{"id":"a2e19d4a-2e4c-4c4a-b63b-4ca16b31ce02","arxiv_id":"2501.01354","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The giant component of supercritical rank-one random graphs has process-level Gaussian fluctuations, with an explicit covariance given by the limiting weight distribution.","lead":"This paper proves that the sizes of the largest component of a large random graph with different vertex weights follow a Gaussian pattern over time, not just at a single moment. It extends a known result for the classic Erdős-Rényi graph to a broader class of inhomogeneous models, which matters for epidemic and network models.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The proof of Theorem 3.1 controls the first excursion of H_n, not the longest: no argument rules out a later excursion being longer, so the link to the giant via Corollary 2.2 is incomplete.","rationale":"The reader's weakest assumption and my own coincide: the proof does not establish that the excursion whose endpoints are controlled is the maximal excursion. This is the load-bearing issue because the final theorem is about the giant component, and the distributional identity in Corollary 2.2 concerns the longest excursion. The missing step is not a routine detail: inequalities (3.4) and (3.5) sandwich the first excursion's end, while a later excursion could in principle be longer. The gap is likely fixable by invoking standard uniqueness results for rank-one random graphs under Assumption 1.2, so the appropriate disposition is conditional rather than rejection. I also note two smaller, cosmetic issues: the covariance formula stated in Theorem 1.3 should read E[W^{p+q} e^{-W t}(1-e^{-W s})] for s≤t instead of the displayed expression, and there are sign typos in Lemma 3.6 and its proof. Since the reader already assigned CONDITIONAL for essentially this reason, my stress-test does not change the verdict.","tokens_in":12877,"tokens_out":33632,"duration_ms":349633,"concrete_test":"Analytic check: rewrite Theorem 3.1 with (g_n,d_n) redefined as the endpoints of the first excursion, namely the one containing the interval from Proposition 3.3, and re-verify the proof. If every step, especially (3.4), (3.5), and the final sandwich, remains valid verbatim, this confirms that maximality is never used and the paper currently proves only the first-excursion version. Then close the gap by adding a proof or citation that the second-largest component has volume o_P(n) under Assumption 1.2, for instance from the uniqueness part of Bollobás-Janson-Riordan; if that o_P(n) gap cannot be established, Theorem 1.3 would require a different argument.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof of Theorem 3.1 defines (g_n,d_n) as the endpoints of the longest excursion of H_n, because Corollary 2.2 identifies d_n-g_n with n^{-1}V_n(λ). But the argument in Section 3.2 actually controls only the first excursion: Proposition 3.3 shows H_n>0 on [εn^{-1/2}, θ(n)+(Ψ_1(λθ(n))-ε)/(β(n)n^{1/2})], so the excursion containing this interval has length at least θ(n)+O(n^{-1/2}); Lemma 3.6 upper-bounds its right endpoint by the first time H_n drops below -εn^{-1/2}. That passage ends the first excursion, and a later excursion could begin there and be longer. No estimate on the second-longest excursion is supplied. Since Theorem 3.1 requires convergence of the longest excursion endpoints, the chain (3.4)-(3.5) proves, at best, convergence of the first excursion. Without an additional argument—e.g., that under Assumption 1.2 the second-largest component has volume o_P(n), so the second-longest excursion length is o_P(1)—the conclusion does not follow.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper claims a process-level functional central limit theorem for the fluctuations of the size and volume of the giant component in a dynamic rank-one Poisson random graph. Under Assumption 1.2 (weak convergence of the empirical weight distribution plus convergence of the second moment), Theorem 1.3 asserts that the joint process (n^{-1/2}(L_n(λ)-ρ^(n)(λ)n), n^{-1/2}(V_n(λ)-θ^(n)(λ)n)) converges in D((λcrit,∞),R^2) to a centered continuous Gaussian process with explicit covariance determined by the limiting weight W. The proof encodes the graph by Limic's breadth-first walk, uses Shorack's weighted empirical process CLT to handle the driving noise processes X_{n,0}, X_{n,1}, and reduces the problem to fluctuations of the endpoints of the longest excursion of H_n(t,λ)=X_{n,1}(λt)-t.","tokens_in":13063,"tokens_out":16191,"duration_ms":148307,"significance":"If the proof is completed, this is a natural and valuable extension of the Erdős-Rényi results of Enriquez-Faraud-Lemaire and Corujo-Lemaire-Limic to rank-one inhomogeneous models under minimal second-moment conditions. The covariance is explicit and derived from the input weight distribution, with no fitted parameters; the argument relies on benchmark tools (Limic's encoding, Shorack's theorem, Skorohod representation) rather than on circular assumptions. The paper is clearly written and the strategy is elegant. The main obstacle is not in the stochastic-process machinery but in the excursion identification, described below; I view it as repairable within the manuscript's scope.","major_comments":[{"comment":"Proposition 3.3 shows that a.s. under the coupling, H_n(s,λ)>0 for s in the interval [εn^{-1/2}, θ^(n)(λ)+(Ψ_1(λθ^(n)(λ))-ε)/(β^(n)(λ)n^{1/2})]. This implies only that some excursion contains this long interval. Since (g_n(λ),d_n(λ)) was fixed in Corollary 2.2 as the longest excursion interval, the inclusion (g_n(λ),d_n(λ)) ⊃ [εn^{-1/2}, θ^(n)(λ)+(Ψ_1(λθ^(n)(λ))-ε)/(β^(n)(λ)n^{1/2})] in (3.4) does not follow: a later excursion at a lower level could have length at least as large. No estimate on the second-longest excursion is given, so Corollary 3.4, Lemma 3.5, the 'In particular' part of Lemma 3.6, and the upper bound in (3.5) apply, at best, to the excursion that contains the controlled positive interval rather than to the longest one. This gap is load-bearing because Theorem 1.3 is derived from Corollary 2.2, which requires the longest excursion. I suggest adding an argument that any excursion other than the controlled one has length o_P(1), for instance from the o_P(n) bound for the second-largest component under Assumption 1.2, or by controlling H_n below its running minimum after the controlled excursion.","section":"Section 3.2, Proposition 3.3 and display (3.4)"},{"comment":"The final displayed inequality in (3.6) reads H_n(...) ≤ ε/(2n^{1/2}); the right-hand side should be -ε/(2n^{1/2}) (or an equivalent negative bound). The preceding line has leading term -ε/n^{1/2} plus o(n^{-1/2}), so the printed inequality is a sign typo, but as it stands it does not imply the required negativity of H_n, and the second statement of the lemma and the upper bound in (3.5) are unsupported without the correction.","section":"Lemma 3.6, display (3.6)"}],"minor_comments":[{"comment":"The section heading and the opening sentence say 'Proof of Theorem 1.1', but the theorem proved in this section is Theorem 1.3.","section":"Section 3, heading and first line"},{"comment":"The phrase 'connected component connected component' contains a duplicated word.","section":"Section 2.1"},{"comment":"The author name 'Limc' should be 'Limic'.","section":"References, item [11]"},{"comment":"The first equality in (3.2) would be easier to read with parentheses: ϕ^(n)_1(x) = E[W_n^2 e^{-W_n ζ_{n,l}(x)}] x, with ζ_{n,l}(x) ∈ [0,x] explicitly stated as in the surrounding text.","section":"Equation (3.2)"},{"comment":"The symbol T=E[W] is introduced but never used in the proof; either use it in the interval definitions or remove it.","section":"Section 3.2, Proposition 3.3"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is a clean extension of known results and the proof strategy is well motivated. The only substantive obstacle is the longest-excursion identification; if the author adds a standard argument that the second-largest component is o_P(n) (or equivalently that the second-longest excursion is o_P(1)), the result should be publishable. The sign typo in (3.6) is easily corrected. No concerns about novelty disclosure or citation patterns beyond the typo in [11]."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nThis paper proves a joint process-level FCLT for the cardinality and volume of the giant component in dynamic rank-one random graphs, under the assumption that the empirical weight distribution converges and its second moment converges. That is a genuine advance: previous process-level results covered Erdős–Rényi and finite-type stochastic blockmodels, and this handles arbitrary limiting weight distributions with an explicit covariance. The approach via Limic's breadth-first walk encoding and Shorack's weighted empirical process is natural, and the proof is mostly well-organized.\n\nThe good parts first. The main theorem is stated cleanly, the covariance formula is explicit, and the reduction to the CLT for the weighted empirical process is sound. The proof of Theorem 2.5 is standard and correct in outline; the omitted cross-covariance computation is indeed routine. The paper is also honest about the literature.\n\nThe soft spots are real but fixable. First, there is a sign typo in Lemma 3.6: the conclusion should be H_n ≤ −ε/(2 n^{1/2}), not the displayed ≤ ε/(2 n^{1/2}). Minor.\n\nSecond, and more substantively, the proof of Theorem 3.1 does not fully justify that the excursion controlled in Proposition 3.3 and Lemma 3.6 is actually the longest one. Proposition 3.3 shows H_n is positive on a long interval; that interval must lie inside some excursion, but nothing rules out a later excursion being longer. The step from (3.4)–(3.5) to convergence of the longest excursion endpoints needs an argument that the second-largest component has volume o_P(n) (equivalently, the second-longest excursion length is o_P(1)). This is standard supercritical behavior for rank-one graphs, but it should be stated or cited. Without it, the proof establishes convergence of the first excursion, not necessarily the longest.\n\nThird, the paper should say explicitly how this relates to the posted Bhamidi–Budhiraja–Sakanaveeti paper [6]: if [6] only covers finite-type stochastic blockmodels, then this is new; if [6] has a comparable general rank-one statement, the novelty claim needs adjustment.\n\nNone of these look fatal. The sign is a typo, the excursion gap is fillable with a standard uniqueness-of-giant argument, and the covariance omission is cosmetic. As it stands, the theorem is probably true and the proof is on the right track, but I would not pass it as-is; it needs a major revision to close the excursion gap.\n\nWho is this for? Random graph theorists and people working on epidemic processes on rank-one graphs. I'd send it to a serious referee. I'd also be happy to discuss it in a reading group, because the excursion identification issue is a good teaching moment.\n\nRecommendation: engage with it; ask the author to fix the typo and add a proof or citation for the second-largest-component bound.\n\nBest","headline":"A clean and useful generalization of process-level Gaussian fluctuations to rank-one random graphs; the proof is plausible but needs a fix to identify the controlled excursion with the longest one.","tokens_in":13637,"tokens_out":11548,"would_cite":true,"duration_ms":101767,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60F17","05C80"],"pacs":[],"model":"deepseek-v4-flash","headline":"The giant component of dynamic rank-one random graphs has explicit process-level Gaussian fluctuations.","keywords":["giant component","rank-one random graph","dynamic random graph","functional central limit theorem","weighted empirical process","breadth-first walk","phase transition"],"falsifier":"Simulate a weight sequence satisfying Assumption 1.2 with a sparse but growing set of very heavy weights, and for each n record whether the longest excursion of H_n(t,$\\lambda$) = X_{n,1}($\\lambda$ t) - t lies inside the interval [epsilon $n^{{-1/2}}$, $theta^{{(n)}}$($\\lambda$) + (Psi_1($\\lambda$ $theta^{{(n)}}$($\\lambda$)) - epsilon)/($beta^{{(n)}}$($\\lambda$) $n^{{1/2}}$)] used in Proposition 3.3. If the proportion of n for which the longest excursion ends outside this interval does not tend to zero, the endpoint control is insufficient and the claimed limit would fail.","tokens_in":12614,"feed_emoji":"📈","tokens_out":12438,"duration_ms":103525,"temperature":0.7,"pith_summary":"Above its critical point, the dynamic rank-one random graph has a giant component, and this paper proves that the random fluctuation of that giant — measured simultaneously by the number of vertices it contains and by the total weight of those vertices — converges, after square-root-of-n scaling, to a single explicit Gaussian process indexed by the edge parameter lambda. The covariance of that process is written directly in terms of the limiting weight distribution of a random vertex, so the result is not merely an existence statement: the fluctuations are described by a formula one can compute. This extends to a large family of inhomogeneous random graphs a conclusion that was previously known only for the classical Erdős-Rényi graph, where all weights are equal to 1. A reader should care because it locates the giant's random behaviour inside the fluctuations of the whole weighted empirical process, and because the process-level statement allows one to discuss the joint distribution of the giant at different values of lambda at once.","feed_headline":"Giant fluctuation law: an explicit Gaussian process","feed_subtitle":"Above criticality, giant cardinality and volume vary jointly as one Gaussian process fixed by the limiting weight law.","key_machinery":"The central mechanism is the simultaneous breadth-first walk H_n(t,$\\lambda$) = X_{n,1}($\\lambda$ t) - t, where X_{n,1}($\\lambda$ t) = $n^{{-1}}$ sum_{j=1}^n w_j 1_{xi_j <= $\\lambda$ t} with independent exponential clocks xi_j of rate w_j. By the excursion encoding of the multiplicative coalescent, the longest excursion interval (g_n($\\lambda$), d_n($\\lambda$)) of this walk has length equal to the volume of the largest component, and the vertices whose clocks fall inside that interval form that component, so its cardinality is n(X_{n,0}($\\lambda$ d_n) - X_{n,0}($\\lambda$ g_n)). The proof couples H_n to its deterministic backbone $phi_1^{{(n)}}$($\\lambda$ t) - t plus $n^{{-1/2}}$ Psi_1($\\lambda$ t) through an almost-sure coupling, then uses a weighted empirical process CLT and Taylor expansion around the deterministic root $theta^{{(n)}}$($\\lambda$) to show that $n^{{1/2}}$ g_n -> 0 and $n^{{1/2}}$(d_n - $theta^{{(n)}}$) -> $beta^{{-1}}$ Psi_1($\\lambda$ $\\theta$($\\lambda$)) locally uniformly. The slope $\\beta$($\\lambda$) = -partial_t(phi_1($\\lambda$ t) - t)|_{t=$\\theta$($\\lambda$)} = 1 - $\\lambda$ E[$W^{2}$ $e^{{-W lambda theta(lambda)}}$] is what turns a one-unit horizontal shift of the walk's endpoint into a predictable vertical change, and its strict positivity is what makes the endpoint fluctuations Gaussian with the stated covariance.","core_discovery":"Under Assumption 1.2, the paper proves the joint convergence in distribution, as a process in $\\lambda$ > lambda_crit, of ($n^{{-1/2}}$(L_n($\\lambda$) - $rho^{{(n)}}$($\\lambda$)n), $n^{{-1/2}}$(V_n($\\lambda$) - $theta^{{(n)}}$($\\lambda$)n)) to an $R^{2}$-valued centered continuous Gaussian process X($\\lambda$) = (Psi_0($\\lambda$ $\\theta$($\\lambda$)) + ($\\lambda$ phi'_0($\\lambda$ $\\theta$($\\lambda$))/$\\beta$($\\lambda$)) Psi_1($\\lambda$ $\\theta$($\\lambda$)), $\\beta$($\\lambda$)^{-1} Psi_1($\\lambda$ $\\theta$($\\lambda$))). Here L_n and V_n are the cardinality and volume of the most voluminous connected component, $rho^{{(n)}}$ and $theta^{{(n)}}$ are the deterministic proportions solving the fixed-point equations (1.3), and Psi_0, Psi_1 are the limiting centered Gaussian processes of the weighted empirical processes X_{n,0}, X_{n,1} of vertex weights, with covariance E[Psi_p(s)Psi_q(t)] = E[$W^{{p+q}}$ $e^{{-Ws}}$(1-$e^{{-Wt}}$)] for s <= t and p,q in {0,1}. The parameter $\\beta$($\\lambda$) = 1 - $\\lambda$ E[$W^{2}$ $e^{{-W lambda theta(lambda)}}$] is strictly positive above criticality and acts as the slope that converts horizontal fluctuations of the excursion endpoint into vertical fluctuations of the counting process.","pith_inferences":["The excursion-based method is flexible enough that analogous process-level CLTs should hold for other additive functionals of the giant, such as the number of surplus edges or the size of the 2-core, in rank-one models, by applying a functional delta-method to weighted empirical processes.","The theorem suggests a universality statement: two weight sequences with the same limiting distribution W should have indistinguishable giant fluctuations, even if their detailed weight profiles differ; this is testable by simulation and stronger than the paper's necessary conditions alone.","The paper's own remark that the barely supercritical regime requires different assumptions points to a natural next problem: establishing the analogous process-level CLT for lambda = lambda_crit + t epsilon_n with n^{1/3} epsilon_n -> infinity for rank-one graphs, where heavier-tailed weights are expected to change the limit.","If the second-moment convergence in Assumption 1.2 were relaxed, the weighted empirical process would no longer be asymptotically Gaussian, so the giant's fluctuations would presumably acquire a non-Gaussian component; that would be a concrete way to test how load-bearing the second-moment assumption really is."],"forward_implications":["For any fixed lambda above criticality, the cardinality and the volume of the largest component are asymptotically jointly normal, with variances and covariance read off from the process covariance at s = t = lambda theta(lambda).","The process-level convergence permits constructing confidence bands for the growth curve of the giant over any compact interval of lambda above the critical point, not just at isolated parameter values.","Any rank-one Poisson random graph whose empirical weight sequence converges weakly with converging second moment inherits the same Gaussian fluctuation type, so the result covers inhomogeneous models well beyond the equal-weight Erdős-Rényi case.","Because the limit is expressed through the same weighted empirical processes that describe the whole vertex-weight profile, the fluctuations of the giant are determined by the fluctuations of that profile and nothing else.","The explicit covariance makes it possible to evaluate asymptotic expressions for related functionals, such as the location of a maximum of the fluctuation process, by standard Gaussian-process calculus."],"supporting_citations":[{"why":"Provides the excursion encoding of the multiplicative coalescent that identifies the longest excursion of the breadth-first walk with the largest-volume component.","marker":"[16]"},{"why":"Supplies the weighted empirical process CLT for row-independent triangular arrays used to prove convergence of X_{n,0} and X_{n,1}.","marker":"[27]"},{"why":"Gives the random-walk approach to asymptotic normality of the giant that the paper adapts for its endpoint analysis.","marker":"[8]"},{"why":"Provides the recent random-walk proof of the Erdős-Rényi process-level result that the paper generalises.","marker":"[11]"},{"why":"Establishes the process-level Gaussian fluctuations for the Erdős-Rényi giant that Theorem 1.3 extends to rank-one models.","marker":"[12]"}],"fun_headline_variants":["Giant fluctuations: explicit Gaussian process for size and volume","Inhomogeneous random graphs: giant fluctuations are Gaussian","Poisson random graph giant: joint Gaussian limit process","Explicit Gaussian limit for giant component fluctuations"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof presumes that the deepest dip of the breadth-first walk, the one whose endpoints are controlled in the estimates, is in fact the longest dip, and that no other dip far away from the controlled interval can become longer as n grows.","fun_headline_variants_meta":{"raw":{"variants":["Giant fluctuations: explicit Gaussian process for size and volume","Inhomogeneous random graphs: giant fluctuations are Gaussian","Poisson random graph giant: joint Gaussian limit process","Explicit Gaussian limit for giant component fluctuations"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000664,"raw_usage":{"total_tokens":3015,"prompt_tokens":913,"completion_tokens":2102,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":529,"completion_tokens_details":{"reasoning_tokens":2040}},"tokens_in":529,"tokens_out":2102,"duration_ms":16076,"temperature":1.0,"reasoning_tokens":2040,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T22:29:35.235994+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Simulate a weight sequence satisfying Assumption 1.2 with a sparse but growing set of very heavy weights, and for each n record whether the longest excursion of H_n(t,$\\lambda$) = X_{n,1}($\\lambda$ t) - t lies inside the interval [epsilon $n^{{-1/2}}$, $theta^{{(n)}}$($\\lambda$) + (Psi_1($\\lambda$ $theta^{{(n)}}$($\\lambda$)) - epsilon)/($beta^{{(n)}}$($\\lambda$) $n^{{1/2}}$)] used in Proposition 3.3. If the proportion of n for which the longest excursion ends outside this interval does not tend to zero, the endpoint control is insufficient and the claimed limit would fail.","supporting_citations":[{"cited_title":"4A, 2479–2507","cited_arxiv_id":null,"evidence_quote":"Provides the excursion encoding of the multiplicative coalescent that identifies the longest excursion of the breadth-first walk with the largest-volume component."},{"cited_title":"4, 169–189","cited_arxiv_id":null,"evidence_quote":"Supplies the weighted empirical process CLT for row-independent triangular arrays used to prove convergence of X_{n,0} and X_{n,1}."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the random-walk approach to asymptotic normality of the giant that the paper adapts for its endpoint analysis."},{"cited_title":"A novel approach to the giant component fluctuations","cited_arxiv_id":"2412.06995","evidence_quote":"Provides the recent random-walk proof of the Erdős-Rényi process-level result that the paper generalises."}],"review_version":1}