## Abstract

We provide a complete description of the giant component of the Erdos-Rényi random graph G(n, p) as soon as it emerges from the scaling window, i.e., for p = (1+ε)/n where ε^{3}n →∞ and ε = o(1). Our description is particularly simple for ε = o(n^{-1/4}), where the giant component C_{1} is contiguous with the following model (i.e., every graph property that holds with high probability for this model also holds w.h.p. for C_{1}). Let Z be normal with mean 2\3ε^{3}n and variance ε^{3}n, and let K be a random 3-regular graph on 2[Z] vertices. Replace each edge of K by a path, where the path lengths are i.i.d. geometric with mean 1/ε. Finally, attach an independent Poisson(1-ε)-Galton-Watson tree to each vertex. A similar picture is obtained for larger ε = o(1), in which case the random 3-regular graph is replaced by a random graph with N_{k} vertices of degree k for k ≥ 3, where N_{k} has mean and variance of order ε^{k}n. This description enables us to determine fundamental characteristics of the supercritical random graph. Namely, we can infer the asymptotics of the diameter of the giant component for any rate of decay of ε, as well as the mixing time of the random walk on C_{1}.

Original language | English (US) |
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Pages (from-to) | 139-178 |

Number of pages | 40 |

Journal | Random Structures and Algorithms |

Volume | 39 |

Issue number | 2 |

DOIs | |

State | Published - Sep 2011 |

## Keywords

- Contiguity
- Giant component
- Near critical random graph
- Poisson cloning

## ASJC Scopus subject areas

- Software
- Mathematics(all)
- Computer Graphics and Computer-Aided Design
- Applied Mathematics