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Subgraph counts in random graphs using incomplete U-statistics methods

Publiceringsår: 1988
Språk: Engelska
Sidor: 299-310
Publikation/Tidskrift/Serie: Discrete Mathematics
Volym: 72
Dokumenttyp: Artikel i tidskrift
Förlag: Elsevier


The random graph Kn,p is constructed on n labelled vertices by inserting each of the (n2) possible edges independently with probability p, 0> p < 1. For a fixed graph G, the threshold function for existence of a subgraph of Kn,p isomorphic to G has been determined by Erdös and Rényi [8] and Bollobás [3]. Bollobás [3] and Karo ski [14] have established asymptotic Poisson and normal convergence for the number of subgraphs of Kn,p isomorphic to G for sequences of p(n)→0 which are slightly greater than the threshold function. We use techniques from asymptotic theory in statistics, designed to study sums of dependent random variables known as U-statistics. We note that a subgraph count has the form of an incomplete U-statistics, and prove asymptotic normality of subgraph counts for a wide range of values of p, including any constant p and sequences of p(n) tending to 0 or 1 sufficiently slowly.


  • Probability Theory and Statistics


  • ISSN: 0012-365X

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