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Short cycle distribution in random regular graphs recursively generated by pegging

We introduce a new method of generating random d‐regular graphs by repeatedly applying an operation called pegging. The pegging operation is abstracted from a type of basic operation applied in a type of peer‐to‐peer network called the SWAN network. We prove that for the resulting graphs, the limiting joint distribution of the numbers of short cycles is independent Poisson. We also use coupling to bound the rate at which the distribution approaches its limit. The coupling involves two different, though quite similar, Markov chains that are not time‐homogeneous. © 2008 Wiley Periodicals, Inc. Random Struct. Alg., 2009