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A random graph model for power law graphs. (English) Zbl 0971.05100
This paper introduces a random graph model of the world wide web. Fix \(\alpha\) and \(\beta\), and a distribution \(P(\alpha, \beta)\) assigns a uniform probability to a class of multigraphs such that, for any such multigraph, the number of vertices of a degree \(d\) is \(\lfloor e^{\alpha}/d^{\beta} \rfloor\). This model approximates the observed structure of the world wide web in some aspects. In studying the asymptotics of this model, \(\beta > 0\) is fixed, while \(\alpha\) is permitted to grow (and as \(\alpha\) increases, so does the expected order and size of the multidigraphs). The results in this paper concern the sizes of the connected components of random graphs in the spaces \(P(\alpha, \beta)\), for various \(\beta\), as \(\alpha \to \infty\).
(1) If \(\beta > 3.47875 \ldots\), then a.s. there is no giant component. They present estimates of the sizes of the small components. (2) If \(\beta < 3.47875 \ldots\), then a.s. the graph has a giant component. (3) If \(2 < \beta < 3.47875 \ldots\), the second largest component is a.s. log-sized; if \(\beta = 2\), the second largest component is a.s. log/loglog-sized; if \(2 > \beta > 1\), the second component is a.s. size \(\Theta(1)\). (4) If \(\beta < 1\), then the graph is a.s. connected.
The proofs rely on martingale arguments. While this model and its properties are interesting, expository problems occasionally make the meaning of passages in the proofs difficult to ascertain.
Reviewer: G.L.McColm (Tampa)

05C80 Random graphs (graph-theoretic aspects)
05C07 Vertex degrees
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