Electronic Journal of Probability

Critical Random Graphs: Limiting Constructions and Distributional Properties

Louigi Addario-Berry, Nicolas Broutin, and Christina Goldschmidt

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We consider the Erdös-Rényi random graph $G(n,p)$ inside the critical window, where $p=1/n+\lambda n^{-4/3}$ for some $\lambda\in\mathbb{R}$. We proved in [1] that considering the connected components of $G(n,p)$ as a sequence of metric spaces with the graph distance rescaled by $n^{-1/3}$ and letting $n\to\infty$ yields a non-trivial sequence of limit metric spaces $C=(C_1,C_2,\ldots)$. These limit metric spaces can be constructed from certain random real trees with vertex-identifications. For a single such metric space, we give here two equivalent constructions, both of which are in terms of more standard probabilistic objects. The first is a global construction using Dirichlet random variables and Aldous' Brownian continuum random tree. The second is a recursive construction from an inhomogeneous Poisson point process on $\mathbb{R}_+$. These constructions allow us to characterize the distributions of the masses and lengths in the constituent parts of a limit component when it is decomposed according to its cycle structure. In particular, this strengthens results of [29] by providing precise distributional convergence for the lengths of paths between kernel vertices and the length of a shortest cycle, within any fixed limit component

Article information

Electron. J. Probab. Volume 15 (2010), paper no. 25, 741-775.

Accepted: 24 May 2010
First available in Project Euclid: 1 June 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 05C80: Random graphs [See also 60B20]
Secondary: 60C05: Combinatorial probability

random graph real tree scaling limit Gromov--Hausdorff distance Brownian excursion continuum random tree Poisson process urn model

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Addario-Berry, Louigi; Broutin, Nicolas; Goldschmidt, Christina. Critical Random Graphs: Limiting Constructions and Distributional Properties. Electron. J. Probab. 15 (2010), paper no. 25, 741--775. doi:10.1214/EJP.v15-772. https://projecteuclid.org/euclid.ejp/1464819810

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  • Addario-Berry, Louigi; Broutin, Nicolas; Goldschmidt, Christina. The continuum limit of critical random graphs. arXiv:0903.4730v1 [math.PR], 2009+.
  • Aldous, David. The continuum random tree. I. Ann. Probab. 19 (1991), no. 1, 1–28.
  • Aldous, David. The continuum random tree. II. An overview. Stochastic analysis (Durham, 1990), 23–70, London Math. Soc. Lecture Note Ser., 167, Cambridge Univ. Press, Cambridge, 1991.
  • Aldous, David. The continuum random tree. III. Ann. Probab. 21 (1993), no. 1, 248–289.
  • Aldous, David. Recursive self-similarity for random trees, random triangulations and Brownian excursion. Ann. Probab. 22 (1994), no. 2, 527–545.
  • Aldous, David. Brownian excursions, critical random graphs and the multiplicative coalescent. Ann. Probab. 25 (1997), no. 2, 812–854.
  • Aldous, David J.; Pitman, Jim. Brownian bridge asymptotics for random mappings. Random Structures Algorithms 5 (1994), no. 4, 487–512.
  • Aldous, David; Miermont, Grégory; Pitman, Jim. Weak convergence of random $p$-mappings and the exploration process of inhomogeneous continuum random trees. Probab. Theory Related Fields 133 (2005), no. 1, 1–17.
  • Aldous, David J. Exchangeability and related topics. École d'été de probabilités de Saint-Flour, XIII–-1983, 1–198, Lecture Notes in Math., 1117, Springer, Berlin, 1985.
  • Athreya, Krishna B.; Ney, Peter E. Branching processes. Die Grundlehren der mathematischen Wissenschaften, Band 196. Springer-Verlag, New York-Heidelberg, 1972. xi+287 pp.
  • Bertoin, Jean; Goldschmidt, Christina. Dual random fragmentation and coagulation and an application to the genealogy of Yule processes. Mathematics and computer science. III, 295–308, Trends Math., Birkhäuser, Basel, 2004.
  • Bertoin, Jean; Pitman, Jim. Path transformations connecting Brownian bridge, excursion and meander. Bull. Sci. Math. 118 (1994), no. 2, 147–166.
  • Blackwell, David; Kendall, David. The martin boundary of Pólya's urn scheme, and an application to stochastic population growth. J. Appl. Probability 1 1964 284–296.
  • Bollobás, Béla. Random graphs. Second edition. Cambridge Studies in Advanced Mathematics, 73. Cambridge University Press, Cambridge, 2001. xviii+498 pp. ISBN: 0-521-80920-7; 0-521-79722-5
  • Chaumont, Loic.; Yor, Marc. Exercises in probability. A guided tour from measure theory to random processes, via conditioning. Cambridge Series in Statistical and Probabilistic Mathematics, 13. Cambridge University Press, Cambridge, 2003. xvi+236 pp. ISBN: 0-521-82585-7
  • de Finetti, Bruno. Funzione caratteristica di un fenomeno aleatorio. Atti della R. Academia Nazionale dei Lincei, Serie 6. Memorie, Classe di Scienze Fisiche, Mathematice e Naturale, 4 1931 251–-299.
  • Ding, Jian; Kim, Jeong Han; Lubetzky, Eyal; Peres, Yuval. Anatomy of a young giant component in the random graph. arXiv:0906.1839v1 [math.CO], 2009.
  • Dufresne, Daniel. Algebraic properties of beta and gamma distributions, and applications. Adv. in Appl. Math. 20 (1998), no. 3, 285–299.
  • F. Eggenberger and G. Pólya. Uber die Statistik verketteter Vorgange. Zeitschrift fur Angewandte Mathematik und Mechanik, 3: 1923 279–289.
  • Erdós, P.; Rényi, A. On the evolution of random graphs. Magyar Tud. Akad. Mat. Kutat' Int. Közl. 5 1960 17–61. (23 #A2338)
  • Evans, Steven N.; Pitman, Jim; Winter, Anita. Rayleigh processes, real trees, and root growth with re-grafting. Probab. Theory Related Fields 134 (2006), no. 1, 81–126.
  • Feller, William. An introduction to probability theory and its applications. Vol. II. Second edition John Wiley & Sons, Inc., New York-London-Sydney 1971 xxiv+669 pp.
  • Freedman, David A. Bernard Friedman's urn. Ann. Math. Statist 36 1965 956–970.
  • Gordon, Louis. A stochastic approach to the gamma function. Amer. Math. Monthly 101 (1994), no. 9, 858–865.
  • Janson, Svante; Knuth, Donald E.; Łuczak, Tomasz; Pittel, Boris. The birth of the giant component. With an introduction by the editors. Random Structures Algorithms 4 (1993), no. 3, 231–358.
  • Janson, Svante; Łuczak, Tomasz; Rucinski, Andrzej. Random graphs. Wiley-Interscience Series in Discrete Mathematics and Optimization. Wiley-Interscience, New York, 2000. xii+333 pp. ISBN: 0-471-17541-2
  • Johnson, Norman L.; Kotz, Samuel; Balakrishnan, N. Continuous univariate distributions. Vol. 1. Second edition. Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics. A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1994. xxii+756 pp. ISBN: 0-471-58495-9
  • Le Gall, Jean-François. Random trees and applications. Probab. Surv. 2 (2005), 245–311 (electronic).
  • Łuczak, Tomasz; Pittel, Boris; Wierman, John C. The structure of a random graph at the point of the phase transition. Trans. Amer. Math. Soc. 341 (1994), no. 2, 721–748.
  • Peres, Yuval; Revelle, David. Mixing times for random walks on finite lamplighter groups. Electron. J. Probab. 9 (2004), no. 26, 825–845 (electronic).
  • Pitman, Jim. Brownian motion, bridge, excursion, and meander characterized by sampling at independent uniform times. Electron. J. Probab. 4 (1999), no. 11, 33 pp. (electronic).
  • Pitman, J. Combinatorial stochastic processes. Lectures from the 32nd Summer School on Probability Theory held in Saint-Flour, July 7–24, 2002. With a foreword by Jean Picard. Lecture Notes in Mathematics, 1875. Springer-Verlag, Berlin, 2006. x+256 pp. ISBN: 978-3-540-30990-1; 3-540-30990-X
  • Rémy, Jean-Luc. Un procédé itératif de dénombrement d'arbres binaires et son application à leur génération aléatoire. (French) [An iterative procedure for enumerating binary trees and its application to their random generation] RAIRO Inform. Théor. 19 (1985), no. 2, 179–195.
  • Schweinsberg, Jason. The loop-erased random walk and the uniform spanning tree on the four-dimensional discrete torus. Probab. Theory Related Fields 144 (2009), no. 3-4, 319–370.
  • S.S. Wilks. Certain generalizations in the analysis of variance. Biometrika, 24 (1932) 471–494.