Diameters in supercritical random graphs via first passage percolation
Combinatorics, Probability and Computing
The diameter of sparse random graphs
Combinatorics, Probability and Computing
The evolution of the cover time
Combinatorics, Probability and Computing
Anatomy of the giant component: The strictly supercritical regime
European Journal of Combinatorics
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We provide a complete description of the giant component of the Erdős-Rényi random graph \documentclass{article} \usepackage{mathrsfs} \usepackage{amsmath} \pagestyle{empty} \begin{document} \begin{align*}{\mathcal{G}}(n,p)\end{align*} \end{document} **image** 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 \documentclass{article} \usepackage{mathrsfs} \usepackage{amsmath} \pagestyle{empty} \begin{document} \begin{align*}{\mathcal{C}_1}\end{align*} \end{document} **image** 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 \documentclass{article} \usepackage{mathrsfs} \usepackage{amsmath} \pagestyle{empty} \begin{document} \begin{align*}{\mathcal{C}_1}\end{align*} \end{document} **image** ). Let Z be normal with mean \documentclass{article} \usepackage{mathrsfs} \usepackage{amsmath} \pagestyle{empty} \begin{document} \begin{align*}\frac{2}{3} \varepsilon^3 n\end{align*} \end{document} **image** and variance ε3n, and let \documentclass{article} \usepackage{mathrsfs} \usepackage{amsmath} \pagestyle{empty} \begin{document} \begin{align*}\mathcal{K}\end{align*} \end{document} **image** be a random 3-regular graph on \documentclass{article} \usepackage{mathrsfs} \usepackage{amsmath} \pagestyle{empty} \begin{document} \begin{align*}2\left\lfloor Z\right\rfloor\end{align*} \end{document} **image** vertices. Replace each edge of \documentclass{article} \usepackage{mathrsfs} \usepackage{amsmath} \pagestyle{empty} \begin{document} \begin{align*}\mathcal{K}\end{align*} \end{document} **image** 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 \documentclass{article} \usepackage{mathrsfs} \usepackage{amsmath} \pagestyle{empty} \begin{document} \begin{align*}{\mathcal{C}_1}\end{align*} \end{document} **image** . © 2010 Wiley Periodicals, Inc. Random Struct. Alg., 39, 139–178, 2011 © 2011 Wiley Periodicals, Inc.