Anatomy of a young giant component in the random graph

Details Anatomy of a young giant component in the random graph Anatomy of a young giant component in the random graph

2009-06-10

arxiv.org/abs/0906.1839v2

Mathematics

Combinatorics

42 pages

Scientific paper

We provide a complete description of the giant component of the Erd\H{o}s-R\'enyi random graph $G(n,p)$ as soon as it emerges from the scaling window, i.e., for $p = (1+\epsilon)/n$ where $\epsilon^3 n \to \infty$ and $\epsilon=o(1)$. Our description is particularly simple for $\epsilon = o(n^{-1/4})$, 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 $\frac23 \epsilon^3 n$ and variance $\epsilon^3 n$, and let $K$ be a random 3-regular graph on $2\lfloor Z\rfloor$ vertices. Replace each edge of $K$ by a path, where the path lengths are i.i.d. geometric with mean $1/\epsilon$. Finally, attach an independent Poisson($1-\epsilon$)-Galton-Watson tree to each vertex. A similar picture is obtained for larger $\epsilon=o(1)$, in which case the random 3-regular graph is replaced by a random graph with $N_k$ vertices of degree $k$ for $k\geq 3$, where $N_k$ has mean and variance of order $\epsilon^k n$. 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 $\epsilon$, as well as the mixing time of the random walk on $C_1$.

Affiliated with

Also associated with

No associations

Rating

Anatomy of a young giant component in the random graph does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Anatomy of a young giant component in the random graph, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Anatomy of a young giant component in the random graph will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-206195