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- W76207694 abstract "Those with experience in random graphs should not be surprised that the Zero-One Law does not hold when p = n -1/3. Take a strictly balanced H with e edges, υ vertices and υ/ e = 1/3 — more specifically, take H = K 7. The number X of copies of K 7 has E [ X ] = ( n 7 ) p 21 → 1/7! . From Janson’s Inequality 5.0.4, or even the original methods of Erdös and Rényi, Pr[X = 0] → e –1/7!. Here X = 0 is the first order ¬∃K 7 and e -1/7! is certainly neither zero nor one. One might expect (as, indeed, this author once conjectured) that for any first order A lim n→ ∞ Pr [G(n, n -1/3) ⊨ A] would exist and one might even hope for a description of the limiting probabilities as in Section 3.6. We shall see in this chapter that very much the opposite is true, that there are first order A for which Pr [A] behaves very strangely at and near n -1/3.KeywordsRandom GraphThreshold FunctionRational PowerCommon NeighborOrder WorldThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves." @default.
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- W76207694 date "2001-01-01" @default.
- W76207694 modified "2023-09-25" @default.
- W76207694 title "Near Rational Powers of n" @default.
- W76207694 doi "https://doi.org/10.1007/978-3-662-04538-1_9" @default.
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