Electronic Journal of Probability
- Electron. J. Probab.
- Volume 15 (2010), paper no. 54, 1682-1702.
Scaling Limits for Critical Inhomogeneous Random Graphs with Finite Third Moments
Shankar Bhamidi, Remco van der Hofstad, and Johan van Leeuwaarden
Full-text: Open access
Abstract
We identify the scaling limit for the sizes of the largest components at criticality for inhomogeneous random graphs with weights that have finite third moments. We show that the sizes of the (rescaled) components converge to the excursion lengths of an inhomogeneous Brownian motion, which extends results of Aldous (1997) for the critical behavior of Erdös-Rényi random graphs. We rely heavily on martingale convergence techniques, and concentration properties of (super)martingales. This paper is part of a programme initiated in van der Hofstad (2009) to study the near-critical behavior in inhomogeneous random graphs of so-called rank-1.
Article information
Source
Electron. J. Probab., Volume 15 (2010), paper no. 54, 1682-1702.
Dates
Accepted: 2 November 2010
First available in Project Euclid: 1 June 2016
Permanent link to this document
https://projecteuclid.org/euclid.ejp/1464819839
Digital Object Identifier
doi:10.1214/EJP.v15-817
Mathematical Reviews number (MathSciNet)
MR2735378
Zentralblatt MATH identifier
1228.60018
Subjects
Primary: 60C05: Combinatorial probability
Secondary: 05C80: Random graphs [See also 60B20] 90B15: Network models, stochastic
Keywords
critical random graphs phase transitions inhomogeneous networks Brownian excursions size-biased ordering martingale techniques
Rights
This work is licensed under aCreative Commons Attribution 3.0 License.
Citation
Bhamidi, Shankar; van der Hofstad, Remco; van Leeuwaarden, Johan. Scaling Limits for Critical Inhomogeneous Random Graphs with Finite Third Moments. Electron. J. Probab. 15 (2010), paper no. 54, 1682--1702. doi:10.1214/EJP.v15-817. https://projecteuclid.org/euclid.ejp/1464819839
References
- Aldous,
David. Brownian excursions, critical random graphs and the
multiplicative coalescent. Ann. Probab. 25 (1997), no. 2, 812–854.Mathematical Reviews (MathSciNet): MR1434128
Zentralblatt MATH: 0877.60010
Digital Object Identifier: doi:10.1214/aop/1024404421
Project Euclid: euclid.aop/1024404421 - Bhamidi,
Shankar; van der Hofstad, Remco; van Leeuwaarden, Johan S.H.. Novel
scaling limits for critical inhomogeneous random graphs. Preprint
(2009).Zentralblatt MATH: 1257.05157
Digital Object Identifier: doi:10.1214/11-AOP680
Project Euclid: euclid.aop/1351258728 - Billingsley, Patrick. Convergence of probability measures. Second edition. Wiley Series in Probability and Statistics: Probability and Statistics. A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1999. x+277 pp. ISBN: 0-471-19745-9
- 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.
- Bollobás, Béla; Janson, Svante; Riordan, Oliver. The phase transition in inhomogeneous random graphs. Random Structures Algorithms 31 (2007), no. 1, 3–122.
- Britton, Tom; Deijfen, Maria; Martin-Löf, Anders. Generating simple random graphs with prescribed degree distribution. J. Stat. Phys. 124 (2006), no. 6, 1377–1397.
- Chung, Fan; Lu, Linyuan. The average distances in random graphs with given expected degrees. Proc. Natl. Acad. Sci. USA 99 (2002), no. 25, 15879–15882 (electronic).
- Chung, Fan; Lu, Linyuan. Connected components in random graphs with given expected degree sequences. Ann. Comb. 6 (2002), no. 2, 125–145.
- Chung,
Fan; Lu, Linyuan. The average distance in a random graph with given
expected degrees. Internet Math. 1 (2003), no. 1, 91–113.Zentralblatt MATH: 1065.05084
Digital Object Identifier: doi:10.1080/15427951.2004.10129081
Project Euclid: euclid.im/1057768561 - Chung, Fan; Lu, Linyuan. Complex graphs and networks. CBMS Regional Conference Series in Mathematics, 107. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2006. viii+264 pp. ISBN: 978-0-8218-3657-6; 0-8218-3657-9
- Chung, Fan; Lu, Linyuan. The volume of the giant component of a random graph with given expected degrees. SIAM J. Discrete Math. 20 (2006), no. 2, 395–411 (electronic).
- Dudley, R. M. Real analysis and probability. Revised reprint of the 1989 original. Cambridge Studies in Advanced Mathematics, 74. Cambridge University Press, Cambridge, 2002. x+555 pp. ISBN: 0-521-00754-2.
- van den Esker, Henri; van der Hofstad, Remco; Hooghiemstra, Gerard. Universality for the distance in finite variance random graphs. J. Stat. Phys. 133 (2008), no. 1, 169–202.
- Ethier,
Stewart N.; Kurtz, Thomas G. Markov processes. Characterization and
convergence. Wiley Series in Probability and Mathematical Statistics:
Probability and Mathematical Statistics. John Wiley & Sons, Inc.,
New York, 1986. x+534 pp. ISBN: 0-471-08186-8.Zentralblatt MATH: 0592.60049
- Gut, Allan. Probability: a graduate course. Springer Texts in Statistics. Springer, New York, 2005. xxiv+603 pp. ISBN: 0-387-22833-0.
- van der Hofstad, Remco. Critical behavior in inhomogeneous random graphs. Preprint (2009).
- van
der Hofstad, Remco; Kager, Wouter; Müller, Tobias. A local limit
theorem for the critical random graph. Electron. Commun. Probab. 14
(2009), 122–131.Mathematical Reviews (MathSciNet): MR2481672
Digital Object Identifier: doi:10.1214/ECP.v14-1451
Project Euclid: euclid.ecp/1465234721 - Janson, Svante. Asymptotic equivalence and contiguity of some random graphs. Random Structures Algorithms 36 (2010), no. 1, 26–45.
- 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.
- Kallenberg, Olav. Foundations of modern probability. Second edition. Probability and its Applications (New York). Springer-Verlag, New York, 2002. xx+638 pp. ISBN: 0-387-95313-2.
- Martin-Löf,
Anders. The final size of a nearly critical epidemic, and the first
passage time of a Wiener process to a parabolic barrier. J. Appl.
Probab. 35 (1998), no. 3, 671–682.Zentralblatt MATH: 0919.92034
Digital Object Identifier: doi:10.1239/jap/1032265215
Project Euclid: euclid.jap/1032265215 - Nachmias, Asaf; Peres, Yuval. Critical percolation on random regular graphs. Random Structures Algorithms 36 (2010), no. 2, 111–148.
- Norros, Ilkka; Reittu, Hannu. On a conditionally Poissonian graph process. Adv. in Appl. Probab. 38 (2006), no. 1, 59–75.Zentralblatt MATH: 1096.05047
Digital Object Identifier: doi:10.1017/S000186780000080X
Project Euclid: euclid.aap/1143936140 - Pittel, Boris. On the largest component of the random graph at a nearcritical stage. J. Combin. Theory Ser. B 82 (2001), no. 2, 237–269.
- Rogers, L. C. G.; Williams, David. Diffusions, Markov processes, and martingales. Vol. 1. Foundations. Second edition. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. John Wiley & Sons, Ltd., Chichester, 1994. xx+386 pp. ISBN: 0-471-95061-0.
- Turova, Tatyana. Diffusion approximation for the components in critical inhomogeneous random graphs of rank 1. Preprint (2009).Mathematical Reviews (MathSciNet): MR3124693
Zentralblatt MATH: 1278.05227
Digital Object Identifier: doi:10.1002/rsa.20503
The Institute of Mathematical Statistics and the Bernoulli Society

- You have access to this content.
- You have partial access to this content.
- You do not have access to this content.
More like this
- Novel scaling limits for critical inhomogeneous random graphs
Bhamidi, Shankar, van der Hofstad, Remco, and van Leeuwaarden, Johan S. H., The Annals of Probability, 2012 - Critical window for the configuration model: finite third moment degrees
Dhara, Souvik, van der Hofstad, Remco, van Leeuwaarden, Johan S.H., and Sen, Sanchayan, Electronic Journal of Probability, 2017 - Component sizes for large quantum Erdős–Rényi graph near criticality
Dembo, Amir, Levit, Anna, and Vadlamani, Sreekar, The Annals of Probability, 2019
- Novel scaling limits for critical inhomogeneous random graphs
Bhamidi, Shankar, van der Hofstad, Remco, and van Leeuwaarden, Johan S. H., The Annals of Probability, 2012 - Critical window for the configuration model: finite third moment degrees
Dhara, Souvik, van der Hofstad, Remco, van Leeuwaarden, Johan S.H., and Sen, Sanchayan, Electronic Journal of Probability, 2017 - Component sizes for large quantum Erdős–Rényi graph near criticality
Dembo, Amir, Levit, Anna, and Vadlamani, Sreekar, The Annals of Probability, 2019 - The component sizes of a critical random graph with given degree sequence
Joseph, Adrien, The Annals of Applied Probability, 2014 - Cycle structure of percolation on high-dimensional tori
van der Hofstad, Remco and Sapozhnikov, Artëm, Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, 2014 - Critical Random Graphs: Limiting Constructions and Distributional Properties
Addario-Berry, Louigi, Broutin, Nicolas, and Goldschmidt, Christina, Electronic Journal of Probability, 2010 - Combinatorial approach to the interpolation method and scaling limits in sparse random graphs
Bayati, Mohsen, Gamarnik, David, and Tetali, Prasad, The Annals of Probability, 2013 - The densest subgraph problem in sparse random graphs
Anantharam, Venkat and Salez, Justin, The Annals of Applied Probability, 2016 - Universality of trap models in the ergodic time scale
Jara, M., Landim, C., and Teixeira, A., The Annals of Probability, 2014 - Convergence of mixing times for sequences of random walks on finite graphs
Croydon, David, Hambly, Ben, and Kumagai, Takashi, Electronic Journal of Probability, 2012