Abstract
We consider the Erdős–Rényi random graph G(n, p) inside the critical window, that is when p = 1/n + λn −4/3, for some fixed \({\lambda \in \mathbb{R}}\) . We prove that the sequence of connected components of G(n, p), considered as metric spaces using the graph distance rescaled by n −1/3, converges towards a sequence of continuous compact metric spaces. The result relies on a bijection between graphs and certain marked random walks, and the theory of continuum random trees. Our result gives access to the answers to a great many questions about distances in critical random graphs. In particular, we deduce that the diameter of G(n, p) rescaled by n −1/3 converges in distribution to an absolutely continuous random variable with finite mean.
References
Addario-Berry, L., Broutin, N., Reed, B.: The diameter of the minimum spanning tree of a complete graph. In: Proceedings of Fourth Colloquium on Mathematics and Computer Science Algorithms, Trees, Combinatorics and Probabilities, vol. 2 (2006)
Addario-Berry L., Broutin N., Reed B.: Critical random graphs and the structure of a minimum spanning tree. Random Struct. Algorithms 35(3), 323–347 (2009)
Addario-Berry L., Broutin N., Goldschmidt C.: Critical random graphs: limiting constructions and distributional properties. Electron. J. Probab. 15, 741–775 (2010)
Aldous D.: The continuum random tree I. Ann. Probab. 19(1), 1–28 (1991) ISSN 0091-1798
Aldous, D.: The continuum random tree II: an overview. In: Stochastic Analysis (Durham, 1990), volume 167 of London Math. Soc. Lecture Note Ser., pp. 23–70. Cambridge University Press, Cambridge (1991)
Aldous D.: The continuum random tree III. Ann. Probab. 21(1), 248–289 (1993) ISSN 0091-1798
Aldous D.: Brownian excursions, critical random graphs and the multiplicative coalescent. Ann. Probab. 25, 812–854 (1997)
Aldous D., Pittel B.: On a random graph with immigrating vertices: emergence of the giant component. Random Struct. Algorithms 17, 79–102 (2000)
Andersen E.S.: Fluctuations of sums of random variables. Math. Scand. 1, 263–285 (1953)
Billingsley P.: Convergence of Probability Measures. Wiley, New York (1968)
Bollobás, B.: Random graphs, 2nd edn. Cambridge Studies in Advanced Mathematics. Cambridge University Press (2001)
Bollobás B.: The evolution of random graphs. Trans. Am. Math. Soc. 286(1), 257–274 (1984)
Borel E.: Sur l’emploi du théorème de Bernouilli pour faciliter le calcul d’une infinité de coefficients. Application au problème de l’attente à un guichet. C.R. Math. Acad. Sci. Paris 214, 452–456 (1942)
Burago, D., Burago, Y., Ivanov, S.: A Course in Metric Geometry, volume 33 of Graduate Studies in Mathematics. American Mathematical Society, Providence (2001)
Chung F., Lu L.: The diameter of random sparse graphs. Adv. Appl. Math. 26, 257–279 (2001)
Cormen T.H., Leiserson C.E., Rivest R.L., Stein C.: Introduction to Algorithms, 2nd edn. MIT Press, Cambridge (2001)
Corneil D.G., Krueger R.M.: A unified view of graph searching. SIAM J. Discrete Math. 22, 1259–1276 (2008)
Cox D.R., Isham V.: Point Processes. Chapman & Hall, London (1980)
Ding, J., Kim, J.H., Lubetzky, E., Peres, Y.: Diameters in supercritical random graphs via first passage percolation. arXiv:0906.1840 [math.CO] (2009)
Duquesne, T., Le Gall, J.-F.: Random trees, Lévy processes and spatial branching processes. Astérisque 281, vi+147 (2002). ISSN 0303-1179
Dwass M.: The total progeny in a branching process and a related random walk. J. Appl. Probab. 6(3), 682–686 (1969)
Erdős P., Rényi A.: On the evolution of random graphs. Publ. Math. Inst. Hungar. Acad. Sci. 5, 17–61 (1960)
Evans, S.N.: Probability and real trees. In: Lecture Notes in Mathematics, vol. 1920. Springer, Berlin (2008). ISBN 978-3-540-74797-0. Lectures from the 35th Summer School on Probability Theory held in Saint-Flour, July 6–23, 2005
Flajolet P., Odlyzko A.: The average height of binary trees and other simple trees. J. Comput. System Sci. 25, 171–213 (1982)
Flajolet P., Gao Z., Odlyzko A., Richmond B.: The distribution of heights of binary trees and other simple trees. Combin. Probab. Comput. 2, 145–156 (1993)
Gessel I., Wang D.-L.: Depth-first search as a combinatorial correspondence. J. Combin. Theory Ser. A 26, 308–313 (1979)
Goldschmidt C.: Critical random hypergraphs: the emergence of a giant set of identifiable vertices. Ann. Probab. 33, 1573–1600 (2005)
Janson S.: Brownian excursion area, Wright’s constants in graph enumeration, and other Brownian areas. Probab. Surv. 4, 80–145 (2007) ISSN 1549-5787
Janson S., Spencer J.: A point process describing the component sizes in the critical window of the random graph evolution. Combin. Probab. Comput. 16, 631–658 (2007)
Janson S., Łuczak T., Ruciński A.: Random Graphs. Wiley, New York (2000)
Kallenberg, O.: Foundations of modern probability. In: Probability and its Applications, 2nd edn. Springer Verlag, Berlin (2003)
Kennedy D.P.: The distribution of the maximum Brownian excursion. J. Appl. Probab. 13, 371–376 (1976)
Khorunzhiy O., Marckert J.-F.: Uniform bounds for exponential moments of maximum of Dyck paths. Electron. Commun. Probab. 14, 327–333 (2009)
Kingman, J.F.C. Poisson processes. In: Oxford Studies in Probability, vol. 3. Oxford University Press (1992)
Le Gall J.-F.: Random trees and applications. Probab. Surv. 2, 245–311 (2005) ISSN 1549-5787
Le Gall J.-F.: Random real trees. Ann. Fac. Sci. Toulouse Math. (6) 15(1), 35–62 (2006) ISSN 0240-2963
Łuczak T.: Component behavior near the critical point of the random graph process. Random Struct. Algorithms 1, 287–310 (1990)
Łuczak T.: The number of trees with a large diameter. J. Aust. Math. Soc. Ser. A 58, 298–311 (1995)
Łuczak T.: Random trees and random graphs. Random Struct. Algorithms 13, 485–500 (1998)
Łuczak T., Pittel B., Wierman J.C.: The structure of random graphs at the point of transition. Trans. Am. Math. Soc. 341, 721–748 (1994)
Marckert J.-F., Mokkadem A.: The depth first processes of Galton–Watson trees converge to the same Brownian excursion. Ann. Probab. 31, 1655–1678 (2003)
Nachmias A., Peres Y.: Critical random graphs: diameter and mixing time. Ann. Probab. 36, 1267–1286 (2008)
Nachmias A., Peres Y.: The critical random graph, with martingales. Isr. J. Math. 176(1), 29–41 (2010a)
Nachmias A., Peres Y.: Critical percolation on random regular graphs. Random Struct. Algorithms 36(2), 111–148 (2010b)
Pittel B.: On the largest component of the random graph at a nearcritical stage. J. Combin. Theory Ser. B 82, 237–269 (2001)
Rényi A., Szekeres G.: On the height of trees. J. Aust. Math. Soc. 7, 497–507 (1967)
Revuz D., Yor M.: Continuous Martingales and Brownian Motion, 3rd edn. Springer, Berlin (2004)
Riordan, O., Wormald, N.C.: The diameter of sparse random graphs. arXiv:0808.4067 [math.PR] (2008)
Rogers, L.C.G., Williams, D.: Diffusions, Markov Processes and Martingales: Itô Calculus, vol. 2, 2nd edn. Cambridge University Press, Cambridge (2000)
Scott A.D., Sorkin G.B.: Solving sparse random instances of Max Cut and Max 2-CSP in linear expected time. Combin. Probab. Comput. 15, 281–315 (2006)
Spencer J.: Enumerating graphs and Brownian motion. Commun. Pure Appl. Math. 50, 291–294 (1997)
Takács L.: A combinatorial theorem for stochastic processes. Bull. Am. Math. Soc. 71, 649–650 (1965)
Tarjan R.: Depth first search and linear graph algorithms. SIAM J. Comput. 1, 146–160 (1972)
Wright E.M.: The number of connected sparsely edged graphs. J. Graph Theory 1(4), 317–330 (1977)
Author information
Authors and Affiliations
Corresponding author
Additional information
L. Addario-Berry was supported by an NSERC Discovery Grant throughout the research and writing of this paper. C. Goldschmidt was funded by EPSRC Postdoctoral Fellowship EP/D065755/1.
Rights and permissions
About this article
Cite this article
Addario-Berry, L., Broutin, N. & Goldschmidt, C. The continuum limit of critical random graphs. Probab. Theory Relat. Fields 152, 367–406 (2012). https://doi.org/10.1007/s00440-010-0325-4
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00440-010-0325-4
Keywords
- Random graphs
- Gromov–Hausdorff distance
- Scaling limits
- Continuum random tree
- Diameter
Mathematics Subject Classification (2000)
- 05C80
- 60C05