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Component structure of the configuration model: Barely supercritical case
Eindhoven Univ Technol, Dept Math & Comp Sci, Eindhoven, Netherlands.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
Univ Melbourne, Sch Math & Stat, Parkville, Vic, Australia.
2019 (English)In: Random structures & algorithms (Print), ISSN 1042-9832, E-ISSN 1098-2418, Vol. 55, no 1, p. 3-55Article in journal (Refereed) Published
Abstract [en]

We study near-critical behavior in the configuration model. Let D-n be the degree of a random vertex and nu n=E[Dn(Dn-1)]/E[Dn]; we consider the barely supercritical regime, where nu(n)-> 1 as n ->infinity, but nu n-1 n-1/3(E[Dn3])2/3. Let Dn* denote the size-biased version of D-n. We prove that there is a unique giant component of size n rho nEDn(1+o(1)), where rho(n) denotes the survival probability of a branching process with offspring distribution Dn*-1. This extends earlier results of Janson and Luczak, as well as those of Janson, Luczak, Windridge, and House, to the case where the third moment of D-n is unbounded. We further study the size of the largest component in the critical regime, where nu n-1=O(n-1/3(EDn3)2/3), extending and complementing results of Hatami and Molloy.

Place, publisher, year, edition, pages
WILEY , 2019. Vol. 55, no 1, p. 3-55
Keywords [en]
percolation, phase transition, random graphs, scaling window
National Category
Computer Sciences
Identifiers
URN: urn:nbn:se:uu:diva-390078DOI: 10.1002/rsa.20837ISI: 000471321600001OAI: oai:DiVA.org:uu-390078DiVA, id: diva2:1340874
Funder
Knut and Alice Wallenberg FoundationAvailable from: 2019-08-06 Created: 2019-08-06 Last updated: 2019-08-06Bibliographically approved

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