Concentration of measure for the number of isolated vertices in the Erdos-Rényi random graph by size bias couplings

Subhankar Ghosh, Larry Goldstein and Martin Raic

Statistics & Probability Letters, 2011, vol. 81, issue 11, 1565-1570

Abstract: A concentration of measure result is proved for the number of isolated vertices Y in the Erdos-Rényi random graph model on n edges with edge probability p. When [mu] and [sigma]2 denote the mean and variance of Y respectively, P((Y-[mu])/[sigma]>=t) admits a bound of the form e-kt2 for some constant positive k under the assumption p[set membership, variant](0,1) and np-->c[set membership, variant](0,[infinity]) as n-->[infinity]. The left tail inequality holds for all n[set membership, variant]{2,3,...},p[set membership, variant](0,1) and t>=0. The results are shown by coupling Y to a random variable Ys having the Y-size biased distribution, that is, the distribution characterized by E[Yf(Y)]=[mu]E[f(Ys)] for all functions f for which these expectations exist.

Keywords: Large; deviations; Graph; degree; Size; biased; couplings (search for similar items in EconPapers)
Date: 2011
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