{"id":"becc8fbf-65c9-46b3-8138-2b8f3475d7d8","arxiv_id":"2507.23111","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"BiGG-E is an autoregressive generative model that jointly samples graph topology and positive edge weights with O((n+m) log n) complexity.","lead":"A new AI model, BiGG-E, generates weighted graphs where both the connections and the edge weights are produced together. It extends the scalable BiGG model so that large, sparse weighted graphs can be created efficiently, which matters for network data in biology, chemistry, and social science.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Equation 3 gives the wrong softplus-normal likelihood: the printed coefficient 1/(2σ²) instead of 1/(σ√(2π)) means that, if implemented as written, the weight loss is not maximum likelihood and σ estimates are biased.","rationale":"The reader's weakest assumption (canonical ordering) is legitimate, but it is a known limitation inherited from BiGG and is only partially tested by the MMD metrics, which are node-order-invariant. The likelihood formula in Eq. 3 is more load-bearing because it concerns whether the training objective is a valid likelihood at all. If the printed formula is used, the σ parameter is biased and the model is not optimizing the distribution used for sampling. The paper's strong empirical weight matches suggest the code may already implement the correct softplus-normal density, which is why I frame this as a concrete check rather than a rejection. If the code confirms the correct density, the Eq. 3 issue reduces to a typo and the remaining concerns are the empirical ones the reader already identified; if not, the central claim about the joint weighted-graph likelihood is degraded. The verdict therefore remains conditional, pending this verification.","tokens_in":22152,"tokens_out":18807,"duration_ms":252024,"concrete_test":"Inspect the public BiGG-E repository's weight loss. The correct log-density term, up to θ-independent constants, is −log σ − (1/(2σ²))(log(e^w−1)−μ)²; Eq. 3 as printed implies −2log σ − (1/(2σ²))(log(e^w−1)−μ)². If the code matches Eq. 3, run a controlled recovery experiment: draw 10,000 weights from Softplus(N(0,1)), train only the weight head with that loss, and compare the estimated σ² to 1. A recovered σ² near 0.5 would confirm the bias; a value near 1 with the correct coefficient in code would show Eq. 3 is a typographical error. Also check whether the code includes the log-Jacobian, though it is not needed for MLE.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Section 3.1 defines W = Softplus(ε) with ε|e ~ N(μ, σ²), and Eq. 3 states pθ(w|e) ∝ 1/(2σ²) exp[−1/(2σ²)(log(e^w−1)−μ)²]. The actual density is f_W(w) = (1/√(2π)σ) · e^w/(e^w−1) · exp[−1/(2σ²)(log(e^w−1)−μ)²]. The missing Jacobian e^w/(e^w−1) depends only on w and so does not affect maximum likelihood estimation of θ, but the printed σ coefficient is not a harmless constant: 1/(2σ²) versus 1/(√(2π)σ) changes the dependence on the variance parameter. For a single observation z = log(e^w−1)−μ, the printed objective has stationary point σ² = z²/2, whereas the true likelihood has σ² = z². Because the weight log-likelihood in Eq. 14 (Appendix A.1) is the training signal for the weight state, an implemented Eq. 3 would mean BiGG-E is not maximizing the likelihood of the softplus-normal distribution it claims to sample from, directly undermining the central claim that it learns a joint distribution over weights and topology. The close empirical standard deviations in Table 2 suggest the public code may quietly use the correct density, so this concern is checkable rather than definitive.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes BiGG-E, an autoregressive generative model that extends BiGG to weighted sparse graphs. The model factorizes each weighted edge as p(e, w) = p(e)p(w|e), models positive edge weights through a softplus transformation of a Gaussian, and maintains separate Fenwick-tree-based states for topology and for previously sampled weights so that topology and weights are generated jointly. The paper claims an O((n + m) log n) generation time for sparse weighted graphs, reports training and sampling scalability on trees up to 15K nodes, and compares against Adj-LSTM, BiGG-MLP, BiGG+GCN, and an Erdős–Rényi baseline on Erdős–Rényi, tree, 3D point cloud, lobster, and joint tree datasets. The central empirical claim is that BiGG-E \"best captures distributions over weighted graphs while remaining scalable and computationally efficient.\"","tokens_in":22507,"tokens_out":8455,"duration_ms":103210,"significance":"If the central claims are substantiated, the paper fills a genuine gap: most deep graph generative models are unweighted, and the few weighted variants do not jointly model topology and continuous weights at scale. The paper's explicit use of sparsity to avoid O(n^2) work, its public code release, and its inclusion of a joint-tree stress test designed to expose topology-weight coupling are strengths. The significance is currently moderated by a likelihood error in the stated density, by the absence of uncertainty quantification on the MMD comparisons, and by an ambiguity in the complexity argument for the weight-state construction. These issues are fixable, but they bear directly on the two headline claims: that the model learns the stated joint distribution and that it does so with the stated complexity.","major_comments":[{"comment":"Equation (3) is not a valid probability density for the softplus-normal model described in the text. If W = Softplus(ε) with ε|e ~ N(μ, σ²), the density of W is f_W(w) = (1/(√(2π)σ)) * (e^w/(e^w − 1)) * exp[−(log(e^w − 1) − μ)²/(2σ²)] for w > 0. The printed coefficient 1/(2σ²) is not a harmless normalization: for a single observation z = log(e^w − 1) − μ, maximizing the printed expression gives σ̂² = z²/2, whereas the true maximum-likelihood estimate is σ̂² = z². Since Equation (14) uses this term as the weight log-likelihood, an implementation of Equation (3) would not maximize the likelihood of the claimed generative distribution. The omitted Jacobian e^w/(e^w − 1) does not depend on θ and is therefore harmless for maximum-likelihood training, but the variance coefficient must be corrected. Please correct the equation and state whether the released code uses the corrected density.","section":"§3.1, Eq. (3)"},{"comment":"The text states that BiGG+GCN holds a \"slight advantage\" in unweighted degree MMD for the 3D Point Cloud dataset, but the reported values are 7.40e-3 for BiGG-E versus 4.37e-4 for BiGG+GCN: a 17-fold gap, not a slight one. This is difficult to reconcile with the abstract's claim that BiGG-E \"best captures distributions over weighted graphs.\" The claim should be qualified to the metrics that BiGG-E actually leads, or the discrepancy should be analyzed and explained.","section":"§4.1, Table 1 (3D Point Cloud row)"},{"comment":"All reported MMD values, error rates, means, and standard deviations are single point estimates; no standard errors, confidence intervals, or repeated-seed results are given. This is a load-bearing issue for the empirical claim that BiGG-E \"consistently outperforms\" the alternatives, because several differences are small (e.g., Table 1 Tree Deg. 2.94e-6 vs 8.47e-6) and some comparisons go the other direction (e.g., Table 2 Erdős–Rényi MMDWT 6.83e-3 for BiGG-E vs 3.01e-3 for the ER baseline). Please report variability across seeds or otherwise provide a statistical basis for the comparative claims.","section":"Tables 1–3 and Section 4.1"},{"comment":"The notation w0_k = LSTMθ(wk) is ambiguous. If this is the standard recurrent LSTM, the sequence of weight embeddings has a sequential dependence over k, so computing all w0_k during training would require O(m) time, which contradicts the claimed O(log n) training time. If instead the operation is a per-weight feedforward embedding that happens to use LSTM-style gates on a single input, the paper should say so explicitly and not describe it as a recurrent LSTM. Please clarify the computation and provide the training-time complexity of the weight Fenwick-tree construction under the intended interpretation.","section":"§3.4.1 and Algorithm 1"}],"minor_comments":[{"comment":"The canonical-ordering approximation p(G) ≈ p(|V| = n)p(W_π(G)) is acknowledged but not discussed as a limitation. Since the learned distribution is not permutation invariant under arbitrary node orderings, it would be helpful to state this explicitly and comment on when the canonical ordering is meaningful for the benchmark datasets.","section":"§2.1"},{"comment":"The notation ε|e ∼ N(µ, σ) should specify the variance parameter: either N(µ, σ²) or N(µ, σ) with σ denoting standard deviation, and then Equation (3) should use the same convention consistently.","section":"§3.1"},{"comment":"The text says \"In Figure 4, we empirically demonstrate that all BiGG extensions remain efficient,\" but Figure 4 shows weight distributions; the scalability plots are in Figure 5. Please correct the cross-reference.","section":"§4.1, Scalability paragraph"},{"comment":"The return value \"{edge indext represents}\" appears to be a garbled placeholder. It should be replaced with a concrete symbolic representation of the sampled edge index.","section":"Algorithm 1, line 9"},{"comment":"There are formatting issues in the tables, including \"0 .76\", \"0 .065\", and \"Erdos Renyi\" in Table 2. These should be cleaned up for the final version.","section":"Tables 1 and 2"}],"recommendation":"major_revision","confidential_remarks":"The Equation (3) issue is the most serious technical concern, but it may be a manuscript typo if the public implementation uses the correct softplus-normal density. I recommend asking the authors to verify the implementation against the corrected density and to add seed-level variability to the MMD comparisons. The paper falls within the scope of cs.LG and makes a plausible contribution if these points are resolved. I do not see grounds for rejection, but the central claims currently rest on an incorrect stated likelihood and on point estimates without uncertainty quantification."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper is a legitimate extension of BiGG to weighted graphs, and the scalability story is real. The main thing to know is that Equation 3, the softplus-normal likelihood, is not a valid density as printed, and that needs to be fixed before the claims about learning the joint distribution can be taken at face value.\n\nWhat's new: they keep BiGG's topology tree and add a separate Fenwick tree that summarizes previously sampled edge weights, merging the two states with Tree-LSTM cells both when deciding where edges go and when sampling weights. That preserves the O((n+m) log n) sampling complexity for sparse graphs. The empirical work is broader than most: ER, tree, lobster, 3D point cloud, and a joint-tree dataset where topology and weights are coupled. Scaling goes to 15K nodes. Code is on GitHub, and the appendix is honest about overtraining on weights and the heuristic loss scaling.\n\nThe Equation 3 problem is real. For W = softplus(ε), ε ~ N(μ,σ²), the density is (1/(σ√(2π))) * e^w/(e^w-1) * exp(-(log(e^w-1)-μ)²/(2σ²)). The printed version drops the Jacobian (harmless for MLE, since it doesn't depend on θ) and replaces 1/(σ√(2π)) with 1/(2σ²). That second change is not a constant: it shifts the MLE for σ² by a factor of 2. If the code implements what's printed, σ estimates are biased. The empirical weight SDs in Table 2 look close to the true values, so I suspect the code uses the correct density, but the paper needs to say so and fix the equation.\n\nMinor issues: no error bars or multiple seeds, so the performance gaps are hard to interpret. The abstract says BiGG-E 'best captures distributions over weighted graphs,' but on ER graphs the baseline is essentially tied, and on 3D point clouds BiGG+GCN has a better unweighted degree MMD. The canonical-ordering assumption is standard for this family and acknowledged, so I won't penalize it.\n\nBottom line: this is a solid subfield contribution for people working on generative models for weighted graphs, especially large sparse ones. It deserves a serious referee, with the expectation of a correction to Eq. 3, a code check, and ideally error bars. I'd send it to review.","headline":"A useful, scalable BiGG extension for weighted graphs, but Eq. 3's mis-specified softplus-normal likelihood needs fixing and the abstract overclaims slightly.","tokens_in":22951,"tokens_out":5085,"would_cite":true,"duration_ms":53551,"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":"Generates weighted graphs, edge weights included, in near-linear time.","keywords":["weighted graph generation","autoregressive models","sparse graphs","edge weights","joint distribution","Fenwick tree","scalable graph generation","softplus-normal distribution"],"falsifier":"Compute degree and weighted-Laplacian MMD on held-out graphs whose node labels have been randomly permuted; if BiGG-E's error rises sharply while a permutation-equivariant baseline stays flat, the canonical-ordering assumption is the bottleneck.","tokens_in":21978,"feed_emoji":"🕸️","tokens_out":7207,"duration_ms":82317,"temperature":0.7,"pith_summary":"This paper introduces BiGG-E, an autoregressive generative model for weighted graphs that treats topology and edge weights as a single joint distribution instead of adding weights after generating the structure. The authors aim to show that a sparse-graph generator can be extended to continuous edge weights without giving up scalability: sampling a weighted graph with $n$ nodes and $m$ edges costs $O((n + m)\\log n)$ time. On simulated and benchmark data, BiGG-E matches or beats unweighted-graph models and two-stage baselines on topological metrics, and it best captures marginal and joint weight distributions among the compared models. If correct, this gives practitioners a generative model that works on large weighted graphs in domains such as neuroscience, social networks, and phylogenetics, where edge weights carry real meaning.","feed_headline":"Generates weighted graphs, edge weights included, in near-linear time","feed_subtitle":"An autoregressive model samples topology and weights jointly, keeping $O((n+m)\\log n)$ runtime for sparse graphs.","key_machinery":"The load-bearing object is the factorization of the weighted adjacency matrix likelihood into edge-existence probabilities and conditional weight densities, paired with a Fenwick weight tree, a hierarchical binary summary that encodes all previously sampled weights so the next weight can be conditioned on them in logarithmic time. BiGG-E keeps the topology state from BiGG and adds this separate weight state, merging the two with Tree-LSTM cells before each binary decision and before each draw of $\\mu$ and $\\sigma^2$ for the softplus-normal weight distribution. That dual-state separation is what preserves $O((n + m)\\log n)$ sampling while allowing topology and weights to influence each other.","core_discovery":"At its core, the paper claims that a weighted graph distribution can be learned autoregressively at the same asymptotic cost as a sparse unweighted graph distribution. Each potential edge is factorized as $p_\\theta(e, w) = p_\\theta(e) p_\\theta(w \\mid e)$, with the weight density taken to be a softplus-transformed normal; the model then generates the weighted edge set in the order defined by BiGG's row-wise decision trees. A second Fenwick tree summarizes all previously sampled weights, and Tree-LSTM cells merge this weight state with the topology state before every existence decision and before every weight draw, so uncertainty flows both ways between structure and weights. The paper's empirical claim is that this joint design best captures distributions over weighted graphs in its comparisons while remaining scalable and computationally efficient.","pith_inferences":["The canonical-ordering approximation means BiGG-E is not permutation equivariant; a natural extension would be to evaluate how much probability mass is lost on isomorphic relabelings, and to train with randomized orderings if that loss is large.","The softplus-normal conditional restricts each marginal weight distribution to one parametric family; datasets with heavy-tailed or bounded weights would likely need a different density, which the factorization supports by swapping $p_\\theta(w \\mid e)$.","The joint-modeling mechanism predicts that the advantage over two-stage generation grows as the coupling between topology and weights strengthens; the paper's joint-tree experiment is consistent with this, but a dataset with even tighter coupling would test it directly.","Because generation is sparse and autoregressive, BiGG-E could be embedded inside larger regression or prediction models that output a whole network as a random effect, an application the paper lists as future work."],"forward_implications":["Weighted graphs with thousands of nodes can be sampled in near-linear time, orders of magnitude faster than adjacency-matrix LSTMs and sparse diffusion baselines.","Joint modeling of topology and weights improves mixed measures such as weighted degree and weighted Laplacian MMD relative to two-stage pipelines that generate structure first and weights second.","Keeping the weight state separate from the topology state reduces memory use by about 20 percent relative to the entangled-state variant and avoids the unstable weight generation that occurs when one state must serve both tasks.","Because the model is application-agnostic, the same architecture can generate brain-connectivity networks, road networks, molecular graphs with continuous features, and phylogenetic trees."],"supporting_citations":[{"why":"Supplies the BiGG architecture, the sparse edge-set autoregressive factorization, and the $O((n+m)\\log n)$ sampling complexity that BiGG-E extends to weighted graphs.","marker":"Dai et al. (2020)"},{"why":"Supplies the Fenwick tree data structure used for row summaries in BiGG and, in BiGG-E, for the separate weight-state summaries.","marker":"Fenwick (1994)"},{"why":"Supplies the Tree-LSTM cell used to summarize subtrees and to merge the topology and weight states before predictions.","marker":"Tai et al. (2015)"},{"why":"Supplies the canonical node-ordering approximation that BiGG-E adopts in place of summing over all node permutations.","marker":"Liao et al. (2019)"},{"why":"Supplies the R-MAT recursive interval-splitting procedure used to construct each row's binary decision tree.","marker":"Chakrabarti et al. (2004)"},{"why":"Motivates the row-wise adjacency likelihood and the Adj-LSTM baseline that BiGG-E is compared against.","marker":"Li et al. (2018)"},{"why":"Provides SparseDiff, the scalable diffusion baseline whose training time and memory BiGG-E is compared with in the scalability experiments.","marker":"Qin et al. (2024)"},{"why":"Provides the LSTM cell used for embedding sampled weights into the Fenwick weight tree and for the Adj-LSTM baseline.","marker":"Hochreiter & Schmidhuber (1997)"}],"fun_headline_variants":["Autoregressive model learns joint distribution of graph structure and weights","Near-linear time generation of weighted graphs with joint topology-weight modeling","BiGG-E: scalable autoregressive generation of weighted graphs with Fenwick trees","Jointly samples edges and weights in O((n+m)log n) time for sparse graphs"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The single load-bearing premise is that a fixed canonical node ordering adequately represents the graph distribution, so graphs with no meaningful node order or with structure defined only up to permutation may be modeled poorly.","fun_headline_variants_meta":{"raw":{"variants":["Autoregressive model learns joint distribution of graph structure and weights","Near-linear time generation of weighted graphs with joint topology-weight modeling","BiGG-E: scalable autoregressive generation of weighted graphs with Fenwick trees","Jointly samples edges and weights in O((n+m)log n) time for sparse graphs"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000824,"raw_usage":{"total_tokens":3561,"prompt_tokens":863,"completion_tokens":2698,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":479,"completion_tokens_details":{"reasoning_tokens":2617}},"tokens_in":479,"tokens_out":2698,"duration_ms":23188,"temperature":1.0,"reasoning_tokens":2617,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T11:01:46.407247+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute degree and weighted-Laplacian MMD on held-out graphs whose node labels have been randomly permuted; if BiGG-E's error rises sharply while a permutation-equivariant baseline stays flat, the canonical-ordering assumption is the bottleneck.","supporting_citations":[],"review_version":1}