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The Cut Metric, Random Graphs, and Branching Processes
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Applied Mathematics.
2010 (English)In: Journal of statistical physics, ISSN 0022-4715, E-ISSN 1572-9613, Vol. 140, no 2, 289-335 p.Article in journal (Refereed) Published
Abstract [en]

In this paper we study the component structure of random graphs with independence between the edges. Under mild assumptions, we determine whether there is a giant component, and find its asymptotic size when it exists. We assume that the sequence of matrices of edge probabilities converges to an appropriate limit object (a kernel), but only in a very weak sense, namely in the cut metric. Our results thus generalize previous results on the phase transition in the already very general inhomogeneous random graph model introduced by the present authors in Random Struct. Algorithms 31:3-122 (2007), as well as related results of Bollobas, Borgs, Chayes and Riordan (Ann. Probab. 38:150-183, 2010), all of which involve considerably stronger assumptions. We also prove corresponding results for random hypergraphs; these generalize our results on the phase transition in inhomogeneous random graphs with clustering (Random Struct. Algorithms, 2010, to appear).

Place, publisher, year, edition, pages
2010. Vol. 140, no 2, 289-335 p.
Keyword [en]
Random graph, Phase transition, Branching process
National Category
Mathematics Physical Sciences
URN: urn:nbn:se:uu:diva-136139DOI: 10.1007/s10955-010-9982-zISI: 000279085600004OAI: oai:DiVA.org:uu-136139DiVA: diva2:376507
Available from: 2010-12-10 Created: 2010-12-10 Last updated: 2011-03-01Bibliographically approved

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