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The size of the giant component of a random graph with a given degree sequence. (English) Zbl 0916.05064
Summary: Given a sequence of nonnegative real numbers \(\lambda_0, \lambda_1, \dots\) that sum to 1, we consider a random graph having approximately \(\lambda_in\) vertices of degree \(i\). In [Random Struct. Algorithms 6, 161-180 (1995)] the authors essentially show that if \(\sum i(i-2)\lambda_i>0\) then the graph a. s. has a giant component, while if \(\sum i(i-2)\lambda_i <0\) then a. s. all components in the graph are small. In this paper we analyse the size of the giant component in the former case, and the structure of the graph formed by deleting that component. We determine \(\varepsilon, \lambda_0', \lambda_1',\dots\) such that a. s. the giant component, \(C\), has \(\varepsilon n+o(n)\) vertices, and the structure of the graph remaining after deleting \(C\) is basically that of a random graph with \(n'=n-| C|\) vertices, and with \(\lambda_i'n'\) of them of degree \(i\).

MSC:
05C80 Random graphs (graph-theoretic aspects)
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