### 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) |
---|---|

Pages (from-to) | 139-178 |

Number of pages | 40 |

Journal | Random Structures and Algorithms |

Volume | 39 |

Issue number | 2 |

DOIs | |

State | Published - Sep 2011 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

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

### Cite this

*Random Structures and Algorithms*,

*39*(2), 139-178. https://doi.org/10.1002/rsa.20342

**Anatomy of a young giant component in the random graph.** / Ding, Jian; Kim, Jeong Han; Lubetzky, Eyal; Peres, Yuval.

Research output: Contribution to journal › Article

*Random Structures and Algorithms*, vol. 39, no. 2, pp. 139-178. https://doi.org/10.1002/rsa.20342

}

TY - JOUR

T1 - Anatomy of a young giant component in the random graph

AU - Ding, Jian

AU - Kim, Jeong Han

AU - Lubetzky, Eyal

AU - Peres, Yuval

PY - 2011/9

Y1 - 2011/9

N2 - 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 ε3n →∞ and ε = o(1). Our description is particularly simple for ε = o(n-1/4), where the giant component C1 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 C1). Let Z be normal with mean 2\3ε3n and variance ε3n, 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 Nk vertices of degree k for k ≥ 3, where Nk has mean and variance of order εkn. 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 C1.

AB - 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 ε3n →∞ and ε = o(1). Our description is particularly simple for ε = o(n-1/4), where the giant component C1 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 C1). Let Z be normal with mean 2\3ε3n and variance ε3n, 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 Nk vertices of degree k for k ≥ 3, where Nk has mean and variance of order εkn. 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 C1.

KW - Contiguity

KW - Giant component

KW - Near critical random graph

KW - Poisson cloning

UR - http://www.scopus.com/inward/record.url?scp=79960525085&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=79960525085&partnerID=8YFLogxK

U2 - 10.1002/rsa.20342

DO - 10.1002/rsa.20342

M3 - Article

VL - 39

SP - 139

EP - 178

JO - Random Structures and Algorithms

JF - Random Structures and Algorithms

SN - 1042-9832

IS - 2

ER -