Electronic Journal of Probability

Scaling Limits for Critical Inhomogeneous Random Graphs with Finite Third Moments

Shankar Bhamidi, Remco van der Hofstad, and Johan van Leeuwaarden

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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


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