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2007 A Phase Transition for the Diameter of the Configuration Model
Remco van der Hofstad, Gerard Hooghiemstra, Dmitri Znamenski
Internet Math. 4(1): 113-128 (2007).

Abstract

In this paper, we study the configuration model (CM) with independent and identically-distributed (i.i.d.) degrees. We establish a phase transition for the diameter when the power-law exponent $\tau$ of the degrees satisfies $\tau \in (2,3)$. Indeed, we show that for $\tau>2$ and when vertices with degree $1$ or $2$ are present with positive probability, the diameter of the random graph is, with high probability, bounded from below by a constant times the logarithm of the size of the graph. On the other hand, assuming that all degrees are 3 or more, we show that, for $\tau \in (2,3)$, the diameter of the graph is, with high probability, bounded from above by a constant times the $\log \log$ of the size of the graph.

Citation

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Remco van der Hofstad. Gerard Hooghiemstra. Dmitri Znamenski. "A Phase Transition for the Diameter of the Configuration Model." Internet Math. 4 (1) 113 - 128, 2007.

Information

Published: 2007
First available in Project Euclid: 27 May 2009

zbMATH: 1167.05048
MathSciNet: MR2492177

Rights: Copyright © 2007 A K Peters, Ltd.

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Vol.4 • No. 1 • 2007
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