{"id":"a102e444-2b0b-46d6-9266-a6fcb00c52fb","arxiv_id":"2606.02472","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":7.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":1,"one_line_summary":"A consistent estimator for the correlation parameter alpha is constructed for a new model of correlated uniform attachment trees as their size tends to infinity.","lead":"The paper introduces a model of two uniform attachment trees grown in parallel with attachment choices correlated by a fixed probability alpha. It constructs a consistent estimator for alpha from the final unlabeled trees by combining Jordan centrality with fringe subtree sizes across time scales.","discovery_kind":"new_method","skeptic_critique":{"model":"grok-4.3","headline":"Jordan centrality identification of common early vertices may fail to yield sufficient overlap for consistent α estimation","rationale":"The reader's weakest assumption directly names the Jordan-centrality step and the constant-α modeling choice; both are load-bearing for the unlabeled consistency claim. No other internal inconsistency (e.g., in the fringe-subtree approximation or the time-scale discretization) appears more fragile once the overlap condition is granted. The verdict therefore stays UNVERDICTED pending verification of the overlap probability.","tokens_in":1726,"tokens_out":375,"duration_ms":15821,"concrete_test":"Generate 100 independent pairs of correlated UA trees with n=10^5, α=0.3 and α=0.7; for each pair compute the Jordan-central subsets of size roughly n/2, count the number of shared vertices among the first √n time-labeled nodes, and check whether this count exceeds 10 with probability >1-1/n; if the count is typically <5, the overlap assumption fails.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The consistency proof requires that Jordan-central subsets S1, S2 of the two unlabeled trees satisfy |S1 ∩ S2 ∩ early vertices| ≫ 1 with high probability, so that fringe-subtree sizes can recover attachment labels across multiple time scales. The novel bounds on the fraction of early vertices that remain central are stated to be of independent interest, yet they are derived under the uniform-attachment dynamics; the sprinkled correlation (probability α of matching time labels) perturbs the degree and subtree-size distributions in a way that could shrink the central overlap below the threshold needed for the approximation error to vanish as n→∞. If the overlap is only O(1) or o(log n), the estimator cannot distinguish the correlated attachments from noise.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The paper introduces a model of correlated uniform attachment (UA) trees in which two UA processes grow in parallel, with attachment choices correlated by a fixed parameter α (match with probability α, independent otherwise). Given only the two unlabeled trees, the main result constructs a consistent estimator for α as n→∞. The estimator first applies Jordan centrality to extract subsets S1, S2 whose intersection is claimed to contain sufficiently many common early vertices, then uses fringe-subtree sizes across multiple time scales to recover approximate attachment labels and thereby estimate α. Novel quantitative bounds on the fraction of early vertices that remain Jordan-central are derived and stated to be of independent interest.","tokens_in":1867,"tokens_out":576,"duration_ms":15933,"significance":"If the central consistency claim holds, the work supplies the first explicit estimator for the correlation parameter in this generative model from unlabeled data and supplies new tail bounds on centrality that may be useful in network archaeology more broadly. The two-step construction (centrality identification followed by multi-scale fringe analysis) is a concrete methodological contribution.","major_comments":[{"comment":"The consistency argument requires that the Jordan-central subsets S1 and S2 satisfy |S1 ∩ S2 ∩ early vertices| ≫ 1 with high probability so that fringe-subtree statistics can distinguish correlated attachments from noise at multiple time scales. The novel bounds on the fraction of early vertices that remain central are derived under the pure UA dynamics; it is not shown that the same quantitative control continues to hold once the α-sprinkled correlation perturbs the degree and subtree-size distributions.","section":"Main result (construction of the estimator and the centrality bounds)"},{"comment":"The error analysis for the fringe-subtree label recovery step must control the approximation error uniformly over the time scales used. If the central overlap is only O(1) or o(log n) under the correlated dynamics, the variance of the resulting estimator for α does not vanish as n→∞; an explicit lower bound on the overlap probability that accounts for α is therefore load-bearing.","section":"Analysis of the estimator (fringe-subtree step)"}],"minor_comments":[{"comment":"The abstract states that the centrality bounds are 'of independent interest' but does not indicate the precise section or theorem number in which the bounds are stated and proved.","section":"Abstract"},{"comment":"Notation for the Jordan-central subsets (S1, S2) and the time-scale discretization should be introduced once and used consistently; several passages refer to 'early vertices' without a formal definition tied to the growth process.","section":"Section introducing the estimator"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the careful reading and constructive comments on the consistency proof. We address each major comment below and will revise the manuscript to close the identified gaps.","responses":[{"response":"We agree that the quantitative centrality bounds must be established under the correlated dynamics. In the revised version we will extend the tail bounds to the α-correlated model by showing that the correlation induces only a multiplicative perturbation of order 1+α to the attachment probabilities; the same concentration arguments then carry through uniformly in α ∈ [0,1], yielding the required overlap size ≫1 with high probability.","revision_made":"yes","referee_comment":"The consistency argument requires that the Jordan-central subsets S1 and S2 satisfy |S1 ∩ S2 ∩ early vertices| ≫ 1 with high probability so that fringe-subtree statistics can distinguish correlated attachments from noise at multiple time scales. The novel bounds on the fraction of early vertices that remain central are derived under the pure UA dynamics; it is not shown that the same quantitative control continues to hold once the α-sprinkled correlation perturbs the degree and subtree-size distributions."},{"response":"We will add an explicit lemma giving a lower bound on P(|S1 ∩ S2 ∩ early vertices| ≥ log log n) that is strictly positive for each fixed α > 0 and tends to 1 as n → ∞. With this overlap size the multi-scale fringe analysis controls the approximation error uniformly over the O(log log n) time scales, ensuring the estimator variance vanishes.","revision_made":"yes","referee_comment":"The error analysis for the fringe-subtree label recovery step must control the approximation error uniformly over the time scales used. If the central overlap is only O(1) or o(log n) under the correlated dynamics, the variance of the resulting estimator for α does not vanish as n→∞; an explicit lower bound on the overlap probability that accounts for α is therefore load-bearing."}],"tokens_in":1453,"tokens_out":436,"duration_ms":16987,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The main contribution is a new generative model in which two UA trees grow in parallel and at each step attach with probability alpha to the same time-labeled vertex. From the two unlabeled trees the authors build a statistic that recovers alpha consistently as n goes to infinity. The construction first uses Jordan centrality to select subsets likely to share many early vertices, then examines fringe-subtree sizes across several time scales to recover attachment information.\n\nThe model itself and the estimator are new relative to the UA-tree literature cited in the abstract. The quantitative bounds on the fraction of early vertices that remain central are presented as reusable for network archaeology and look like the most portable piece.\n\nThe high-level strategy is coherent and the estimator is built from observable tree properties rather than fitted parameters, so there is no obvious circularity. The constant-alpha assumption is standard and reasonable for a first model.\n\nThe soft spot is whether the correlation perturbs the degree and subtree distributions enough to shrink the common central overlap below the threshold needed for the approximation error to vanish. The stress-test note flags exactly this issue: the novel bounds are stated under pure UA dynamics, yet the alpha-matching introduces dependence that could reduce overlap to O(1) or o(log n). If the proofs fully control this perturbation the result stands; otherwise consistency may fail. Without the detailed error analysis it is impossible to judge how tight the bounds are under the actual dynamics.\n\nThe work is aimed at people studying inference on random trees or network archaeology. It is coherent enough and the construction non-obvious enough that it deserves a serious referee who can check the overlap bounds.","headline":"This paper introduces a sprinkled-correlation model for pairs of uniform attachment trees and constructs a consistent estimator for alpha from unlabeled trees via Jordan centrality plus multi-scale fringe analysis.","tokens_in":2390,"tokens_out":401,"would_cite":false,"duration_ms":26708,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"Unlabeled correlated uniform attachment trees admit a consistent estimator for the correlation parameter α as size grows to infinity.","keywords":["correlated uniform attachment trees","consistent estimator","Jordan centrality","fringe subtrees","unlabeled trees","network archaeology","correlation parameter","growing networks"],"falsifier":"Generate many pairs of trees from the model with a fixed known α and large n; if the proposed statistic fails to approach that α, the consistency claim is false.","tokens_in":2621,"feed_emoji":"","tokens_out":677,"duration_ms":18471,"temperature":0.7,"pith_summary":"The paper introduces a model of two uniform attachment trees that grow in parallel with attachments correlated by a fixed parameter α at each step. It establishes that a statistic built from the pair of unlabeled trees converges in probability to the true α. The construction identifies candidate early vertices via Jordan centrality in each tree and then matches attachment patterns using the sizes of their fringe subtrees across multiple scales. A reader would care because the result turns two anonymous growing networks into a source of information about their hidden correlation structure. The analysis also supplies quantitative bounds on how many early vertices remain central, which stand alone in the study of network reconstruction.","feed_headline":"Unlabeled trees recover correlation parameter α consistently","feed_subtitle":"Jordan centrality and fringe sizes turn two anonymous uniform attachment trees into an estimator that converges to the hidden α.","key_machinery":"Jordan centrality to locate subsets containing many common early vertices, together with fringe-subtree size comparisons at several time scales to approximate birth-time labels.","core_discovery":"In the correlated uniform attachment model, two trees are built by adding one vertex to each at every time step; with probability α the new vertices attach to vertices bearing the same birth time, and otherwise the attachments are chosen independently. Given only the two final unlabeled trees, the fraction of early vertices that remain central can be bounded from below, and the sizes of the fringe subtrees rooted at those central vertices supply enough information to recover the value of α consistently as the trees become large.","pith_inferences":["The technique could be applied to real-world pairs of networks suspected to share an early common history, such as duplicated social graphs or biological interaction networks.","If the constant-α assumption is relaxed to slowly varying correlation, the same centrality-plus-fringe approach might still yield local estimates at different epochs.","The bounds on central early vertices may be useful for single-tree network archaeology problems even without a second correlated copy."],"forward_implications":["The estimator converges in probability to α as the number of vertices tends to infinity.","Quantitative lower bounds hold on the fraction of early vertices that remain Jordan-central in each tree.","Fringe-subtree sizes at multiple scales suffice to match a positive fraction of the common early vertices across the two trees.","The same centrality and fringe-size ideas apply to other questions of detection and estimation between unlabeled growing trees."],"fun_headline_variants":["Unlabeled UA trees yield consistent α estimate","Correlation α estimated from paired attachment trees","Central vertices and fringes recover α consistently","Paired UA trees allow consistent α estimation"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The correlation parameter stays constant for the whole growth process and Jordan centrality on the unlabeled trees succeeds in isolating subsets that share enough early vertices.","fun_headline_variants_meta":{"raw":{"variants":["Unlabeled UA trees yield consistent α estimate","Correlation α estimated from paired attachment trees","Central vertices and fringes recover α consistently","Paired UA trees allow consistent α estimation"]},"model":"grok-4.3","cost_usd":0.005941,"raw_usage":{"total_tokens":2830,"prompt_tokens":692,"num_sources_used":0,"completion_tokens":51,"cost_in_usd_ticks":59412000,"prompt_tokens_details":{"text_tokens":692,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":2087,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":692,"tokens_out":51,"duration_ms":15093,"temperature":1.0,"reasoning_tokens":2087,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-28T12:36:00.785843+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"Generate many pairs of trees from the model with a fixed known α and large n; if the proposed statistic fails to approach that α, the consistency claim is false.","supporting_citations":[],"review_version":1}