Critical random hypergraphs: The emergence of a giant set of identifiable vertices

Details Critical random hypergraphs: The emergence of a giant set of identifiable vertices Critical random hypergraphs: The emergence of a giant set of identifiable vertices

2004-01-16

arxiv.org/abs/math/0401208v3

Annals of Probability 2005, Vol. 33, No. 4, 1573-1600

Mathematics

Probability

Published at http://dx.doi.org/10.1214/009117904000000847 in the Annals of Probability (http://www.imstat.org/aop/) by the Ins

Scientific paper

10.1214/009117904000000847

We consider a model for random hypergraphs with identifiability, an analogue of connectedness. This model has a phase transition in the proportion of identifiable vertices when the underlying random graph becomes critical. The phase transition takes various forms, depending on the values of the parameters controlling the different types of hyperedges. It may be continuous as in a random graph. (In fact, when there are no higher-order edges, it is exactly the emergence of the giant component.) In this case, there is a sequence of possible sizes of ``components'' (including but not restricted to N^{2/3}). Alternatively, the phase transition may be discontinuous. We are particularly interested in the nature of the discontinuous phase transition and are able to exhibit precise asymptotics. Our method extends a result of Aldous [Ann. Probab. 25 (1997) 812-854] on component sizes in a random graph.

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