{"id":"29aa97e1-935a-4ce0-80ca-3ddbdad09479","arxiv_id":"2412.12416","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"high","formal_verification":"none","parameter_count":8,"one_line_summary":"DeepSN combines a sheaf reaction-diffusion GNN with Louvain-based seed allocation to learn and optimize influence spread under IC, LT, and SIS models.","lead":"DeepSN is a new machine-learning framework for influence maximization that models influence spread as a sheaf reaction-diffusion process and picks seed sets with a community-based search. It reports large gains over prior learning-based methods, especially for non-progressive SIS dynamics.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Table 1 contradicts the paper's headline SIS claim: at 20% budget on Digg, DeepSN reports 33.2% influence vs DeepIM's 36.3%, so 'substantial improvement over all baseline methods' is not supported by the paper's own data.","rationale":"The reader identified the Louvain overlap-minimization assumption as the weakest point and also noted missing code/error bars, but did not flag the direct contradiction inside Table 1. That contradiction is more decisive: the paper's headline empirical claim about SIS is not merely unverified, it is contradicted by the authors' own reported numbers. The theoretical proof issues and Louvain assumption are secondary because they do not by themselves invalidate the empirical results; however, a false or unsupported central empirical claim does. The paper could potentially be repaired by rerunning the experiments, correcting the table, and restating the claim, but as submitted the central claim should not be accepted.","tokens_in":19920,"tokens_out":5948,"duration_ms":53703,"concrete_test":"Recompute the Digg SIS 20% row of Table 1 using the exact SIS parameters (infection rate, recovery rate, simulation horizon, 100 repetitions) and the same seed sets for DeepSN and DeepIM. If the reported values stand, then DeepSN (33.2%) is below DeepIM (36.3%), falsifying the claim of improvement over all baselines; if the values change, correct the table and rerun the full SIS comparison with released code and error bars.","verdict_should_be":"REJECT","load_bearing_attack":"The paper's stated central advantage is that 'DeepSN achieves a substantial improvement over all baseline methods for the SIS diffusion model' (Exp-1). Its own Table 1, SIS block, Digg row, contradicts this: at a 20% seed budget, DeepSN reports 33.2% total influence while DeepIM reports 36.3%; SubSIM reports 32.6%, OPIM 32.3%, and IMM 32.6%. Thus, on at least one dataset and budget combination, DeepSN does not beat all baselines, and at 20% it is worse than DeepIM. This is not a subtle modeling assumption but an internal inconsistency in the main empirical result. It is load-bearing because the SIS improvement is the paper's primary claimed advantage over prior learning-based IM methods. The issue is compounded by the absence of code, error bars, and a precise SIS evaluation protocol: the appendix states only that simulation runs 'until the diffusion process halts, or converges to a steady-state,' which is ambiguous for a non-progressive SIS process, and baseline numbers are taken from Ling et al. (2023) rather than recomputed. Even if the authors intended 'on average across datasets,' the unqualified statement in Exp-1 is not supported by the table as reported.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes DeepSN, a two-phase framework for influence maximization. In the first phase, a sheaf reaction-diffusion GNN is trained to estimate vertex activation probabilities under IC, LT, and SIS diffusion models. In the second phase, seed sets are selected by applying Louvain community detection to a graph weighted by learned sheaf coefficients and then allocating seeds across subgraphs via an MLP. The authors claim that DeepSN and a sparsified variant DeepSNSP outperform or match state-of-the-art baselines, with a substantial improvement under SIS, and provide stability and separability lemmas and propositions in support of the architecture.","tokens_in":20226,"tokens_out":7306,"duration_ms":65204,"significance":"If the claims held, the paper would make a useful contribution to learning-based influence maximization: it targets non-progressive SIS dynamics, scales to graphs of roughly 280,000 vertices, and offers a computationally cheaper variant. Strengths include the breadth of experiments across six datasets and three diffusion models, ablations of the reaction terms and seed-allocation components, and a complexity analysis showing only a constant overhead for the feature dimensions used. The manuscript is also candid about its current limitation of supporting only two-state diffusion models. However, the theoretical support contains load-bearing errors, the main SIS claim is contradicted by the paper's own Table 1, and the seed-selection principle rests on an unvalidated assumption. These issues need to be resolved before the empirical results can be taken at face value.","major_comments":[{"comment":"The statement that 'DeepSN achieves a substantial improvement over all baseline methods for the SIS diffusion model' is contradicted by the numbers reported in Table 1. In the Network Science SIS block at a 1% seed budget, DeepSN reports 2.0% total influence versus DeepIM's 2.7%. In the Digg SIS block at a 20% seed budget, DeepSN reports 33.2% and DeepSNSP reports 33.1% versus DeepIM's 36.3%. Thus the unqualified 'all baselines' claim is not supported by the paper's own data. Please revise the claim, report uncertainty or error bars, and specify the exact SIS evaluation protocol.","section":"Experiments (Exp-1), Table 1"},{"comment":"The proof of Lemma 3 derives ||ΔF X†||2 ≤ C and then uses the largest eigenvalue λmax of ΔF to conclude ||X†||2 ≤ C/(|α|λmax). This is a direction error: the operator norm bound ||ΔF X†|| ≤ λmax ||X†|| cannot be inverted to upper-bound ||X†||. The correct argument would use the smallest eigenvalue, giving ||X†|| ≤ ||ΔF X†||/λmin when ΔF is symmetric positive definite. As written, the proof of boundedness of the fixed point is invalid.","section":"Appendix, proof of Lemma 3"},{"comment":"The proof of Proposition 3 divides the fixed-point equation by X† and treats αΔF as a scalar to impose the condition 0 < αΔF < a/b + c/d. Here X† is a vector (or stacked matrix), ΔF is an operator, and the reaction terms are not scalar functions of X†. The monotonicity argument for f(X†) does not establish existence or uniqueness of a fixed point for the resulting vector-valued nonlinear equation. Consequently Proposition 3 and the separability conclusion of Remark 2 are unproven.","section":"Appendix, proof of Proposition 3"},{"comment":"The paper asserts that applying Louvain to the sheaf-weighted graph Gw 'minimizes the overlap of influence between vertices in different subgraphs.' Louvain maximizes modularity, which is a structural objective; no argument or experiment is provided connecting modularity communities to low influence overlap under IC, LT, or SIS. In addition, the constraint |Si| ≤ (k/n)|Vi| forces seed counts proportional to community size, which is not obviously optimal when communities differ in influence potential. This is a load-bearing premise of the seed-selection phase and should be validated or substantially weakened.","section":"Optimizing Seed Selection"},{"comment":"The evaluation protocol says each diffusion model is simulated 'until the diffusion process halts, or converges to a steady-state.' For a non-progressive SIS model this stopping rule is ambiguous: the process can persist indefinitely, and the infection and recovery rates are not specified in the paper. Since the SIS baseline numbers are taken from Ling et al. (2023) rather than recomputed, the comparability of DeepSN's SIS results depends on exactly matching an unspecified protocol. Please provide the SIS parameters and the precise stopping criterion, and ideally recompute baselines under the same protocol.","section":"Appendix, Experimental Setups"}],"minor_comments":[{"comment":"The tables are difficult to read because values run together without visual separators; for example, in Table 1 the IMINfECTOR IC row contains '50.92.1' where two distinct numbers are merged, and similar merged entries appear for DeepSN in the Power Grid IC row. Please regenerate the tables with clear cell boundaries.","section":"Table 1 and Table 4"},{"comment":"The proof of Lemma 2 compares Φ1_v,i X / (κ + |X|) with Φ1_v,i as though Φ1_v,i is nonnegative; if these coefficients can be negative, the inequality should use |Φ1_v,i| or explicit sign conditions.","section":"Lemma 2 proof"},{"comment":"The proof relies on Proposition 9 of Bodnar et al. (2022) and on 'symmetric sheaves,' but the latter notion is not defined in this manuscript; please add a definition or a precise citation.","section":"Proposition 2 proof"},{"comment":"The training objective for the seed-selection network Tφ involves the influence σ of the selected seed set, but the paper does not explain how gradients flow through the discrete selection operation or how Tφ is trained in practice; a short training description would clarify the method.","section":"Equation (11), training of Tφ"},{"comment":"The reference to Caralt et al. is incomplete (no year or venue), and the Limitations section is a single sentence; please expand both.","section":"References and formatting"}],"recommendation":"major_revision","confidential_remarks":"The paper's core idea is interesting and the empirical scope is substantial, but the current version overclaims the SIS result and contains several mathematical gaps in the appendix. I would encourage the editor to send the revised version back to the same or a similarly qualified reviewer, especially to check the corrected eigenvalue bound and the revised Proposition 3 proof. Code release and recomputed baselines would also materially increase confidence in the empirical comparisons."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things you should know: (1) the paper is a genuine empirical contribution—sheaf diffusion-reaction GNN for influence estimation, plus Louvain-based seed selection on learned sheaf weights—and its numbers on SIS are often much better than baselines; (2) the headline claim that DeepSN achieves a 'substantial improvement over all baseline methods' for SIS is contradicted by its own Table 1, and the appendix proofs have real math errors.\n\nWhat's actually new: combining sheaf diffusion with reaction operators (pointwise and coupled) to capture non-progressive dynamics, and a seed-allocation strategy using Louvain on the sheaf-coefficient weighted graph. That combination doesn't appear in the cited sheaf GNN, reaction-diffusion GNN, or learned-IM literature. The ablation on reaction operators shows each component helps, and the gains over DeepIM on SIS are large on most datasets.\n\nThe soft spots are serious but not fatal. First, Table 1's SIS block on Digg at a 20% budget shows DeepSN at 33.2% vs DeepIM's 36.3%; DeepSNSP also loses there (33.1). So 'over all baseline methods' is false; the claim must be qualified to 'on most datasets/budgets'. Second, Lemma 3's proof bounds ||X†|| using the largest eigenvalue of ∆F; the correct bound uses the smallest eigenvalue via the inverse. Proposition 3's proof divides both sides of a matrix equation by X†, treating vectors as scalars, and then argues about a scalar function. These invalid steps undermine the stated boundedness and uniqueness results. The ε-shift in Eq. (3) also conflicts with Proposition 1's convergence to H^0. Third, there are no error bars, no code or data release, baselines are taken from Ling et al. (2023) rather than recomputed, and the SIS simulation stopping criterion ('until the diffusion process halts') is ambiguous for a non-progressive model. Fourth, the claim that Louvain subgraphs minimize overlap of influence is asserted but never tested; the DeepSN-WSA ablation is indirect evidence at best.\n\nWho this is for: researchers in learning-based IM, especially those working on non-progressive models. The architecture is worth knowing. My recommendation: send it to peer review rather than desk-reject, but require the authors to fix or remove the flawed proofs, soften the SIS claim, release code and data, and add error bars or explain their absence.","headline":"New sheaf diffusion-reaction architecture for influence maximization with strong SIS numbers, but the headline claim is internally contradicted and the theoretical appendix needs correction.","tokens_in":20790,"tokens_out":4708,"would_cite":false,"duration_ms":38893,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"DeepSN claims a sheaf reaction-diffusion GNN with Louvain-based seed allocation matches or beats state-of-the-art influence maximization, with large SIS gains.","keywords":["influence maximization","sheaf neural networks","graph neural networks","reaction-diffusion","seed selection","SIS model","Louvain community detection","diffusion models"],"falsifier":"Take a real network such as Cora-ML under the IC model, compute the Louvain partition of the learned weighted graph $G_w$, and measure the expected number of vertices reachable from seeds in two different subgraphs that are also reachable from one another. If that cross-subgraph overlap is large, or if replacing the per-subgraph constraint $|S_i| \\leq (k/n)|V_i|$ with global seed selection gives equal or larger total influence, the central seed-allocation claim fails.","tokens_in":19643,"feed_emoji":"📈","tokens_out":6779,"duration_ms":53911,"temperature":0.7,"pith_summary":"This paper introduces DeepSN, a graph neural network that treats influence propagation as a sheaf reaction-diffusion process and learns to pick seed sets that maximize spread. The central claim is that combining a sheaf Laplacian with learnable vertex-confidence weights and two reaction operators lets the model capture both progressive (IC, LT) and non-progressive (SIS) diffusion dynamics, where standard GNNs oversmooth and fail. For seed selection, DeepSN builds a weighted graph from the learned sheaf coefficients, partitions it with Louvain, and allocates seeds proportionally to subgraph sizes, shrinking the combinatorial search space. In experiments on graphs up to roughly 280,000 vertices, DeepSN matches or beats state-of-the-art methods on IC and LT and reports substantial gains on SIS. If the claims hold, learning-based influence maximization can be made both diffusion-agnostic and scalable to real networks.","feed_headline":"Sheaf GNN beats baselines on influence spread, especially SIS","feed_subtitle":"A sheaf reaction-diffusion network learns IC, LT, and SIS dynamics and picks seeds via Louvain subgraphs.","key_machinery":"The load-bearing objects are the nonlinear sheaf Laplacian with learnable trust coefficients and the two reaction operators that turn pure diffusion into reaction-diffusion. The Laplacian measures, per edge, the disagreement between two vertices' public opinions after translation by the restriction maps; the reaction terms add bounded, non-monotonic updates and remove the forced equality of transformed features at the fixed point ($F_{v\\unlhd e}x_v = F_{u\\unlhd e}x_u$), which is what gives the model separability and protects against oversmoothing. On the optimization side, the key mechanism is building the weighted graph $G_w$ from the learned sheaf coefficients, applying Louvain to obtain subgraphs that are assumed to have low cross-subgraph influence overlap, and allocating seed budget proportionally to subgraph size.","core_discovery":"The paper's central claim is that influence maximization can be solved end-to-end by a sheaf reaction-diffusion GNN coupled to a subgraph-based seed optimizer. Influence propagation is modeled as a sheaf diffusion process: each vertex carries a private opinion vector, each edge carries a public-opinion space, and restriction maps translate private opinions to edge level; a learnable coefficient $\\psi_{vu}$ in the nonlinear sheaf Laplacian encodes how much vertex $v$ trusts neighbor $u$. Two reaction operators, pointwise dynamics for vertex-intrinsic state changes and coupled dynamics for the combined effect of activated and susceptible neighbors, make the evolution non-monotonic and, the paper argues, capture non-progressive models such as SIS. After training on ground-truth activation probabilities, the learned sheaf coefficients define a weighted graph $G_w$, which is partitioned by Louvain into subgraphs; a small MLP then selects seeds within each subgraph subject to $|S_i| \\leq (k/n)|V_i|$, and total influence is estimated by the trained sheaf GNN. The reported outcome is that DeepSN and its sparsified variant DeepSN_SP match or outperform IMM, OPIM, SubSIM, IMINFECTOR, PIANO, ToupleGDD, and DeepIM across IC, LT, and SIS, with the largest relative gains on SIS.","pith_inferences":["If the Louvain overlap assumption is weak on real networks, proportional seed allocation could underperform global selection; a natural extension is to learn per-subgraph budgets instead of fixing $|S_i| \\leq (k/n)|V_i|$.","The reaction operators resemble epidemic and opinion-dynamics terms, so the architecture may transfer to multi-stage or multi-state models such as SEIR or threshold models with more than two states, which the paper lists as future work.","DeepSN's gains on SIS suggest that other non-progressive phenomena, like recurring misinformation or churn in user activity, could be treated with the same estimator.","Since baseline numbers are taken from a single prior study, an independent replication or an evaluation against freshly run traditional algorithms would clarify whether the improvement comes from the sheaf estimator, the Louvain seed allocation, or the experimental setup."],"forward_implications":["If the sheaf reaction-diffusion model genuinely captures non-progressive dynamics, a single trained DeepSN can be applied to diffusion models with recovery or oscillation without task-specific re-engineering.","The Louvain-based reduction turns the exponential seed search into per-community selection, which is what lets the method scale to graphs of roughly 280,000 vertices.","The relaxation of the sheaf fixed-point condition implies the architecture can keep neighboring vertex features distinct, a property the paper links to stronger estimation of long-range influence.","Because the diffusion model is learned from ground-truth activation data rather than supplied, the same framework can switch among IC, LT, and SIS settings without changing the pipeline."],"supporting_citations":[{"why":"Defines neural sheaf diffusion and the oversmoothing analysis that DeepSN extends; the DeepSN Laplacian reduces to it when all sheaf coefficients are 1.","marker":"Bodnar et al. (2022)"},{"why":"Supplies the H0 convergence theorem used in Proposition 1 to characterize the fixed point of sheaf diffusion.","marker":"Hansen and Ghrist (2021)"},{"why":"Provides the Louvain algorithm used to partition the sheaf-coefficient weighted graph into subgraphs.","marker":"Blondel et al. (2008)"},{"why":"Supplies the experimental setup and baseline results (including DeepIM) that DeepSN is compared against.","marker":"Ling et al. (2023)"},{"why":"Defines the influence maximization problem and the IC model that ground the objective and experiments.","marker":"Kempe, Kleinberg, and Tardos (2003)"},{"why":"Introduces graph neural reaction-diffusion networks that motivate the pointwise and coupled reaction operators.","marker":"Choi et al. (2023)"},{"why":"Formalizes the SIS model used to test non-progressive diffusion.","marker":"d'Onofrio (2008)"}],"fun_headline_variants":["Sheaf diffusion GNN learns IC, LT, SIS and picks seeds","DeepSN: end-to-end influence maximization with sheaf neural nets","Sheaf reaction-diffusion GNN matches or beats strong baselines","Subgraph seed selection with sheaf diffusion beats IMM and more","Learned sheaf Laplacian captures non-progressive influence"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument assumes that partitioning the sheaf-coefficient weighted graph with Louvain actually minimizes the overlap of influence between vertices in different subgraphs, so that allocating seeds proportionally to subgraph size does not sacrifice spread.","fun_headline_variants_meta":{"raw":{"variants":["Sheaf diffusion GNN learns IC, LT, SIS and picks seeds","DeepSN: end-to-end influence maximization with sheaf neural nets","Sheaf reaction-diffusion GNN matches or beats strong baselines","Subgraph seed selection with sheaf diffusion beats IMM and more","Learned sheaf Laplacian captures non-progressive influence"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000752,"raw_usage":{"total_tokens":3392,"prompt_tokens":1039,"completion_tokens":2353,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":655,"completion_tokens_details":{"reasoning_tokens":2260}},"tokens_in":655,"tokens_out":2353,"duration_ms":16088,"temperature":1.0,"reasoning_tokens":2260,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T14:06:41.218235+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a real network such as Cora-ML under the IC model, compute the Louvain partition of the learned weighted graph $G_w$, and measure the expected number of vertices reachable from seeds in two different subgraphs that are also reachable from one another. If that cross-subgraph overlap is large, or if replacing the per-subgraph constraint $|S_i| \\leq (k/n)|V_i|$ with global seed selection gives equal or larger total influence, the central seed-allocation claim fails.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines neural sheaf diffusion and the oversmoothing analysis that DeepSN extends; the DeepSN Laplacian reduces to it when all sheaf coefficients are 1."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the H0 convergence theorem used in Proposition 1 to characterize the fixed point of sheaf diffusion."},{"cited_title":"D.; Guillaume, J.-L.; Lambiotte, R.; and Lefebvre, E","cited_arxiv_id":null,"evidence_quote":"Provides the Louvain algorithm used to partition the sheaf-coefficient weighted graph into subgraphs."},{"cited_title":"T.; Xue, R.; Song, J.; Qiu, M.; and Zhao, L","cited_arxiv_id":null,"evidence_quote":"Supplies the experimental setup and baseline results (including DeepIM) that DeepSN is compared against."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines the influence maximization problem and the IC model that ground the objective and experiments."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces graph neural reaction-diffusion networks that motivate the pointwise and coupled reaction operators."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Formalizes the SIS model used to test non-progressive diffusion."}],"review_version":1}