Fluctuations of the giant of Poisson random graphs

David Clancy Jr

REVIEW 2 major objections 5 minor 30 references

Fluctuations of the giant of Poisson random graphs

T0 review · 2 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read The giant component of dynamic rank-one random graphs has explicit process-level Gaussian fluctuations.

desk verdict 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. read the letter →

arxiv 2501.01354 v1 pith:CWDPISAT submitted 2025-01-02 math.PR

David Clancy Jr This is my paper · ORCID

classification math.PR MSC 60F1705C80

keywords giantcomponentrank-onerandomgraphdynamicfunctionalcentrallimittheoremweightedempiricalprocessbreadth-firstwalkphasetransition

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

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.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Dear colleague,

This 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.

The 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.

The 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.

Second, 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.

Third, 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.

None 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.

Who 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.

Recommendation: engage with it; ask the author to fix the typo and add a proof or citation for the second-largest-component bound.

Best

Referee Report

2 major / 5 minor

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.

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 (2)

[Section 3.2, Proposition 3.3 and display (3.4)] 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.
[Lemma 3.6, display (3.6)] 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.

minor comments (5)

[Section 3, heading and first line] 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 2.1] The phrase 'connected component connected component' contains a duplicated word.
[References, item [11]] The author name 'Limc' should be 'Limic'.
[Equation (3.2)] 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 3.2, Proposition 3.3] The symbol T=E[W] is introduced but never used in the proof; either use it in the interval definitions or remove it.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the Gaussian limit is derived from the weight input via empirical-process and excursion-encoding theorems, with no fitted parameters or self-citation chains.

full rationale

Walking the derivation chain, the central claim (Theorem 1.3) is obtained as follows. Theorem 2.5 derives joint convergence of the weighted empirical processes X_{n,0}, X_{n,1} from Shorack's weighted empirical process theorem, using only Assumption 1.2 (weak convergence of W_n and convergence of the second moment); the limit covariances are computed from the input weight distribution W, not assumed. Corollary 2.2 transfers the volume and cardinality of the largest component to the longest excursion of H_n(t,λ)=X_{n,1}(λt)-t via Limic's external distributional identity. Section 3.2 then uses a Skorohod coupling (3.1) of H_n to the limiting process Ψ_1 and deterministic Taylor estimates around θ^(n) to establish the endpoint fluctuations in Theorem 3.1; the centering θ^(n), ρ^(n) and normalization β^(n) are explicit deterministic functions of w(n), and no parameter is fitted to the data whose fluctuations are predicted. The final covariance formula is a functional composition of the empirical-process limits and deterministic derivatives, so it is not a restatement of the input. No load-bearing self-citations occur: Limic [16] and Shorack [27] are external theorems, and the cited works [11,12] are context rather than proof dependencies. The possible concern about controlling the first excursion rather than the longest is a proof-gap/correctness issue, not circularity, since it does not make the conclusion equivalent to an assumption.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper introduces no new entities, particles, or fitted parameters. The Gaussian processes Ψ_0 and Ψ_1 and X are derived from the input weight distribution and standard empirical-process limits. The main burden is carried by external theorems (Limic, Shorack) and the domain assumption on the weight vector.

assumptions (4)

standard math Limic's encoding theorem (Theorem 2.1) identifies the largest component's volume with the longest excursion length of the associated breadth-first walk.
Used in Section 2.1 to justify Corollary 2.2, which is the basis for all subsequent fluctuation analysis.
standard math Shorack's weighted empirical process convergence theorem (Theorem 2.3) applies to the triangular array of exponential variables with weights w_j^p.
Used in Theorem 2.5 to derive the Gaussian limits Ψ_0 and Ψ_1; conditions (i)-(iii) are checked in the proof.
standard math Skorohod representation theorem allows replacing weak convergence by almost sure locally uniform convergence when the limit is continuous.
Invoked at the start of Section 3 to obtain the coupling (3.1) used throughout Section 3.2.
domain assumption Assumption 1.2: W_n converges weakly to W and E[W_n^2] converges to E[W^2], with P(W>0)=1 and E[W^2]<∞.
This is the stated hypothesis of the main theorem; it ensures the critical time, the deterministic asymptotics, and the covariance limits are well defined.

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Cite this review

Pith. "Pith review of Fluctuations of the giant of Poisson random graphs." pith.science (2026). https://pith.science/paper/CWDPISAT

@misc{pith2026250101354,
  author       = {Pith},
  title        = {Pith review of: Fluctuations of the giant of Poisson random graphs},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CWDPISAT}},
  note         = {Machine review of arXiv:2501.01354}
}

read the original abstract

Enriquez, Faraud, and Lemaire (2023) have established process-level fluctuations for the giant of the dynamic Erd\H{o}s-R\'{e}nyi random graph above criticality and show that the limit is a centered Gaussian process with continuous sample paths. A random walk proof was recently obtained by Corujo, Limic and Lemaire (2024). We show that a similar result holds for rank-one inhomogeneous models whenever the empirical weight distribution converges to a limit and its second moment converges as well.

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [6]

Shankar Bhamidi, Amarjit Budhiraja, and Akshay Sakanaveeti, Functional central limit theorems for microscopic and macroscopic functionals of inhomogeneous random graphs, 2024

work page 2024

[2] [1]

David Aldous, Brownian excursions, critical random graphs and the multip licative coalescent , Ann. Probab. 25 (1997), no. 2, 812–854. MR 1434128

work page 1997

[3] [2]

David Aldous and Vlada Limic, The entrance boundary of the multiplicative coalescent , Electron. J. Probab. 3 (1998), no. 3, 59. MR 1491528

work page 1998

[4] [3]

Frank Ball and Peter Neal, The asymptotic variance of the giant component of conﬁgurat ion model random graphs, Ann. Appl. Probab. 27 (2017), no. 2, 1057–1092. MR 3655861

work page 2017

[5] [4]

2, 1046–1069

AD Barbour and Adrian R¨ ollin,Central limit theorems in the conﬁguration model , The Annals of Applied Probability 29 (2019), no. 2, 1046–1069

work page 2019

[6] [5]

Barraez, S

D. Barraez, S. Boucheron, and W. Fernandez de la Vega, On the ﬂuctuations of the giant component , Combin. Probab. Comput. 9 (2000), no. 4, 287–304. MR 1786919

work page 2000

[7] [7]

1, 3–122

B´ ela Bollob´ as, Svante Janson, and Oliver Riordan, The phase transition in inhomogeneous random graphs, Random Structures Algorithms 31 (2007), no. 1, 3–122. MR 2337396

work page 2007

[8] [8]

B´ ela Bollob´ as and Oliver Riordan,Asymptotic normality of the size of the giant component via a random walk, J. Combin. Theory Ser. B 102 (2012), no. 1, 53–61. MR 2871766

work page 2012