The problem of efficiently sampling from a set of (undirected, or directed) graphs with a given degree sequence has many applications. One approach to this problem uses a simple Markov chain, which we call the switch chain, to perform the sampling. The switch chain is known to be rapidly mixing for regular degree sequences, both in the undirected and directed setting. We prove that the switch chain for undirected graphs is rapidly mixing for any degree sequence with minimum degree at least 1 and with maximum degree @math which satisfies @math , where @math is the sum of the degrees. The mixing time bound obtained is only a factor @math larger than that established in the regular case, where @math is the number of vertices. Our result covers a wide range of degree sequences, including power-law graphs with parameter @math and sufficiently many edges. For directed degree sequences such that the switch chain is irreducible, we prove that the switch chain is rapidly mixing when all in-degrees and out-degrees are positive and bounded above by @math , where @math is the number of arcs, and not all in-degrees and out-degrees equal 1. The mixing time bound obtained in the directed case is an order of @math larger than that established in the regular case.
@cite_24: We develop a new approach for uniform generation of combinatorial objects, and apply it to derive a uniform sampler REG for @math -regular graphs. REG can be implemented such that each graph is generated in expected time @math , provided that @math . Our result significantly improves the previously best uniform sampler, which works efficiently only when @math , with essentially the same running time for the same @math . We also give a linear-time approximate sampler REG*, which generates a random @math -regular graph whose distribution differs from the uniform by @math in total variation distance, when @math .
Gao and Wormald @cite_24 have recently described an extremely efficient expected polynomial time algorithm for exactly uniform sampling @math -regular undirected graphs, where @math . The expected running time of their algorithm is @math . They also describe a variant of their algorithm with expected running time @math such that the total variation distance of the output distribution from uniform is @math , again when @math .