The problem of efficiently sampling from a set of(undirected) 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. We prove that the switch chain 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 an order of @math larger than that established in the regular case, where @math is the number of vertices.
There are several approaches to the problem of sampling graphs with a given degree sequence, though none is known to be efficient for all degree sequences. The configuration model of Bollob ' a s @cite_8 gives expected polynomial time uniform sampling if @math . McKay and Wormald @cite_7 adapted the configuration model to give an algorithm which performs uniform sampling from @math in expected polynomial time when @math .
@cite_7 @cite_8
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We present a nearly-linear time algorithm for counting and randomly generating simple graphs with a given degree sequence in a certain range. For degree sequence (d i )i=1n with maximum degree d max =O(m 1 4−τ ), our algorithm generates almost uniform random graphs with that degree sequence in time O(md max ) where @math is the number of edges in the graph and τ is any positive constant. The fastest known algorithm for uniform generation of these graphs (McKay and Wormald in J. Algorithms 11(1):52–67, 1990) has a running time of O(m 2 d max 2). Our method also gives an independent proof of McKay’s estimate (McKay in Ars Combinatoria A 19:15–25, 1985) for the number of such graphs. We also use sequential importance sampling to derive fully Polynomial-time Randomized Approximation Schemes (FPRAS) for counting and uniformly generating random graphs for the same range of d max =O(m 1 4−τ ). Moreover, we show that for d=O(n 1 2−τ ), our algorithm can generate an asymptotically uniform d-regular graph. Our results improve the previous bound of d=O(n 1 3−τ ) due to Kim and Vu (Adv. Math. 188:444–469, 2004) for regular graphs. 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. In 1999 Kannan, Tet ali and Vempala proposed a MCMC method to uniformly sample all possible realizations of a given graphical degree sequence and conjectured its rapidly mixing nature. Recently their conjecture was proved affirmative for regular graphs (by Cooper, Dyer and Greenhill, 2007), for regular directed graphs (by Greenhill, 2011) and for half-regular bipartite graphs (by Miklos, Erdős and Soukup, 2013). Several heuristics on counting the number of possible realizations exist (via sampling processes), and while they work well in practice, so far no approximation guarantees exist for such an approach. This paper is the first to develop a method for counting realizations with provable approximation guarantee. In fact, we solve a slightly more general problem; besides the graphical degree sequence a small set of forbidden edges is also given. We show that for the general problem (which contains the Greenhill problem and the Miklos, Erdős and Soukup problem as special cases) the derived MCMC process is rapidly mixing. Further, we show that this new problem is self-reducible therefore it provides a fully polynomial randomized approximation scheme (a.k.a. FPRAS) for counting of all realizations. The problem of efficiently sampling from a set of (undirected) 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. We prove that the switch chain is rapidly mixing for any degree sequence with minimum degree at least 1 and with maximum degree dmax which satisfies 3 ≤ dmax ≤ 1 4 [EQUATION]M, where M is the sum of the degrees. The mixing time bound obtained is only an order of n larger than that established in the regular case, where n is the number of vertices.