The Annals of Applied Probability

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

B. G. Pittel

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A uniformly random graph on n vertices with a fixed degree sequence, obeying a γ subpower law, is studied. It is shown that, for γ>3, in a subcritical phase with high probability the largest component size does not exceed n1/γ+ɛn, ɛn=O(ln ln n/ln n), 1/γ being the best power for this random graph. This is similar to the best possible n1/(γ−1) bound for a different model of the random graph, one with independent vertex degrees, conjectured by Durrett, and proved recently by Janson.

Article information

Ann. Appl. Probab., Volume 18, Number 4 (2008), 1636-1650.

First available in Project Euclid: 21 July 2008

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Zentralblatt MATH identifier

Primary: 60C05: Combinatorial probability 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)

Random graph degree sequence power law largest cluster pairing process martingale asymptotic bounds


Pittel, B. G. On the largest component of a random graph with a subpower-law degree sequence in a subcritical phase. Ann. Appl. Probab. 18 (2008), no. 4, 1636--1650. doi:10.1214/07-AAP493.

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