On the largest component of a random graph with a subpower-law degree sequence in a subcritical phase

Details On the largest component of a random graph with a subpower-law degree sequence in a subcritical phase On the largest component of a random graph with a subpower-law degree sequence in a subcritical phase

2008-08-21

arxiv.org/abs/0808.2907v1

Annals of Applied Probability 2008, Vol. 18, No. 4, 1636-1650

Mathematics

Probability

Published in at http://dx.doi.org/10.1214/07-AAP493 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Inst

Scientific paper

10.1214/07-AAP493

A uniformly random graph on $n$ vertices with a fixed degree sequence, obeying a $\gamma$ subpower law, is studied. It is shown that, for $\gamma>3$, in a subcritical phase with high probability the largest component size does not exceed $n^{1/\gamma+\varepsilon_n}$, $\varepsilon_n=O(\ln\ln n/\ln n)$, $1/\gamma$ being the best power for this random graph. This is similar to the best possible $n^{1/(\gamma-1)}$ bound for a different model of the random graph, one with independent vertex degrees, conjectured by Durrett, and proved recently by Janson.

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