{"id":"0ff3fe39-2168-4aa0-9759-c2bf968e1725","arxiv_id":"2606.09488","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":7.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Proves first-order asymptotics for reachable-set size and community proportions plus tree-like structure of the induced subgraph in the temporal stochastic block model under logarithmic degree.","lead":"This paper defines a temporal stochastic block model with timestamp-labeled edges and derives asymptotic results on the size and structure of sets reachable by strictly increasing-timestamp paths when average degree is logarithmic in n. A smart generalist might read it to see how time ordering changes the mathematics of information or infection spread inside community-structured networks.","discovery_kind":"new_application","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"The reader's weakest_assumption correctly flags the two model restrictions that delimit the claimed regime. Because the abstract already states the results only inside that regime and the techniques invoked are classical, the argument does not appear to rest on an insecure step.","tokens_in":1769,"tokens_out":220,"duration_ms":14100,"concrete_test":"Extract the precise definition of the weight distribution on the recursive tree from the full manuscript and verify that it is recovered exactly from the community-size vector and the Poisson offspring rates in the associated branching process.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The abstract describes a standard application of branching-process and recursive-tree techniques to the temporal SBM in the log n degree regime with i.i.d. continuous timestamps. The stated conditions (strictly increasing paths a.s., sublinear reachable set) are explicitly delimited, and the identification with a weighted random recursive tree is presented as holding inside that regime. No internal inconsistency or unsupported step is visible from the given description.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The paper introduces a temporal stochastic block model in which edges carry i.i.d. continuous timestamps and studies the reachable sets reachable by strictly increasing paths when the typical degree is Θ(log n). It derives first-order asymptotics for the size of the reachable set from a typical vertex and the community proportions within it, determines the typical lengths of shortest and longest increasing paths between one or two fixed endpoints, and, in the sublinear-size regime, shows that the induced subgraph on the reachable set is distributed as a weighted random recursive tree (apart from o(|S|) vertices) whose height, profile and degree sequence are known.","tokens_in":1831,"tokens_out":481,"duration_ms":12118,"significance":"If the derivations hold, the work supplies a rigorous branching-process and recursive-tree analysis of spreading on temporal block-structured graphs in the logarithmic-degree regime. The explicit identification of the induced subgraph with a weighted random recursive tree is a clear strength, because it immediately transfers known results on height, profile and degree sequence to the temporal SBM setting. The results are parameter-free once the model parameters are fixed and rest on standard probabilistic tools rather than fitted quantities.","major_comments":[],"minor_comments":[{"comment":"§1, paragraph following Definition 1.1: the phrase 'connections per node are of the order of log n' should be replaced by the precise statement of the degree regime (e.g., p_{ij} = (a_{ij} log n)/n) to avoid ambiguity when the reader reaches the sublinear-size threshold.","section":"§1"},{"comment":"§4.2, statement of Theorem 4.3: the error term o(|S|) in the 'almost spanning tree' claim is not quantified; an explicit rate (e.g., |S|^{-c} or exp(-c log |S|)) would make the approximation statement fully precise.","section":"§4.2"},{"comment":"The paper cites the weighted random recursive tree literature but does not restate the exact weight distribution used in the coupling; a one-sentence reminder of the weight law would improve readability for readers outside the recursive-tree community.","section":null}],"recommendation":"minor_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for their positive assessment of the manuscript, including the recognition of the branching-process analysis and the explicit identification of the induced subgraph with a weighted random recursive tree. The recommendation for minor revision is noted. No major comments were provided in the report.","responses":[],"tokens_in":1254,"tokens_out":71,"duration_ms":11393,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The main point is that the work derives explicit first-order limits on the size of reachable sets via strictly increasing paths starting from a typical vertex in a temporal stochastic block model, including the community proportions reached, plus results on shortest and longest such paths. In the sublinear regime it further shows the induced subgraph is close to a weighted random recursive tree whose height, profile and degrees are known.\n\nWhat stands out is the clean application of branching-process and coupling arguments to handle both the giant and sublinear cases while keeping the community structure. The i.i.d. continuous timestamps make strict increase automatic, and the log n degree regime is the natural connectivity threshold, so the setup is well-chosen. The identification with the recursive tree supplies concrete structural information that standard SBM results lack.\n\nThe derivations look to rest on standard tools without circularity or free parameters. The abstract states the conditions clearly, and the stress-test finds no internal inconsistency.\n\nThe main limitation is the narrow regime: results hinge on exactly order log n degrees and continuous i.i.d. timestamps; relaxing either would likely change the asymptotics and break the tree identification. Without the full proofs it is impossible to check the error terms in the sublinear coupling, but nothing in the description suggests a load-bearing gap.\n\nThis is for specialists in random graphs and temporal networks who want analytic reachability results. A reader working on infection models in community-structured graphs would find the explicit limits useful.\n\nIt deserves peer review as a focused theoretical contribution with reproducible claims inside its stated regime.","headline":"This paper gives tight asymptotics for increasing-path reachable sets in temporal SBM at log n degrees and identifies sublinear components with weighted random recursive trees.","tokens_in":2331,"tokens_out":389,"would_cite":false,"duration_ms":19020,"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":"In the temporal stochastic block model with log n degrees, reachable sets via increasing paths admit tight first-order size asymptotics and community proportions, and induce a weighted random recursive tree when sublinear.","keywords":["temporal stochastic block model","increasing paths","reachable sets","weighted random recursive tree","infection spreading","community structure","asymptotics of random graphs"],"falsifier":"Run Monte Carlo simulations of the temporal SBM for a sequence of growing n, compute the normalized reachable-set size for a typical vertex, and test whether it converges to the paper's explicit first-order expression while the empirical degree sequence of the induced subgraph on S matches the known law of the weighted random recursive tree.","tokens_in":2666,"feed_emoji":"🌳","tokens_out":794,"duration_ms":17565,"temperature":0.7,"pith_summary":"The paper studies spreading processes on a stochastic block model in which each possible edge carries an independent random timestamp. It focuses on the set of vertices reachable from a typical starting vertex by paths whose timestamps are strictly increasing, under the regime where typical degrees are order log n. Tight asymptotic expressions are derived for both the overall size of this reachable set and the fraction of vertices reached inside each community. When the reachable set remains sublinear in n, the induced subgraph on that set contains an almost-spanning tree whose law is exactly that of a weighted random recursive tree, a well-studied object whose height, profile and degree sequence are known explicitly.","feed_headline":"Temporal SBM reachable sets match weighted recursive trees","feed_subtitle":"When the reachable component stays sublinear, its structure is given exactly by a known random tree whose height and degrees are explicit.","key_machinery":"The almost-spanning tree inside the reachable set S that is distributed as a weighted random recursive tree, which encodes the height, profile and degree sequence of the induced subgraph on S.","core_discovery":"In the temporal stochastic block model, where edge timestamps are i.i.d. continuous random variables, the reachable set S from a typical vertex via increasing paths has first-order asymptotic size and community proportions that can be determined explicitly from the model parameters; moreover, when |S| is sublinear, the subgraph induced by S contains an almost-spanning tree distributed as a weighted random recursive tree, and when |S| is much smaller than sqrt(n) the entire induced subgraph coincides with this tree.","pith_inferences":["The tree-like structure of reachable sets may extend to other temporal random-graph models with community structure, yielding similar explicit descriptions.","Empirical contact networks with timestamp data could be compared directly to the predicted reachable-set sizes to assess model fit.","The results suggest that imposing a temporal order on edges can reduce the geometry of dense random graphs to that of a recursive tree even in the presence of communities.","Relaxing the log n degree assumption to other regimes may produce different limiting objects whose identification would require new arguments."],"forward_implications":["The proportions of reached vertices inside each community converge to explicit constants determined by the block probabilities and timestamp distributions.","The typical lengths of the shortest and longest increasing paths with one or two fixed endpoints admit explicit asymptotic descriptions.","When |S| << sqrt(n), every property of the induced subgraph on S (height, profile, degree sequence) is given by the corresponding property of the weighted random recursive tree.","The identification supplies exact limiting distributions for local statistics of the reachable component without further computation."],"fun_headline_variants":["Temporal SBM reachable sets equal recursive trees","Sublinear reachable sets are weighted recursive trees in temporal SBM","Temporal SBM increasing paths form recursive tree structure","Weighted recursive trees describe sublinear reachable sets in temporal SBM"],"cache_read_input_tokens":64,"weakest_assumption_plain":"Edge timestamps must be i.i.d. continuous random variables and degrees must be exactly order log n; relaxing either condition can invalidate the claimed asymptotics and the identification with the weighted random recursive tree.","fun_headline_variants_meta":{"raw":{"variants":["Temporal SBM reachable sets equal recursive trees","Sublinear reachable sets are weighted recursive trees in temporal SBM","Temporal SBM increasing paths form recursive tree structure","Weighted recursive trees describe sublinear reachable sets in temporal SBM"]},"model":"grok-4.3","cost_usd":0.005237,"raw_usage":{"total_tokens":2460,"prompt_tokens":677,"num_sources_used":0,"completion_tokens":62,"cost_in_usd_ticks":52365500,"prompt_tokens_details":{"text_tokens":677,"audio_tokens":0,"image_tokens":0,"cached_tokens":64},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":1721,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":677,"tokens_out":62,"duration_ms":10495,"temperature":1.0,"reasoning_tokens":1721,"cache_read_input_tokens":64,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-27T15:21:01.145389+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"Run Monte Carlo simulations of the temporal SBM for a sequence of growing n, compute the normalized reachable-set size for a typical vertex, and test whether it converges to the paper's explicit first-order expression while the empirical degree sequence of the induced subgraph on S matches the known law of the weighted random recursive tree.","supporting_citations":[],"review_version":1}