builds a small vicinity graph around each node or node pair, filters it by a distance function, and records the births and deaths of connected components and loops in an extended persistence diagram. These diagrams, compressed into persistence images and augmented with simple counts, are fed into a graph neural network as extra features or as message weights. If the paper is right, ordinary message-passing GNNs can become strictly more expressive than their usual 1-WL ceiling, separating graph pairs that even high levels of the classic Weisfeiler-Leman hierarchy, namely 3-WL and 4-WL, cannot separate, and the added structure shows up as higher accuracy on node classification and link prediction benchmarks.","feed_headline":"Loop features let GNNs beat 3-WL and 4-WL on hard pairs","feed_subtitle":"Persistent-homology features on node neighborhoods sharpen node classification and link prediction.","key_machinery":"The central object is the extended persistence diagram (EPD) of a vicinity graph filtered by a distance function. Ordinary persistence records when a loop or component is born but not when it dies, so the paper uses extended persistence, which tracks each structure across both ascending and descending filtrations and captures the whole life span of loops and of the connected component. The EPD is computed on the hop-bounded neighborhood of a node for node classification, and on the intersection of the two endpoints' neighborhoods for link prediction, with distance measured under a graph metric. Because diagrams live in an infinite-dimensional space, the paper vectorizes them into persistence images (Gaussian-smoothed pixel grids) and augments the vector with counts of nodes at each distance layer, intra-layer edges, and crossing edges to form PI+; these vectors then reweight message passing in the Persistence Enhanced Graph Network (PEGN).","core_discovery":"The paper claims that localized extended persistence diagrams (EPDs) are a rich and provably powerful representation of local graph structure. With the shortest-path distance as the filter function, the EPD of a node's hop-bounded vicinity graph counts the cycles formed by that node's neighbors and other layer-wise connectivity statistics; this lets the features separate the Shrikhande graph from the 4×4 Rook graph, a pair 3-WL cannot separate, and separate a CFI pair (non-isomorphic graphs built to defeat low-dimensional Weisfeiler-Leman tests) that 4-WL cannot separate. The same EPD is simultaneously bounded: with shortest-path-distance filters, 1D EPDs are less powerful than 4-WL overall, and for most pairs of sufficiently large regular graphs of a fixed degree an EPD with a bounded number of hops tells the two graphs apart. The paper further claims that vectorized versions of these features, namely persistence images plus hand-added layer and edge counts (PI+), improve node classification and link prediction beyond strong baselines when used to reweight GNN messages, and that end-to-end learning of the filter function is feasible but mainly pays off on large, dense link-prediction benchmarks.","pith_inferences":["The expressiveness theorems are proved for exact EPDs with shortest-path-distance filters, while the deployed model uses lossy persistence images and an Ollivier-Ricci-curvature-based graph metric; a direct comparison on the Theorem 1 and Theorem 2 counterexample pairs would show whether the vectorized feature retains the proved separation power.","The PI+ design points to a general recipe: pair any lossy topological vectorization with hand-added structural counts that the vectorization is known to blur, and the same pattern could be extended to higher-order recoverable statistics such as numbers of short cycles through a node.","A cheap ablation replacing Gaussian persistence images with exact-diagram-derived counts or Dirac peaks would isolate whether the empirical gains come from the smoothed persistence image or from the explicitly appended structural counts, which the paper does not run."],"forward_implications":["Augmented GNNs can tell apart the Shrikhande graph and the 4×4 Rook graph, so their distinguishing power on that pair exceeds 3-WL.","A CFI pair that 4-WL cannot separate is separated by the same localized EPD construction, although Theorem 3 keeps 1D EPDs below 4-WL overall.","PI+ restores layer and edge counts that plain persistence images blur, giving node and edge embeddings access to local structural statistics that standard message passing cannot easily recover.","End-to-end learning of localized topological features is computationally heavy; the benchmarks show it helps mainly on large and dense link-prediction graphs, while precomputed curvature-based filters usually win for node classification."],"supporting_citations":[{"why":"Introduces extended persistent homology, the construction that lets loops and the whole connected component be born and killed, so EPDs capture full lifetimes.","marker":"Cohen-Steiner et al., 2009"},{"why":"Defines persistence images, the fixed-size vectorization used to turn EPDs into GNN-ready features.","marker":"Adams et al., 2017"},{"why":"Supplies the recoverability results for layer sizes, intra-layer and inter-layer edge counts, and clustering coefficients that motivate the distance filter and the PI+ augmentation.","marker":"Tian and Wang, 2019"},{"why":"The preliminary node-level persistence-enhanced GNN that this paper extends with expressiveness analysis and end-to-end learning.","marker":"Zhao et al., 2020"},{"why":"The preliminary edge-level persistent-homology link prediction work that this paper extends.","marker":"Yan et al., 2021"},{"why":"Establishes that 3-WL cannot separate the 4×4 Rook and Shrikhande graphs, the counterexample pair used in Theorem 1.","marker":"Arvind et al., 2020"},{"why":"The CFI graph construction whose variants yield the pair that 4-WL cannot separate in Theorem 2.","marker":"Cai et al., 1992"},{"why":"Provides the subgraph-WL bound used to show that 1D EPDs are less powerful than 4-WL in Theorem 3.","marker":"Huang et al., 2023a"},{"why":"The random regular graph analysis used in Theorem 4's proof that EPDs distinguish almost all regular graph pairs.","marker":"Bollobás, 1982"},{"why":"Supplies the distance-function filter used in Theorem 2 and the pair-wise neighborhood formulation adopted for link prediction.","marker":"Zhang and Chen, 2018"}],"fun_headline_variants":["Topological features give GNNs power beyond 3-WL and 4-WL","Persistent homology features sharpen GNNs beyond WL tests","Local topological loops give GNNs a stronger separation power","Graph neural nets get a topological upgrade for node and link tasks","Persistent homology features make GNNs more expressive"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The weakest load-bearing premise is that the lossy vectorization used in the deployed model, namely persistence images plus hand-added counts, preserves the distinguishing power that the paper proves for exact persistence diagrams; the paper does not prove this transfer.","fun_headline_variants_meta":{"raw":{"variants":["Topological features give GNNs power beyond 3-WL and 4-WL","Persistent homology features sharpen GNNs beyond WL tests","Local topological loops give GNNs a stronger separation power","Graph neural nets get a topological upgrade for node and link tasks","Persistent homology features make GNNs more expressive"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000671,"raw_usage":{"total_tokens":3048,"prompt_tokens":926,"completion_tokens":2122,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":542,"completion_tokens_details":{"reasoning_tokens":2034}},"tokens_in":542,"tokens_out":2122,"duration_ms":15145,"temperature":1.0,"reasoning_tokens":2034,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T20:11:03.346523+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take the Shrikhande and 4×4 Rook graphs and the CFI pair, compute the implemented 25-pixel persistence image with the paper's default Gaussian smoothing plus the PI+ counts, and check whether the feature vectors, and the trained GNN embeddings, are identical up to symmetry; if they coincide, the separation proved for exact EPDs has been lost in vectorization and the implemented method's expressiveness claim fails.","supporting_citations":[],"review_version":1}