{"id":"9e015224-cbe3-4c00-8929-7e8ff883ee8a","arxiv_id":"2607.12459","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Iterated renormalisation of scaled inhomogeneous random graphs converges to a two-parameter attractor: homogeneous Erdős–Rényi in the light-tailed regime and a stable infinite-mean weight graph in the heavy-tailed regime.","lead":"Repeated coarsening of inhomogeneous random graphs by grouping vertices is shown to converge to a universal two-parameter family of limiting graphs. The result separates light-tailed networks (Erdős–Rényi attractor) from heavy-tailed ones (stable-weight attractor) under renormalisation.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified beyond the abstract-only limitation already flagged by the Reader.","rationale":"The Reader’s weakest-assumption statement already isolates the only load-bearing point that can be read from the abstract: the necessity of regime-specific scalings. Because the full text is unavailable, no deeper analytic gap (e.g., failure of the greedy aggregation to preserve independence, or non-existence of the stable-weight limit) can be confirmed or refuted. Manufacturing an additional concern would violate the good-faith rule. Consequently the UNVERDICTED / LOW-confidence assessment stands; the concrete test simply operationalises the verification that the Reader already indicated is required.","tokens_in":2072,"tokens_out":415,"duration_ms":4214,"concrete_test":"Once the full paper is available, extract the precise scaling hypotheses for the light- and heavy-tailed regimes (the statements that replace the abstract phrase “appropriately scaled connection functions”) and check whether the renormalisation operator maps the scaled class into itself and contracts toward the claimed two-parameter family in the topology used for the convergence statements; if either mapping or contraction fails for a natural choice of connection functions, the attractor claim weakens.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The Reader correctly notes that the abstract alone cannot support a soundness verdict: the dual-regime attractor claim (light-tailed ER vs heavy-tailed stable infinite-mean weights with exponential disconnection) rests on regime-specific scalings of the connection functions and weight law that keep the renormalisation map inside the claimed basin. Without the full text one cannot verify that those scalings are well-defined, that the greedy aggregation map is continuous in the appropriate topology, or that the two-parameter family is indeed attractive under iteration. That is an information gap, not an internal inconsistency visible from the abstract. No further load-bearing technical flaw can be isolated from the given material.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The manuscript studies inhomogeneous random graphs with i.i.d. vertex weights, edges present independently with probability given by a bivariate function of the endpoint weights, and self-loops present independently with probability given by a univariate function of the vertex weight. A renormalisation map is defined by greedy aggregation of vertices into equal-sized blocks: an aggregated edge exists between distinct blocks iff at least one cross-edge is present, and an aggregated self-loop exists iff at least one internal self-loop or internal edge is present. The central claim is that, starting from appropriately scaled connection functions, iterated application of this map converges to a two-parameter family of random graphs that acts as an attractor in a universality class. Two regimes are distinguished: a light-tailed regime whose limit is a homogeneous Erdős–Rényi graph, and a heavy-tailed regime whose limit is an inhomogeneous graph with stable infinite-mean weights and an exponential disconnection function. Different scalings are required in each regime; which regime prevails depends on the connection functions and the weight law.","tokens_in":2221,"tokens_out":705,"duration_ms":18572,"significance":"If the claimed convergence and attractor properties are established rigorously, the work would supply a renormalisation-group description of inhomogeneous random graphs under greedy coarse-graining, with an explicit two-parameter universal family and a clean light-tailed/heavy-tailed dichotomy. The identification of a homogeneous ER attractor in the light-tailed regime and of a stable infinite-mean weight law with exponential disconnection in the heavy-tailed regime would be a substantial contribution to the probabilistic theory of complex networks and to universality for discrete structures. The programme is mathematically precise at the level of the abstract and cleanly separates the two regimes.","major_comments":[{"comment":"Only the abstract is available for review. The load-bearing claim that iterated renormalised graphs converge to a two-parameter attractor family (light-tailed ER vs heavy-tailed stable infinite-mean weights with exponential disconnection) cannot be assessed without the proofs, the topology of convergence, error estimates, and technical lemmas. This is an information gap, not an identified internal inconsistency.","section":null},{"comment":"Abstract: the iteration is asserted to remain inside a basin of attraction only after regime-specific scalings of the connection functions and of the weight law (“starting from appropriately scaled connection functions”; “Different scalings are needed for the two regimes”). Without the full text it is impossible to verify that those scalings are well-defined, that the greedy aggregation map is continuous in the relevant topology, or that the stated two-parameter family is attractive under iteration.","section":null},{"comment":"Abstract: the criterion that selects the light-tailed versus heavy-tailed regime is said to depend on the connection functions and the law of the weights, but the precise separation condition and the form of the required scalings are not available for inspection. Any correctness assessment of the dual-regime claim therefore remains provisional.","section":null}],"minor_comments":[],"recommendation":"uncertain","confidential_remarks":"This is an abstract-only review (full text not supplied). A definitive recommendation (accept / minor_revision / major_revision / reject) is not possible until the complete manuscript with proofs is available. On the material provided there is no visible internal contradiction; the limitation is purely informational. Please supply the full text for a proper technical report."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"The one thing you need to know is that this is an abstract-only package: a clean-sounding scaling-limit claim for iterated greedy aggregation of inhomogeneous random graphs, with no theorems, lemmas or topology visible. The punchline they advertise is a two-parameter attractor that splits by regime—light-tailed weights give ordinary ER, heavy-tailed give stable infinite-mean weights plus exponential disconnection—under regime-specific scalings of the connection functions.\n\nWhat looks new is the combination: equal-size greedy aggregation that keeps self-loops, plus an explicit dual-regime universality class rather than a single continuum limit. The abstract is carefully written; the model (i.i.d. weights, independent edges given weights, bi- and uni-variate connection functions) is standard, the renormalisation map is defined without hand-waving, and they flag that different scalings are required for the two regimes. Circularity is low: pure derivation from stated assumptions, no data fitting.\n\nThe soft spot is simply the missing paper. We cannot check that the scalings keep the map inside a basin of attraction, that the aggregation is continuous in a usable topology, or that the two-parameter family is actually attractive under iteration. That is an information gap, not a visible contradiction. The reader’s middle-of-the-road soundness score and the stress-test note both land correctly on that point; I see no reason to invent further flaws.\n\nThis is for people who already work on scaling limits of inhomogeneous graphs or renormalisation ideas in network science. A serious referee who knows the continuum-random-graph and stable-weight literature would get value from it once the full text appears. I would not cite it yet, and I would not bring the abstract alone to reading group. But the programme is coherent enough that a full manuscript deserves peer review rather than a desk reject. Send it out when the proofs arrive.","headline":"Abstract-only claim of a dual-regime renormalisation attractor for inhomogeneous graphs; coherent programme, but no proofs to inspect.","tokens_in":2800,"tokens_out":479,"would_cite":false,"duration_ms":4778,"reading_group":"no","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60C05","05C80","60F05"],"pacs":[],"model":"grok-4.5","headline":"Iterated renormalisation of inhomogeneous random graphs converges to a two-parameter attractor family, with different limits in light- and heavy-tailed regimes.","keywords":["inhomogeneous random graphs","renormalisation","universality","Erdős–Rényi graph","stable weights","heavy tails","light tails","greedy aggregation"],"falsifier":"Choose a concrete connection function and weight law that the paper places in one regime, apply the stated scaling and the greedy aggregation repeatedly on large finite graphs, and check whether the empirical edge probabilities and weight statistics converge to the predicted Erdős–Rényi or stable-weight exponential form; systematic deviation after many iterations would falsify the attractor claim.","tokens_in":2975,"feed_emoji":"🔀","tokens_out":641,"duration_ms":6067,"temperature":0.7,"pith_summary":"This paper studies what happens when you repeatedly coarse-grain an inhomogeneous random graph by greedily aggregating vertices into equal-sized blocks and declaring an edge between blocks whenever any original edge crossed the cut (or a self-loop whenever any internal edge or loop existed). Starting from connection functions and weight distributions that have been scaled in a regime-dependent way, the sequence of renormalised graphs converges to a simple two-parameter family that acts as an attractor. In the light-tailed regime the attractor is an ordinary homogeneous Erdős–Rényi graph; in the heavy-tailed regime it is an inhomogeneous graph whose weights are stable random variables with infinite mean and whose disconnection probabilities decay exponentially. The regime that is selected is completely determined by the original connection functions and the law of the vertex weights. The result therefore supplies a universality picture for how local connection rules and weight tails organise themselves under successive aggregation, explaining why many microscopically different models look the same after enough renormalisation steps.","feed_headline":"Renormalised graphs collapse to a two-parameter attractor","feed_subtitle":"Light tails yield Erdős–Rényi; heavy tails yield stable weights and exponential disconnection","key_machinery":"The renormalisation map that aggregates vertices into equal-sized blocks via a greedy algorithm and defines an aggregated edge (respectively self-loop) whenever at least one original edge crosses the cut (respectively lies inside the block); iterated application of this map, under regime-specific scalings of the connection functions, produces the claimed convergence.","core_discovery":"When an inhomogeneous random graph with i.i.d. vertex weights and suitably scaled connection functions is subjected to iterated greedy block renormalisation, the sequence of renormalised graphs converges to a two-parameter family that is an attractor: a homogeneous Erdős–Rényi graph in the light-tailed regime and an inhomogeneous graph with stable infinite-mean weights and exponential disconnection function in the heavy-tailed regime.","pith_inferences":[],"forward_implications":[],"fun_headline_variants":["Renormalised graphs attract to two-parameter family","Greedy renormalisation yields ER or stable-weight limits","Iterated blocks drive graphs to universal attractors","Light tails to ER; heavy tails to infinite-mean weights","Inhomogeneous graphs renormalise to two-parameter limits"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The connection functions and the law of the vertex weights must be scaled in a regime-specific way so that the renormalisation map stays inside the claimed basin of attraction; without those scalings the iteration need not converge to the stated two-parameter family.","fun_headline_variants_meta":{"raw":{"variants":["Renormalised graphs attract to two-parameter family","Greedy renormalisation yields ER or stable-weight limits","Iterated blocks drive graphs to universal attractors","Light tails to ER; heavy tails to infinite-mean weights","Inhomogeneous graphs renormalise to two-parameter limits"]},"model":"grok-4.5","effort":"low","cost_usd":0.004392,"raw_usage":{"total_tokens":1328,"prompt_tokens":846,"num_sources_used":0,"completion_tokens":80,"cost_in_usd_ticks":43920000,"prompt_tokens_details":{"text_tokens":846,"audio_tokens":0,"image_tokens":0,"cached_tokens":128},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":402,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":846,"tokens_out":80,"duration_ms":3578,"temperature":1.0,"reasoning_tokens":402,"cache_read_input_tokens":128,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-15T05:55:33.835799+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Choose a concrete connection function and weight law that the paper places in one regime, apply the stated scaling and the greedy aggregation repeatedly on large finite graphs, and check whether the empirical edge probabilities and weight statistics converge to the predicted Erdős–Rényi or stable-weight exponential form; systematic deviation after many iterations would falsify the attractor claim.","supporting_citations":[],"review_version":1}