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Diffusion approximation for the components in critical inhomogeneous random graphs of rank 1.

Authors

  • Tatyana S. Turova

    Corresponding author
    1. Department of Mathematical Statistics, Mathematical Center, University of Lund, Lund S-221 00, Sweden
    • Department of Mathematical Statistics, Mathematical Center, University of Lund, Lund S-221 00, Sweden

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  • supported by the swedish natural science research council

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 equation image. 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 equation imageand drift equation image. 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

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