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The diameter of sparse random graphs (2010)
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@MISC{Riordan10thediameter,
author = {Oliver Riordan and Nicholas Wormald},
title = {The diameter of sparse random graphs},
year = {2010}
}
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Abstract
In this paper we study the diameter of the random graph G(n,p), i.e., the largest finite distance between two vertices, for a wide range of functions p = p(n). For p = λ/n with λ> 1 constant we give a simple proof of an essentially best possible result, with an Op(1) additive correction term. Using similar techniques, we establish twopoint concentration in the case that np → ∞. For p = (1 + ε)/n with ε → 0, we obtain a corresponding result that applies all the way down to the scaling window of the phase transition, with an Op(1/ε) additive correction term whose (appropriately scaled) limiting distribution we describe. Combined with earlier results, our new results complete the determination of the diameter of the random graph G(n,p) to an accuracy of the order of its standard deviation (or better), for all functions p = p(n). Throughout we use branching process methods, rather than the more common approach of separate analysis of the 2-core and the trees attached to it. 1 Introduction and main results Throughout, we write diam(G) for the diameter of a graph G, meaning the largest graph
Keyphrases
sparse random graph random graph additive correction term new result simple proof phase transition similar technique scaling window corresponding result twopoint concentration main result separate analysis wide range finite distance standard deviation process method possible result common approach
