Diffusion approximation for the components in critical inhomogeneous random graphs of rank 1.1. Issue 4 (16th July 2013) - Turova, Tatyana S.
Abstract Consider the random graph on n vertices 1, …, n. Each vertex i is assigned a type xi with x1, …, xn being independent identically distributed as a nonnegative random variable X. We assume that EX3< ∞. Given types of all vertices, an edge exists between vertices i and j independent of anything else and with probability <![CDATA[\documentclass{article}\usepackage{mathrsfs, amsmath, amssymb}\pagestyle{empty}\begin{document}\begin{align*}\min \{1, \frac{x_ix_j}{n}\left(1+\frac{a}{n^{1/3}} \right) \}\end{align*} \end{document}]]> <inline-graphic xlink:href="ark:/27927/pgg3qnzwn9r" mimetype="image" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />. We study the critical phase, which is known to take place when EX2 = 1. We prove that normalized by n‐2/3the asymptotic joint distributions of component sizes of the graph equals the joint distribution of the excursions of a reflecting Brownian motion with diffusion coefficient <![CDATA[\documentclass{article}\usepackage{mathrsfs, amsmath, amssymb}\pagestyle{empty}\begin{document}\begin{align*}\sqrt{{\textbf{ E}}X{\textbf{ E}}X^3}\end{align*} \end{document}]]> <inline-graphic xlink:href="ark:/27927/pgg3qnzwnw4" mimetype="image" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />and drift <![CDATA[\documentclass{article}\usepackage{mathrsfs, amsmath, amssymb}\pagestyle{empty}\begin{document}\begin{align*}a-\frac{{\textbf{ E}}X^3}{{\textbf{ E}}X}s\end{align*} \end{document}]]> <inline-graphic xlink:href="ark:/27927/pgg3qnzwp09" mimetype="image" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />. In particular, we conclude that the size of the largest connected component is of order n2/3. © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 43, 486–539, 2013