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.
@cite_18: 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.
@cite_22: Sampling from combinatorial families can be difficult. However, complicated families can often be embedded within larger, simpler ones, for which easy sampling algorithms are known. We take advantage of such a relationship to describe a sampling algorithm for the smaller family, via a Markov chain started at a random sample of the larger family. The utility of the method is demonstrated via several examples, with particular emphasis on sampling labelled graphs with given degree sequence, a well-studied problem for which existing algorithms leave much room for improvement. For graphs with given degrees, with maximum degree @math where @math is the number of edges, we obtain an asymptotically uniform sample in @math steps, which substantially improves upon existing algorithms.
@cite_7: Abstract We show how to generate k -regular graphs on n vertices uniformly at random in expected time O ( nk 3 ), provided k = O(n 1 3 ) . The algorithm employs a modification of a switching argument previously used to count such graphs asymptotically for k = o(n 1 3 ) . The asymptotic formula is re-derived, using the new switching argument. The method is applied also to graphs with given degree sequences, provided certain conditions are met. In particular, it applies if the maximum degree is O(∥E(G)∥ 1 4 ) . The method is also applied to bipartite graphs.
@cite_3: Random regular graphs play a central role in combinatorics and theoretical computer science. In this paper, we analyze a simple algorithm introduced by Steger and Wormald [10] and prove that it produces an asymptotically uniform random regular graph in a polynomial time. Precisely, for fixed d and n with d = O(n1 3−e), it is shown that the algorithm generates an asymptotically uniform random d-regular graph on n vertices in time O(nd2). This confirms a conjecture of Wormald. The key ingredient in the proof is a recently developed concentration inequality by the second author. The algorithm works for relatively large d in practical (quadratic) time and can be used to derive many properties of uniform random regular graphs.
@cite_5: We present a practical algorithm for generating random regular graphs. For all d growing as a small power of n, the d-regular graphs on n vertices are generated approximately uniformly at random, in the sense that all d-regular graphs on n vertices have in the limit the same probability as n → ∞. The expected runtime for these ds is O(nd2).
Steger and Wormald @cite_5 gave an easily-implementable algorithm for sampling regular graphs, and proved that their algorithm performs asymptotically uniform sampling in polynomial time when @math (where @math denotes the degree). Kim and Vu @cite_3 gave a sharper analysis and established that @math suffices for efficient asymptotically uniform sampling. Bayati, Kim and Saberi @cite_18 extended Steger and Wormald's algorithm to irregular degree sequences, giving polynomial-time asymptotically uniform sampling when @math . From this they constructed a sequential importance sampling algorithm for @math . Recently, Zhao @cite_22 described and analysed a similar approach to that of @cite_7 , in a general combinatorial setting. Zhao shows that for sampling from @math , when @math , his algorithm performs asymptotically uniform sampling in time @math .