The component sizes of a critical random graph with given degree sequence. (English) Zbl 1318.60015

The theme of this paper is the limiting distribution of the component sizes of a critical random graph on \(n\) vertices with a given degree distribution. Let \(\nu_k\) denote the fraction of vertices that have \(k\) neighbours. The Molloy-Reed criterion for the existence of a giant component is the sign of the expression \(\sum_{k=1}^\infty k(k-2) \nu_k\). The author of the present paper considers the critical case, where \(\sum_{k=1}^\infty k(k-2) \nu_k = 0\) under the condition that \(\sum_{k>0} k^3 \nu_k < \infty\) and \(\nu_2 < 1\). The result obtained here is analogous to the result of D. Aldous [Ann. Probab. 25, No. 2, 812–854 (1997; Zbl 0877.60010)] on the component sizes distribution of a critical \(G(n,p)\) random graph. The author shows that the ordered vector of the component sizes rescaled by \(n^{-2/3}\) converges to the ordered vector of the excursion intervals of a Brownian motion with parabolic drift.


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


Zbl 0877.60010
Full Text: DOI arXiv Euclid


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