The evolution of random graphs

Abstract

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