## 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.

### MSC:

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

Zbl 0877.60010
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### References:

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