The cut metric, random graphs, and branching processes

Details The cut metric, random graphs, and branching processes The cut metric, random graphs, and branching processes

2009-01-14

arxiv.org/abs/0901.2091v3

J. Statistical Physics 140 (2010), 289--335

Mathematics

Probability

53 pages; minor edits and references updated

Scientific paper

10.1007/s10955-010-9982-z

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 we introduced recently, as well as related results of Bollob\'as, Borgs, Chayes and Riordan, 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.

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