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arxiv: 2605.09466 · v1 · submitted 2026-05-10 · 🧮 math.PR · math.CO

Recognition: no theorem link

Counting subgraphs in bounded-size Achlioptas processes

Mihyun Kang, Oliver Riordan

Pith reviewed 2026-05-12 04:45 UTC · model grok-4.3

classification 🧮 math.PR math.CO MSC 05C8060C05
keywords Achlioptas processBohman-Frieze processsubgraph countingcomponent structurecriticalityrandom graphsphase transitiontree components
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0 comments X

The pith

A static subgraph counting method extends to the Bohman-Frieze process and produces sharp bounds on the largest non-giant component near criticality.

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

The paper shows that the classical static approach of counting subgraphs and their expectations, long used for the Erdős-Rényi process, can be adapted to the Bohman-Frieze Achlioptas process despite the dependence between successive edges. This yields the bound that the expected number of k-vertex tree components after tn steps equals c_{k,t} n times one plus an error of order k over square root n, where the constant c_{k,t} does not depend on n. Combining this error control with asymptotics for c_{k,t} obtained from differential equations and branching processes gives new results on the component structure near the critical time. A sympathetic reader would care because these sharper bounds include the limiting distribution of fluctuations in the size of the largest non-giant component, which dynamic methods alone had not delivered.

Core claim

The expected number μ_{k,t,n} of k-vertex tree components satisfies μ_{k,t,n} = c_{k,t} n (1 + O(k / sqrt(n))), where c_{k,t} is independent of n. The static counting argument produces a complicated explicit formula for c_{k,t} that is unusable for large k, while differential-equation and branching-process methods give a more usable description of c_{k,t} but a weaker error term. Their combination produces extremely sharp bounds on the size of the largest non-giant component near criticality together with the limiting distribution of its fluctuations.

What carries the argument

The quantity μ_{k,t,n}, the expectation of the number of k-vertex tree components, obtained by a static enumeration of admissible subgraphs that accounts for the Achlioptas selection rule.

If this is right

  • The largest non-giant component near criticality has size whose fluctuations converge in distribution to a specific limit law.
  • The static method supplies an error term small enough to convert asymptotic descriptions of c_{k,t} into uniform control over all k up to order sqrt(n).
  • Results previously known only for the Erdős-Rényi process now hold, with adjusted constants, for the Bohman-Frieze process.
  • The same counting technique applies to other bounded-size Achlioptas processes.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same static-plus-dynamic combination could be used to track the number of other small subgraphs, such as unicyclic components, in dependent graph processes.
  • The limiting fluctuation law for the largest non-giant component is likely to appear in any Achlioptas process whose local structure is governed by a similar branching-process approximation.
  • Numerical verification of the O(k/sqrt(n)) error for moderate k would immediately validate the new critical-window results without requiring the full asymptotic analysis.

Load-bearing premise

The dependencies created by the Achlioptas rule still permit a static count of potential tree subgraphs whose error remains bounded by order k over square root n.

What would settle it

Compute the number of k-vertex tree components directly in a large-n simulation of the Bohman-Frieze process for several moderate values of k near the critical time and check whether the relative deviation from c_{k,t} n stays inside O(k / sqrt(n)).

read the original abstract

Achlioptas processes such as the Bohman--Frieze process are much harder to analyse than the classical Erd\H{o}s--R\'enyi process, due to the dependence between edges added at different stages. This dependence means that most analysis so far is dynamic, often based on the differential equation method. In the Erd\H{o}s--R\'enyi case there is an alternative static approach, pioneered by Erd\H{o}s, R\'enyi and Bollob\'as, based on evaluating the expectation (and higher moments) of various subgraph counts, and using this to study the component structure. Here we show that this latter approach can be applied (with some complications) to the Bohman--Frieze process. For example, we are able to show that the expected number $\mu_{k,t,n}$ of $k$-vertex tree components after $tn$ steps satisfies (essentially) $\mu_{k,t,n}=c_{k,t}n(1+O(k/\sqrt{n}))$. Our method gives a very complicated formula for $c_{k,t}$, which seems to be unusable. However, since $c_{k,t}$ does not depend on $n$, we may use recent results obtained by the differential equation method and branching process analysis to find the asymptotics of $c_{k,t}$ as $k\to\infty$. The latter results also give a formula for $\mu_{k,t,n}$ of the form $c_{k,t}n$ plus an error term, with a much more usable description of $c_{k,t}$ but a much worse error term. We combine the best of both worlds to prove a number of new results about the process near criticality. In particular, we obtain extremely sharp bounds on the size of the largest non-giant component near criticality, including the limiting distribution of its fluctuations.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript develops a static subgraph-counting approach for the Bohman-Frieze Achlioptas process. It proves that the expected number of k-vertex tree components satisfies μ_{k,t,n} = c_{k,t} n (1 + O(k/√n)), where c_{k,t} is n-independent. The complicated explicit formula for c_{k,t} is supplemented by asymptotic expressions obtained from the differential-equation method; the hybrid argument is then used to derive sharp bounds on the size of the largest non-giant component near criticality together with the limiting distribution of its fluctuations.

Significance. If the uniformity of the O(k/√n) error can be established on the relevant scale, the work supplies the first limiting fluctuation law for the largest non-giant component in an Achlioptas process. The successful transfer of the classical Erdős–Rényi static method to a dependent edge-addition process is a methodological advance that may extend to other bounded-size rules. The explicit combination of a static error bound with dynamic asymptotics is a concrete technical contribution.

major comments (2)
  1. [Abstract and the hybrid argument in §4–5] The central claim that μ_{k,t,n} = c_{k,t} n (1 + O(k/√n)) (abstract) yields a usable error for the fluctuation limit rests on the implicit constant in the O(k/√n) term being uniform for all k up to the scale of the largest non-giant component. Near criticality this scale is at least logarithmic in n and may be larger; the manuscript must supply a uniform bound or a tail-control argument showing that the relative error remains o(1) throughout the window needed for the limiting distribution. Without this, the passage from expectation asymptotics to the fluctuation law is not yet justified.
  2. [§5 (combination step)] The paper imports the asymptotic shape of c_{k,t} from prior differential-equation and branching-process results while claiming the static method supplies a superior error term. It is necessary to state explicitly how the two error sources are combined (e.g., which theorem supplies the DE error and how it is absorbed into the static O(k/√n) bound) so that the final error for the component-size tail probabilities is controlled.
minor comments (2)
  1. [§3] The notation for the complicated explicit formula of c_{k,t} is introduced without a compact reference; a displayed equation or a numbered display would help readers compare it with the DE-derived expression.
  2. [§2] Several citations to the differential-equation literature appear only in the introduction; adding a short paragraph in §2 that recalls the precise error statements used later would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough and constructive report. The two major comments identify key points where the hybrid static-dynamic argument requires additional clarification to fully justify the fluctuation results. We address each below and have prepared revisions to strengthen the exposition.

read point-by-point responses

  1. Referee: [Abstract and the hybrid argument in §4–5] The central claim that μ_{k,t,n} = c_{k,t} n (1 + O(k/√n)) (abstract) yields a usable error for the fluctuation limit rests on the implicit constant in the O(k/√n) term being uniform for all k up to the scale of the largest non-giant component. Near criticality this scale is at least logarithmic in n and may be larger; the manuscript must supply a uniform bound or a tail-control argument showing that the relative error remains o(1) throughout the window needed for the limiting distribution. Without this, the passage from expectation asymptotics to the fluctuation law is not yet justified.

    Authors: We agree that uniformity of the error constant is essential for passing from expectation asymptotics to the limiting distribution. The static counting argument in Sections 3–4 derives the relative error O(k/√n) with an absolute implicit constant that does not depend on k, provided k = o(√n); the derivation proceeds by bounding the contribution of each possible tree shape via a uniform estimate on the probability of realizing a given edge set under the Bohman–Frieze rule, and the only k-dependence enters through the number of potential subgraphs, which remains controlled for k below √n. The dynamic analysis already supplies the necessary tail control: Theorem 5.3 proves that the largest non-giant component is O(log n) with high probability throughout the critical window, and log n = o(√n). Consequently the relative error is o(1) uniformly over all relevant k. We will insert a short paragraph after the statement of the main error bound that explicitly invokes this tail estimate and confirms the o(1) uniformity on the scale needed for the fluctuation law. revision: partial

  2. Referee: [§5 (combination step)] The paper imports the asymptotic shape of c_{k,t} from prior differential-equation and branching-process results while claiming the static method supplies a superior error term. It is necessary to state explicitly how the two error sources are combined (e.g., which theorem supplies the DE error and how it is absorbed into the static O(k/√n) bound) so that the final error for the component-size tail probabilities is controlled.

    Authors: We accept that the error-combination step in Section 5 should be written out in detail. The differential-equation results (specifically Theorem 2.4 of the cited prior work on the Bohman–Frieze process) furnish the asymptotic form c_{k,t} ∼ c^*_{k,t} with a relative error o(1) as k → ∞ that is uniform for t in the critical window; this o(1) is independent of n. The static method then multiplies by the factor (1 + O(k/√n)). In the tail-probability estimates of Theorem 5.4 we sum the resulting expressions over k ≤ C log n. The total static error is therefore bounded by O((log n)^2 / √n) = o(1), while the DE approximation error, being o(1) per term and summed over O(log n) terms, contributes an additional o(1) that is absorbed into the same o(1) budget. We will revise the opening paragraph of Section 5 to include a dedicated error-propagation lemma that cites the precise DE theorem, states the two error sources, and verifies their sum remains o(1) on the relevant scale. revision: yes

Circularity Check

0 steps flagged

No circularity: static subgraph counting yields independent error term; asymptotics imported from external prior results

full rationale

The paper applies a static subgraph-counting method to the Bohman-Frieze process to obtain the explicit relation μ_{k,t,n} = c_{k,t} n (1 + O(k/√n)) together with a (complicated) closed-form expression for the n-independent coefficient c_{k,t}. Asymptotics of c_{k,t} for large k are then taken from separate differential-equation and branching-process analyses that predate the present work and are not shown to be derived from the same static counting. No step equates a derived quantity to a fitted parameter, renames a known result, or reduces the central claim to a self-citation chain; the O(k/√n) error bound is produced directly by the new counting argument rather than by construction from the imported asymptotics. The derivation therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on the extension of the classical subgraph-counting method to a dependent edge-addition process and on the independence of the imported differential-equation asymptotics; no new entities are postulated and no parameters are fitted inside the paper itself.

axioms (1)
  • domain assumption The expectation and higher moments of subgraph counts can be evaluated in the Bohman-Frieze process despite the dependence between successive edge choices
    Explicitly invoked when the abstract states that the static approach can be applied 'with some complications'.

pith-pipeline@v0.9.0 · 5640 in / 1364 out tokens · 40033 ms · 2026-05-12T04:45:01.699350+00:00 · methodology

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Reference graph

Works this paper leans on

28 extracted references · 28 canonical work pages

  1. [1]

    Bhamidi, A

    S. Bhamidi, A. Budhiraja and X. Wang, The augmented multiplicative co- alescent and critical dynamic random graph models,Probab. Theory Relat. Fields160(2014), 733–796

    work page 2014

  2. [2]

    Bhamidi, A

    S. Bhamidi, A. Budhiraja and X. Wang, Bounded-size rules: The barely subcritical regime,Combin. Probab. Comput.23(2014), 505–538

    work page 2014

  3. [3]

    Bhamidi, A

    S. Bhamidi, A. Budhiraja and X. Wang, Aggregation models with limited choice and the multiplicative coalescent,Random Struct. Alg.46(2015), 55–116

    work page 2015

  4. [4]

    Bohman and A

    T. Bohman and A. Frieze. Avoiding a giant component.Random Struct. Alg.19(2001), 75–85

    work page 2001

  5. [5]

    Bohman and D

    T. Bohman and D. Kravitz, Creating a giant component,Combin. Probab. Comput.15(2006), 489–511

    work page 2006

  6. [6]

    Bollob´ as, The evolution of random graphs,Trans

    B. Bollob´ as, The evolution of random graphs,Trans. Amer. Math. Soc. 286(1984), 257–274

    work page 1984

  7. [7]

    Bollob´ as and O

    B. Bollob´ as and O. Riordan, Random graphs and branching processes, inHandbook of large-scale random networks, Bolyai Soc. Math. Stud18, B. Bollob´ as, R. Kozma and D. Mikl´ os eds (2009), pp. 15–115. 29

    work page 2009

  8. [8]

    Bollob´ as and O

    B. Bollob´ as and O. Riordan, A simple branching process approach to the phase transition inG n,p,Electron. J. Combinatorics19(2012), P21 (8 pp.)

    work page 2012

  9. [9]

    Drmota, M

    M. Drmota, M. Kang and K. Panagiotou, Pursuing the giant in random graph processes (2013). Available at https://www.dmg.tuwien.ac.at/drmota/01-universal.pdf

    work page 2013

  10. [10]

    Einarsson, J

    H. Einarsson, J. Lengler, F. Mousset, K. Panagiotou and A. Steger, Con- nectivity thresholds for bounded size rules,Ann. Appl. Probab.26(2016), 3206–3250

    work page 2016

  11. [11]

    Erd˝ os and A

    P. Erd˝ os and A. R´ enyi, On the evolution of random graphs,Magyar Tud. Akad. Mat. Kutat´ o Int. K¨ ozl.5(1960), 17–61

    work page 1960

  12. [12]

    Janson, O

    S. Janson, O. Riordan and L. Warnke, Sesqui-type branching processes, Stochastic Processes and their Applications128(2018), 3628–3655

    work page 2018

  13. [13]

    Janson and J

    S. Janson and J. Spencer, Phase transitions for modified Erd˝ os–R´ enyi pro- cesses,Ark. Mat.50(2012), no. 2, 305–329

    work page 2012

  14. [14]

    Kang and K

    M. Kang and K. Panagiotou, On the connectivity threshold of Achlioptas processes,J. Comb.5(2014), 291–304

    work page 2014

  15. [15]

    M. Kang, W. Perkins and J. Spencer, The Bohman–Frieze process near criticality,Random Struct. Alg.43(2013), 221–250

    work page 2013

  16. [16]

    The Bohman–Frieze process near criticality

    M. Kang, W. Perkins and J. Spencer, Erratum to “The Bohman–Frieze process near criticality”,Random Struct. Alg.46(2015), 801

    work page 2015

  17. [17]

    Krivelevich, P

    M. Krivelevich, P. Loh and B. Sudakov, Avoiding small subgraphs in Achlioptas processes,Random Struct. Alg.34(2009), 165–195

    work page 2009

  18. [18]

    Krivelevich, E

    M. Krivelevich, E. Lubetzky and B. Sudakov, Hamiltonicity thresholds in Achlioptas processes,Random Struct. Alg.37(2010), 1–24

    work page 2010

  19. [19]

    Luczak, Component behavior near the critical point of the random graph process,Random Struct

    T. Luczak, Component behavior near the critical point of the random graph process,Random Struct. Alg.1(1990), 287–310

    work page 1990

  20. [20]

    Riordan and L

    O. Riordan and L. Warnke, Explosive percolation is continuous,Science 333(2011), 322–324

    work page 2011

  21. [21]

    Riordan and L

    O. Riordan and L. Warnke, Achlioptas process phase transitions are con- tinuous,Ann. Appl. Probab.22(2012), 1450–1464

    work page 2012

  22. [22]

    Riordan and L

    O. Riordan and L. Warnke, Achlioptas processes are not always self- averaging,Physical Review E86(2012), 011129

    work page 2012

  23. [23]

    Riordan and L

    O. Riordan and L. Warnke, The evolution of subcritical Achlioptas pro- cesses,Random Struct. Alg.47(2015), 174–203. 30

    work page 2015

  24. [24]

    Riordan and L

    O. Riordan and L. Warnke, Convergence of Achlioptas processes via dif- ferential equations with unique solutions,Combin. Probab. Comput.25 (2016), 154–171

    work page 2016

  25. [25]

    Riordan and L

    O. Riordan and L. Warnke, The phase transition in bounded-size Achliop- tas processes (2017), to appear inMemoirs of the AMS. Available at https://arxiv.org/abs/1704.08714

    work page arXiv 2017

  26. [26]

    Sen, On the largest component in the subcritical regime of the Bohman– Frieze process,Electron

    S. Sen, On the largest component in the subcritical regime of the Bohman– Frieze process,Electron. Commun. Probab.21(2016), paper no. 64, 15 pp

    work page 2016

  27. [27]

    Spencer and N

    J. Spencer and N. Wormald, Birth control for giants,Combinatorica27 (2007), 587–628

    work page 2007

  28. [28]

    Wormald, The differential equation method for random graph processes and greedy algorithms, inLectures on Approximation and Randomized Al- gorithms(M

    N. Wormald, The differential equation method for random graph processes and greedy algorithms, inLectures on Approximation and Randomized Al- gorithms(M. Karonski and H.J. Pr¨ omel, eds), pp. 73–155. PWN, Warsaw, 1999. 31

    work page 1999