The evolution of random graphs

Bollobás, Béla

doi:10.1090/S0002-9947-1984-0756039-5

The evolution of random graphs
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Trans. Amer. Math. Soc. 286 (1984), 257-274 Request permission

Abstract:

According to a fundamental result of Erdös and Rényi, the structure of a random graph ${G_M}$ changes suddenly when $M \sim n/2$: if $M = \left \lfloor {cn} \right \rfloor$ and $c < \frac {1}{2}$ then a.e. random graph of order $n$ and since $M$ is such that its largest component has $O(\log n)$ vertices, but for $c > \frac {1}{2}$ a.e. ${G_M}$ has a giant component: a component of order $(1-{\alpha _c}+o(1))n$ where ${\alpha _c} < 1$. The aim of this paper is to examine in detail the structure of a random graph ${G_M}$ when $M$ is close to $n/2$. Among others it is proved that if $M = n/2 + s$, $s = o(n)$ and $s \geq {(\log n)^{1/2}}{n^{2/3}}$ then the giant component has $(4 + o(1))s$ vertices. Furthermore, rather precise estimates are given for the order of the $r$th largest component for every fixed $r$.

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