A Phase Transition for the Diameter of the Configuration Model
TLDR
A phase transition for the diameter is established when the power-law exponent τ of the degrees satisfies τ ∈ (2, 3) and it is shown that for τ > 2 and when vertices with degree 1 or 2 are present with positive probability, the diameter of the random graph is bounded from below by a constant times the logarithm of the size of the graph.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 τ of the degrees satisfies τ ∈ (2, 3). Indeed, we show that for τ > 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 τ ∈ (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.read more
Citations
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Some typical properties of the spatial preferred attachment model
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References
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Proceedings ArticleDOI
Random graphs
TL;DR: Some of the major results in random graphs and some of the more challenging open problems are reviewed, including those related to the WWW.