On the number of vertices of given degree in a random graph

IntroductionBarbourfl] invented an ingenious method of establishing the asymptotic distribution of the number X of specified subgraphs of a random graph. The novelty of his method relies on using the first two moments of X only, despite the traditional method of moments that involves all moments of X (compare [8,10,11,14]). He also adjusted that new method for counting isolated trees of a given size in a random graph. (For further applications of Barbour's method see [4] and [10].) The main goal of this paper is to show how this method can be extended to a general setting that enables us to derive asymptotic distributions of subsets of vertices of a random graph with various properties.Throughout the paper we focus our attention on those properties that are determined by the presence or absence of the edges having at least one endpoint in the subset, i.e. lying totally inside it or belonging to its edge-cut. We call such properties semi-induced. For instance, the properties of being an independent set, dominating set, or the vertex set of a connected component are of this type. Also, being a vertex of a given degree is a semi-induced property.In the next section we formally define various classes of properties of subsets of vertices of a graph. Then we develop and refine Barbour's original argument and obtain several estimates for the total variation distance between the distributions of random variables in question and appropriate Poisson distributions.Section 3 is devoted to applications of those general results to a random graph K n v , obtained by independent deletion of the edges of a complete graph on n vertices with probability 1 -p. In particular, we state conditions under which the number of vertices of K n with a given degree has asymptotically normal distribution. Our result is a substantial extension of the normal phase that has been previously found (see, for example, [13]).Finally, using our estimates from Section 2 we establish the asymptotic distributions of the number of independent, 2-independent, and dominating sets of a given size in K n p . Some of these results extend and strengthen earlier results of Burtin (see [5,6]).We write X n^->Po(A) and X n^~> N(0,1) when a sequence of random variables {X n } converges in distribution to the Poisson distribution with expectation A or to the standard normal distribution, respectively. Given a random variable X, EX stands for its expectation, var X for its variance, and X for its standardization, i.e.

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“…Our result is a substantial extension of the normal phase that has been previously found (see, for example, [13]). …”

Section: Introductionsupporting

confidence: 68%

Poisson convergence and semi-induced properties of random graphs

Karoński

Ruciński

1987

Math. Proc. Camb. Phil. Soc.

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“…W e will s t u d y some properties o f ~r(m, n) for large m a n d u as a function p -~ p(m, n) increases f r o m 0 to 1. Similar results concerning the distribution of the n u m b e r of vertices of a given degree in a usual r a n d o m g r a p h Kn, p are p r e s e n t e d in p a p e r s [ 1 ] -- [ 7 ] .…”

Section: Introductionsupporting

confidence: 53%

On the degrees of vertices in A bichromatic random graph

Pałka

1984

Period Math Hung

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“…See also Palka (1984) and Bollobás (1985). Asymptotic normality when πn → c > 0, was obtained by Barbour, Karoński and Ruciński (1989).…”

Section: Graph Degree Countsmentioning

confidence: 95%

Multivariate normal approximations by Stein's method and size bias couplings

Goldstein

Rinott

1996

Journal of Applied Probability

100

134

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Stein's method is used to obtain two theorems on multivariate normal approximation. Our main theorem, Theorem 1.2, provides a bound on the distance to normality for any nonnegative random vector. Theorem 1.2 requires multivariate size bias coupling, which we discuss in studying the approximation of distributions of sums of dependent random vectors. In the univariate case, we briefly illustrate this approach for certain sums of nonlinear functions of multivariate normal variables. As a second illustration, we show that the multivariate distribution counting the number of vertices with given degrees in certain random graphs is asymptotically multivariate normal and obtain a bound on the rate of convergence. Both examples demonstrate that this approach may be suitable for situations involving non-local dependence. We also present Theorem 1.4 for sums of vectors having a local type of dependence. We apply this theorem to obtain a multivariate normal approximation for the distribution of the random p-vector which counts the number of edges in a fixed graph both of whose vertices have the same given color when each vertex is colored by one of p colors independently. All normal approximation results presented here do not require an ordering of the summands related to the dependence structure. This is in contrast to hypotheses of classical central limit theorems and examples, which involve e.g., martingale, Markov chain, or various mixing assumptions.

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On the number of vertices of given degree in a random graph

Cited by 19 publications

References 2 publications

Poisson convergence and semi-induced properties of random graphs

Poisson convergence and semi-induced properties of random graphs

On the degrees of vertices in A bichromatic random graph

Multivariate normal approximations by Stein's method and size bias couplings

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