Scaling limits for critical inhomogeneous random graphs with finite third moments. (English) Zbl 1228.60018

Bhamidi, Shankar; Van Der Hofstad, Remco; Van Leeuwaarden, Johan S. H.

Electron. J. Probab. 15, Paper No. 54, 1682-1702 (2010).

Summary: 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 length of an inhomogeneous Brownian motion, which extends results of D. Aldous [Ann. Probab. 25, No. 2, 812–854 (1997; Zbl 0877.60010)] for the critical behavior of Erdős-Rényi random graphs. We rely heavily on martingale convergence techniques, and concentration properties of (super)martingales. This paper is part of a programme initiated by R. van der Hofstad [“Critical behavior in inhomogeneous random graphs”, arXiv:0902.0216] to study the near-critical behavior in inhomogeneous random graphs of so-called rank-1.

MSC:

60C05	Combinatorial probability
05C80	Random graphs (graph-theoretic aspects)
90B15	Stochastic network models in operations research

Keywords:

critical random graphs; phase transitions; inhomogeneous networks; Brownian excursions; size-biased ordering; martingale techniques

Citations:

Zbl 0877.60010