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The Tail does not Determine the Size of the Giant
Stockholm University, Faculty of Science, Department of Mathematics.
Stockholm University, Faculty of Science, Department of Mathematics.
Stockholm University, Faculty of Science, Department of Mathematics.
Number of Authors: 32018 (English)In: Journal of statistical physics, ISSN 0022-4715, E-ISSN 1572-9613, Vol. 173, no 3-4, p. 736-745Article in journal (Refereed) Published
Abstract [en]

The size of the giant component in the configuration model, measured by the asymptotic fraction of vertices in the component, is given by a well-known expression involving the generating function of the degree distribution. In this note, we argue that the distribution over small degrees is more important for the size of the giant component than the precise distribution over very large degrees. In particular, the tail behavior of the degree distribution does not play the same crucial role for the size of the giant as it does for many other properties of the graph. Upper and lower bounds for the component size are derived for an arbitrary given distribution over small degrees d <= L and given expected degree, and numerical implementations show that these bounds are close already for small values of L. On the other hand, examples illustrate that, for a fixed degree tail, the component size can vary substantially depending on the distribution over small degrees.

Place, publisher, year, edition, pages
2018. Vol. 173, no 3-4, p. 736-745
Keywords [en]
Configuration model, Component size, Degree distribution
National Category
Mathematics
Identifiers
URN: urn:nbn:se:su:diva-162885DOI: 10.1007/s10955-018-2071-4ISI: 000450490500010OAI: oai:DiVA.org:su-162885DiVA, id: diva2:1273604
Available from: 2018-12-21 Created: 2018-12-21 Last updated: 2018-12-21Bibliographically approved

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Deijfen, MariaRosengren, SebastianTrapman, Pieter
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