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arxiv: 2607.00731 · v1 · pith:TWDWKUMOnew · submitted 2026-07-01 · 🧮 math.PR

Sharp Asymptotics for the Largest Component in the Subcritical Regime of Preferential Attachment Without Vertex Growth

Yiming Chen This is my paper

Pith reviewed 2026-07-02 07:12 UTC · model grok-4.3

classification 🧮 math.PR
keywords preferential attachmentlargest componentsubcritical regimerandom graphsasymptoticsconfiguration modelcomponent size
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The pith

In the subcritical regime of preferential attachment on a fixed vertex set, the largest component L1 is (1+o_p(1)) times 2(α+2)/(α+1) times ε^{-2} log(ε³ n).

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

The paper establishes a sharp asymptotic formula for the size of the largest connected component in a preferential attachment random graph model on a fixed set of n vertices. When the number of edges m is set to m_c (1 - ε) where m_c is the critical value and ε tends to zero slowly enough that ε cubed n goes to infinity, this size is shown to be (1 + o_p(1)) times 2(α + 2)/(α + 1) times ε to the minus 2 times the log of ε cubed n. The result holds for fixed α > 0 and extends to cases where α itself depends on n and tends to a positive constant. This provides precise information on component sizes in a model that generalizes the classical Erdős–Rényi random graph.

Core claim

We prove that if m = m_c(1-ε), ε=ε(n)→0, ε³n→∞, then L1=(1+o_p(1)) * 2(α+2)/(α+1) * ε^{-2} log(ε³ n) for every fixed α>0. The same asymptotic holds when α=α(n)→a∈(0,∞]. In particular the constant converges to the Erdős–Rényi value 2 as α→∞. Moreover if m=⌊n/2(1-ε)⌋ and αε→∞ then L1=(2+o_p(1))ε^{-2}log(ε³ n). The upper bound uses that after conditioning on the degree sequence the graph can be treated through the corresponding configuration model; the lower bound follows from tree component asymptotics and a second moment argument.

What carries the argument

Conditioning on the degree sequence to treat the graph as a configuration model for the upper bound on L1, together with tree-component asymptotics and the second-moment method for the lower bound.

If this is right

  • The leading constant 2(α+2)/(α+1) approaches the Erdős–Rényi value of 2 as α tends to infinity.
  • The asymptotic continues to hold when the parameter α grows or shrinks with n as long as it tends to a positive limit.
  • In the special case where αε tends to infinity the size simplifies to (2 + o_p(1)) ε^{-2} log(ε³ n).

Where Pith is reading between the lines

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

  • The comparison to the configuration model indicates that degree conditioning captures the essential randomness for component structure in this attachment process.
  • Similar sharp asymptotics might be derivable for other attachment kernels that produce comparable degree sequences.
  • Numerical checks for moderate n could verify the accuracy of the logarithmic term in the formula.

Load-bearing premise

After conditioning on the degree sequence the preferential attachment graph behaves sufficiently like the configuration model for the purpose of bounding the largest component.

What would settle it

A computation or simulation in which L1 divided by ε^{-2} log(ε³ n) fails to converge in probability to 2(α+2)/(α+1) for sequences where ε→0 and ε³n→∞ would falsify the stated asymptotic.

read the original abstract

We study the size of the largest component in Pittel's preferential attachment process without vertex growth. Starting from the empty graph on a fixed vertex set $[n]$, edges are added one by one with probabilities proportional to $(d_u+\alpha)(d_v+\alpha)$, where $d_u$ and $d_v$ are the current degrees of $u$ and $v$, and $\alpha>0$. Let $L_1$ denote the size of the largest component, and set $m_c:=\frac{\alpha n}{2(\alpha+1)}.$ We prove that if $m=m_c(1-\varepsilon), \varepsilon=\varepsilon(n)\to0, \varepsilon^3 n\to\infty,$ then \[ L_1=(1+o_p(1))\frac{2(\alpha+2)}{\alpha+1}\varepsilon^{-2}\log(\varepsilon^3 n) \] for every fixed $\alpha>0$. More generally, the same asymptotic holds whenever $\alpha=\alpha(n)\to a\in(0,\infty]$. In particular, the constant $2(\alpha+2)/(\alpha+1)$ converges to the Erd\H{o}s--R\'enyi value $2$ as $\alpha\to\infty$. Moreover, if $m=\left\lfloor \frac n2(1-\varepsilon)\right\rfloor$ and $\alpha\varepsilon\to\infty$, then \[ L_1=(2+o_p(1))\varepsilon^{-2}\log(\varepsilon^3 n). \] The subcritical asymptotics for \(L_1\) resolve the problem left open by Janson and Warnke. The upper bound argument relies on the observation that, after conditioning on the degree sequence, the graph can be treated through the corresponding configuration model, the lower bound follows from tree component asymptotics and a second moment argument.

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

1 major / 2 minor

Summary. The manuscript establishes sharp asymptotics for the largest component L1 in Pittel's preferential attachment process on n fixed vertices (no vertex growth). For m = m_c(1-ε) with ε=ε(n)→0 and ε³n→∞, it claims L1=(1+o_p(1)) * [2(α+2)/(α+1)] ε^{-2} log(ε³n) for fixed α>0 (and the same form when α(n)→a∈(0,∞]). The upper bound proceeds by conditioning on the degree sequence and reducing to the configuration model; the lower bound uses tree-component asymptotics plus a second-moment argument. The result resolves an open question of Janson and Warnke and recovers the Erdős–Rényi constant 2 in the α→∞ limit.

Significance. If the configuration-model approximation is controlled with error small enough not to affect the leading constant, the result supplies the first explicit sharp asymptotic for this model in the subcritical window, with a clean dependence on α that interpolates to the known ER case. The combination of degree conditioning, configuration-model tail bounds, and second-moment methods on trees is a standard and potentially reproducible approach once the error terms are verified.

major comments (1)
  1. [Upper bound argument] Upper bound argument (as described after the statement of the main theorem): the assertion that, after conditioning on the degree sequence, “the graph can be treated through the corresponding configuration model” is used to transfer component-size tail bounds that deliver the precise prefactor 2(α+2)/(α+1). No quantitative total-variation or coupling bound is supplied showing that the difference in the probability of long paths (or in the exponential decay rate of component sizes) is o(1) uniformly in the regime ε→0, ε³n→∞. Because the target asymptotic is sharp, any discrepancy of order ε or larger in the decay rate would change the leading constant; an explicit error estimate (e.g., via the evolving attachment probabilities versus the uniform configuration measure) is therefore load-bearing.
minor comments (2)
  1. [Main theorem] The statement of the result for α(n)→a should clarify whether the o_p(1) term is uniform in a or requires a separate argument when a=∞.
  2. [Introduction] Notation: the critical value m_c is defined as αn/(2(α+1)); it would help to recall the exact relation between m_c and the emergence of a giant component in the model.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for identifying the need for quantitative error control in the upper-bound argument. We address the comment below.

read point-by-point responses

  1. Referee: [Upper bound argument] Upper bound argument (as described after the statement of the main theorem): the assertion that, after conditioning on the degree sequence, “the graph can be treated through the corresponding configuration model” is used to transfer component-size tail bounds that deliver the precise prefactor 2(α+2)/(α+1). No quantitative total-variation or coupling bound is supplied showing that the difference in the probability of long paths (or in the exponential decay rate of component sizes) is o(1) uniformly in the regime ε→0, ε³n→∞. Because the target asymptotic is sharp, any discrepancy of order ε or larger in the decay rate would change the leading constant; an explicit error estimate (e.g., via the evolving attachment probabilities versus the uniform configuration measure) is therefore load-bearing.

    Authors: We agree that an explicit quantitative bound is required to justify transferring the sharp tail asymptotics from the configuration model, since any discrepancy of order ε in the decay rate would alter the leading constant. The current manuscript relies on a qualitative statement that the conditioned preferential-attachment graph can be treated via the configuration model but does not supply total-variation or coupling estimates. In the revision we will add a dedicated section deriving such bounds: we will compare the sequential attachment probabilities to the uniform stub-matching measure and show that the difference in the logarithmic asymptotics of the probability of long paths (or of component-size tails) is o(ε) uniformly on the event that the degree sequence lies in the typical set for the subcritical regime. This error is small enough not to affect the prefactor 2(α+2)/(α+1) when ε³n→∞. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation self-contained from model definition

full rationale

The paper states its main result as a theorem proved from the preferential attachment process definition. The upper bound uses conditioning on the degree sequence followed by treatment as a configuration model; the lower bound uses tree-component asymptotics and the second-moment method. These steps are presented as direct applications of the model without any reduction of the target asymptotic to a fitted parameter, self-referential definition, or load-bearing self-citation chain. The cited prior work (Janson-Warnke) is external and the result is obtained via standard probabilistic arguments on the given process, rendering the derivation self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The argument rests on standard tools of random graph theory (configuration model after degree conditioning, tree-component asymptotics, second-moment method) with no new free parameters, invented entities, or ad-hoc axioms introduced in the abstract.

axioms (1)
  • standard math Standard results on configuration models and second-moment calculations for component sizes in random graphs with given degree sequences.
    Invoked for the upper and lower bounds respectively.

pith-pipeline@v0.9.1-grok · 5870 in / 1340 out tokens · 28432 ms · 2026-07-02T07:12:20.357343+00:00 · methodology

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Reference graph

Works this paper leans on

56 extracted references · 47 canonical work pages

  1. [1]

    Brownian excursions, critical random graphs and the multiplicative coalescent

    David Aldous. Brownian excursions, critical random graphs and the multiplicative coalescent. The Annals of Probability, 25(2):812–854, 1997. doi:10.1214/aop/1024404421

    work page doi:10.1214/aop/1024404421 1997

  2. [2]

    Cambridge University Press, Cambridge, 2016

    Albert-L´ aszl´ o Barab´ asi.Network Science. Cambridge University Press, Cambridge, 2016

    2016

  3. [3]

    Emergence of scaling in random networks.Science, 286(5439):509–512, 1999

    Albert-L´ aszl´ o Barab´ asi and R´ eka Albert. Emergence of scaling in random networks.Science, 286(5439):509–512, 1999. doi:10.1126/science.286.5439.509

    work page doi:10.1126/science.286.5439.509 1999

  4. [4]

    Eli Ben-Naim and P. L. Krapivsky. Popularity-driven networking.Europhysics Letters, 97(4):48003, 2012. doi:10.1209/0295-5075/97/48003

    work page doi:10.1209/0295-5075/97/48003 2012

  5. [5]

    Chayes, and Amin Saberi

    Noam Berger, Christian Borgs, Jennifer T. Chayes, and Amin Saberi. Asymptotic behavior and distributional limits of preferential attachment graphs.The Annals of Probability, 42(1):1– 40, 2014. doi:10.1214/12-AOP755

    work page doi:10.1214/12-aop755 2014

  6. [6]

    Shankar Bhamidi, Remco van der Hofstad, and Sanchayan Sen. The multiplicative coalescent, inhomogeneous continuum random trees, and new universality classes for critical random graphs.Probability Theory and Related Fields, 170(1–2):387–474, 2018. doi:10.1007/s00440- 017-0760-6

    work page doi:10.1007/s00440- 2018

  7. [7]

    Cambridge University Press, Cambridge, second edition,

    B´ ela Bollob´ as.Random Graphs. Cambridge University Press, Cambridge, second edition,

  8. [8]

    doi:10.1017/CBO9780511814068

    work page doi:10.1017/cbo9780511814068

  9. [9]

    The phase transition in inhomogeneous random graphs.Random Structures & Algorithms, 31(1):3–122, 2007

    B´ ela Bollob´ as, Svante Janson, and Oliver Riordan. The phase transition in inhomogeneous random graphs.Random Structures & Algorithms, 31(1):3–122, 2007. doi:10.1002/rsa.20168

    work page doi:10.1002/rsa.20168 2007

  10. [10]

    Random graphs and branching processes

    B´ ela Bollob´ as and Oliver Riordan. Random graphs and branching processes. InHandbook of Large-Scale Random Networks, pages 15–115. Springer, Berlin, 2009. doi:10.1007/978-3-540- 69395-6 1

    work page doi:10.1007/978-3-540- 2009

  11. [11]

    The phase transition in the Erd˝ os–R´ enyi random graph process

    B´ ela Bollob´ as and Oliver Riordan. The phase transition in the Erd˝ os–R´ enyi random graph process. InErd˝ os Centennial, pages 59–110. J´ anos Bolyai Mathematical Society, Budapest,

  12. [12]

    doi:10.1007/978-3-642-39286-3 3

    work page doi:10.1007/978-3-642-39286-3

  13. [13]

    The degree sequence of a scale-free random graph process.Random Structures & Algorithms, 18(3):279–290, 2001

    B´ ela Bollob´ as, Oliver Riordan, Joel Spencer, and G´ abor Tusn´ ady. The degree sequence of a scale-free random graph process.Random Structures & Algorithms, 18(3):279–290, 2001. doi:10.1002/rsa.1009. 31

    work page doi:10.1002/rsa.1009 2001

  14. [14]

    S´ os, and Katalin Vesztergombi

    Christian Borgs, Jennifer Chayes, L´ aszl´ o Lov´ asz, Vera T. S´ os, and Katalin Vesztergombi. Limits of randomly grown graph sequences.European Journal of Combinatorics, 32(7):985– 999, 2011. doi:10.1016/j.ejc.2011.03.015

    work page doi:10.1016/j.ejc.2011.03.015 2011

  15. [15]

    Generating simple random graphs with prescribed degree distribution.Journal of Statistical Physics, 124(6):1377–1397, 2006

    Tom Britton, Maria Deijfen, and Anders Martin-L¨ of. Generating simple random graphs with prescribed degree distribution.Journal of Statistical Physics, 124(6):1377–1397, 2006. doi:10.1007/s10955-006-9168-x

    work page doi:10.1007/s10955-006-9168-x 2006

  16. [16]

    Connected components in random graphs with given expected degree sequences.Annals of Combinatorics, 6(2):125–145, 2002

    Fan Chung and Linyuan Lu. Connected components in random graphs with given expected degree sequences.Annals of Combinatorics, 6(2):125–145, 2002. doi:10.1007/PL00012580

    work page doi:10.1007/pl00012580 2002

  17. [17]

    Largest component of subcritical random graphs with given degree sequence.Electronic Journal of Probability, 28:34, 1–28, 2023

    Matthew Coulson and Guillem Perarnau. Largest component of subcritical random graphs with given degree sequence.Electronic Journal of Probability, 28:34, 1–28, 2023. doi:10.1214/23-EJP921

    work page doi:10.1214/23-ejp921 2023

  18. [18]

    Souvik Dhara, Remco van der Hofstad, Johan S. H. van Leeuwaarden, and Sanchayan Sen. Critical window for the configuration model: finite third moment degrees.Electronic Journal of Probability, 22:1–33, 2017. doi:10.1214/17-EJP29

    work page doi:10.1214/17-ejp29 2017

  19. [19]

    Anatomy of a young giant component in the random graph.Random Structures & Algorithms, 39(2):139–178, 2011

    Jian Ding, Jeong Han Kim, Eyal Lubetzky, and Yuval Peres. Anatomy of a young giant component in the random graph.Random Structures & Algorithms, 39(2):139–178, 2011. doi:10.1002/rsa.20342

    work page doi:10.1002/rsa.20342 2011

  20. [20]

    Anatomy of the giant component: The strictly supercritical regime.European Journal of Combinatorics, 35:155–168, 2014

    Jian Ding, Eyal Lubetzky, and Yuval Peres. Anatomy of the giant component: The strictly supercritical regime.European Journal of Combinatorics, 35:155–168, 2014. doi:10.1016/j.ejc.2013.06.004

    work page doi:10.1016/j.ejc.2013.06.004 2014

  21. [21]

    S. N. Dorogovtsev and J. F. F. Mendes.Evolution of Networks. Oxford University Press, Oxford, 2003. doi:10.1093/acprof:oso/9780198515906.001.0001

    work page doi:10.1093/acprof:oso/9780198515906.001.0001 2003

  22. [22]

    S. N. Dorogovtsev, J. F. F. Mendes, and A. N. Samukhin. Principles of statistical mechanics of uncorrelated random networks.Nuclear Physics B, 666(3):396–416, 2003. doi:10.1016/S0550- 3213(03)00504-2

    work page doi:10.1016/s0550- 2003

  23. [23]

    Cambridge University Press, Cambridge, 2010

    Rick Durrett.Random Graph Dynamics. Cambridge University Press, Cambridge, 2010

    2010

  24. [24]

    On random graphs

    Paul Erd˝ os and Alfr´ ed R´ enyi. On random graphs. I.Publicationes Mathematicae Debrecen, 6:290–297, 1959

    1959

  25. [25]

    On the evolution of random graphs.Publications of the Math- ematical Institute of the Hungarian Academy of Sciences, 5:17–61, 1960

    Paul Erd˝ os and Alfr´ ed R´ enyi. On the evolution of random graphs.Publications of the Math- ematical Institute of the Hungarian Academy of Sciences, 5:17–61, 1960

    1960

  26. [26]

    Percolation on sparse random graphs with given degree sequence

    Nikolaos Fountoulakis. Percolation on sparse random graphs with given degree sequence. Internet Mathematics, 4(4):329–356, 2007. doi:10.1080/15427951.2007.10129148

    work page doi:10.1080/15427951.2007.10129148 2007

  27. [27]

    Cambridge University Press, Cambridge, 2016

    Alan Frieze and Micha l Karo´ nski.Introduction to Random Graphs. Cambridge University Press, Cambridge, 2016. doi:10.1017/CBO9781316339831. 32

    work page doi:10.1017/cbo9781316339831 2016

  28. [28]

    The scaling window for a random graph with a given de- gree sequence.Random Structures & Algorithms, 41(1):99–123, 2012

    Hamed Hatami and Michael Molloy. The scaling window for a random graph with a given de- gree sequence.Random Structures & Algorithms, 41(1):99–123, 2012. doi:10.1002/RSA.20394

    work page doi:10.1002/rsa.20394 2012

  29. [29]

    Nongrowing preferential attachment random graphs.Internet Mathematics, 6(4):461–487, 2010

    Tom´ aˇ s Hruˇ z and Uwe Peter. Nongrowing preferential attachment random graphs.Internet Mathematics, 6(4):461–487, 2010. doi:10.1080/15427951.2010.553143

    work page doi:10.1080/15427951.2010.553143 2010

  30. [30]

    Slightly subcritical hypercube percolation.Random Struc- tures & Algorithms, 56(2):557–593, 2020

    Tim Hulshof and Asaf Nachmias. Slightly subcritical hypercube percolation.Random Struc- tures & Algorithms, 56(2):557–593, 2020. doi:10.1002/rsa.20853

    work page doi:10.1002/rsa.20853 2020

  31. [31]

    On percolation in random graphs with given vertex degrees.Electronic Journal of Probability, 14:87–118, 2009

    Svante Janson. On percolation in random graphs with given vertex degrees.Electronic Journal of Probability, 14:87–118, 2009. doi:10.1214/EJP.v14-603

    work page doi:10.1214/ejp.v14-603 2009

  32. [32]

    Susceptibility of random graphs with given vertex degrees.Journal of Com- binatorics, 1(4):357–387, 2010

    Svante Janson. Susceptibility of random graphs with given vertex degrees.Journal of Com- binatorics, 1(4):357–387, 2010. doi:10.4310/JOC.2010.v1.n4.a2

    work page doi:10.4310/joc.2010.v1.n4.a2 2010

  33. [33]

    On edge exchangeable random graphs.Journal of Statistical Physics, 173(3– 4):448–484, 2018

    Svante Janson. On edge exchangeable random graphs.Journal of Statistical Physics, 173(3– 4):448–484, 2018. doi:10.1007/s10955-017-1832-9

    work page doi:10.1007/s10955-017-1832-9 2018

  34. [34]

    Svante Janson and Malwina J. Luczak. A simple solution to thek-core problem.Random Structures & Algorithms, 30(1–2):50–62, 2007. doi:10.1002/rsa.20147

    work page doi:10.1002/rsa.20147 2007

  35. [35]

    Svante Janson and Malwina J. Luczak. A new approach to the giant component problem. Random Structures & Algorithms, 34(2):197–216, 2009. doi:10.1002/rsa.20231

    work page doi:10.1002/rsa.20231 2009

  36. [36]

    Wiley, New York,

    Svante Janson, Tomasz Luczak, and Andrzej Ruci´ nski.Random Graphs. Wiley, New York,

  37. [37]

    doi:10.1002/9781118032718

    work page doi:10.1002/9781118032718

  38. [38]

    Preferential attachment without vertex growth: emer- gence of the giant component.The Annals of Applied Probability, 31(4):1523–1547, 2021

    Svante Janson and Lutz Warnke. Preferential attachment without vertex growth: emer- gence of the giant component.The Annals of Applied Probability, 31(4):1523–1547, 2021. doi:10.1214/20-AAP1610

    work page doi:10.1214/20-aap1610 2021

  39. [39]

    The component sizes of a critical random graph with given degree sequence

    Adrien Joseph. The component sizes of a critical random graph with given degree sequence. The Annals of Applied Probability, 24(6):2560–2594, 2014. doi:10.1214/13-AAP985

    work page doi:10.1214/13-aap985 2014

  40. [40]

    Seierstad

    Mihyun Kang and Tom G. Seierstad. The critical phase for random graphs with a given degree sequence.Combinatorics, Probability and Computing, 17(1):67–86, 2008. doi:10.1017/S096354830700867X

    work page doi:10.1017/s096354830700867x 2008

  41. [41]

    A critical point for random graphs with a given degree se- quence.Random Structures & Algorithms, 6(2–3):161–180, 1995

    Michael Molloy and Bruce Reed. A critical point for random graphs with a given degree se- quence.Random Structures & Algorithms, 6(2–3):161–180, 1995. doi:10.1002/rsa.3240060204

    work page doi:10.1002/rsa.3240060204 1995

  42. [42]

    The size of the giant component of a random graph with a given degree sequence.Combinatorics, Probability and Computing, 7(3):295–305, 1998

    Michael Molloy and Bruce Reed. The size of the giant component of a random graph with a given degree sequence.Combinatorics, Probability and Computing, 7(3):295–305, 1998. doi:10.1017/S0963548398003526

    work page doi:10.1017/s0963548398003526 1998

  43. [43]

    Critical percolation on random regular graphs.Random Structures & Algorithms, 36(2):111–148, 2010

    Asaf Nachmias and Yuval Peres. Critical percolation on random regular graphs.Random Structures & Algorithms, 36(2):111–148, 2010. doi:10.1002/rsa.20277. 33

    work page doi:10.1002/rsa.20277 2010

  44. [44]

    Advances in Applied Probability , author =

    Ilkka Norros and Hannu Reittu. On a conditionally Poissonian graph process.Advances in Applied Probability, 38(1):59–75, 2006. doi:10.1239/aap/1143936140

    work page doi:10.1239/aap/1143936140 2006

  45. [45]

    On a random graph evolving by degrees.Advances in Mathematics, 223(2):619– 671, 2010

    Boris Pittel. On a random graph evolving by degrees.Advances in Mathematics, 223(2):619– 671, 2010. doi:10.1016/j.aim.2009.08.015

    work page doi:10.1016/j.aim.2009.08.015 2010

  46. [46]

    Sudden emergence of a giantk-core in a random graph.Journal of Combinatorial Theory, Series B, 67(1):111–151, 1996

    Boris Pittel, Joel Spencer, and Nicholas Wormald. Sudden emergence of a giantk-core in a random graph.Journal of Combinatorial Theory, Series B, 67(1):111–151, 1996. doi:10.1006/jctb.1996.0036

    work page doi:10.1006/jctb.1996.0036 1996

  47. [47]

    de Solla Price

    Derek J. de Solla Price. A general theory of bibliometric and other cumulative advantage processes.Journal of the American Society for Information Science, 27(5):292–306, 1976. doi:10.1002/ASI.4630270505

    work page doi:10.1002/asi.4630270505 1976

  48. [48]

    Time evolution of dense multigraph limits under edge-conservative pref- erential attachment dynamics.Random Structures & Algorithms, 41(3):365–390, 2012

    Bal´ azs R´ ath. Time evolution of dense multigraph limits under edge-conservative pref- erential attachment dynamics.Random Structures & Algorithms, 41(3):365–390, 2012. doi:10.1002/rsa.20422

    work page doi:10.1002/rsa.20422 2012

  49. [49]

    Multigraph limit of the dense configuration model and the preferential attachment graph.Acta Mathematica Hungarica, 136(3):196–221, 2012

    Bal´ azs R´ ath and L´ aszl´ o Szak´ acs. Multigraph limit of the dense configuration model and the preferential attachment graph.Acta Mathematica Hungarica, 136(3):196–221, 2012. doi:10.1007/s10474-012-0217-4

    work page doi:10.1007/s10474-012-0217-4 2012

  50. [50]

    The phase transition in the configuration model.Combinatorics, Probability and Computing, 21(1–2):265–299, 2012

    Oliver Riordan. The phase transition in the configuration model.Combinatorics, Probability and Computing, 21(1–2):265–299, 2012. doi:10.1017/S0963548311000666

    work page doi:10.1017/s0963548311000666 2012

  51. [51]

    Birth control for giants.Combinatorica, 27(5):587–628,

    Joel Spencer and Nicholas Wormald. Birth control for giants.Combinatorica, 27(5):587–628,

  52. [52]

    doi:10.1007/s00493-007-2163-2

    work page doi:10.1007/s00493-007-2163-2

  53. [53]

    Volume 1

    Remco van der Hofstad.Random Graphs and Complex Networks. Volume 1. Cambridge University Press, Cambridge, 2017. doi:10.1017/9781316779422

    work page doi:10.1017/9781316779422 2017

  54. [54]

    Volume 2

    Remco van der Hofstad.Random Graphs and Complex Networks. Volume 2. Cambridge University Press, Cambridge, 2024. doi:10.1017/9781316795552

    work page doi:10.1017/9781316795552 2024

  55. [55]

    Remco van der Hofstad, Svante Janson, and Malwina J. Luczak. Component structure of the configuration model: barely supercritical case.Random Structures & Algorithms, 55(1):3–55,

  56. [56]

    doi:10.1002/rsa.20837. 34

    work page doi:10.1002/rsa.20837