Fast uniform generation of random graphs with given degree sequences

Abstract

In this paper we provide an algorithm that generates a graph with given degree sequence uniformly at random. Provided that Δ4=O(m), where Δ is the maximal degree and m is the number of edges, the algorithm runs in expected time O(m). Our algorithm significantly improves the previously most efficient uniform sampler, which runs in expected time O(m2Δ2) for the same family of degree sequences. Our method uses a novel ingredient which progressively relaxes restrictions on an object being generated uniformly at random, and we use this to give fast algorithms for uniform sampling of graphs with other degree sequences as well. Using the same method, we also obtain algorithms with expected run time which is (i) linear for power‐law degree sequences in cases where the previous best was O(n4.081), and (ii) O(nd + d4) for d‐regular graphs when d=o(n), where the previous best was O(nd3).

Sampling hypergraphs with given degrees

“…(III) If d max + k = O (M 1/4−τ ) for some positive constant τ then the algorithm from [3] can be applied, as described in (II).…”

“…Indeed, Bayati, Kim and Saberi [4, Theorem 3] used sequential importance sampling to give an algorithm which is close to an FPAUS, except that the runtime is polynomial in 1/ε, while in an FPAUS the dependence should be on log(1/ε). However, this algorithm is valid only when d max = O (M 1/4−τ ) for some τ > 0 and no longer has the advantage of simplicity, and so it is surpassed by the fast, precisely uniform sampling algorithm of Arman, Gao and Wormald [3], described in (II) above. (Other authors, such as Chen et al [8], have used sequential importance sampling to sample bipartite graphs with given degrees, but without rigorous analysis.)…”

Sampling hypergraphs with given degrees

“…(III) If d max +r = O(M 1/4−τ ) for some positive constant τ then the algorithm from [3] can be applied, as described in (II). An alternative is to use the sampling algorithm of Bayati, Kim and Saberi [4, Theorem 1], which has expected runtime…”

“…Indeed, Bayati, Kim and Saberi [4, Theorem 3] used sequential importance sampling to give an algorithm which is close to an FPAUS, except that the runtime is polymial in 1/ε, while in an FPAUS the dependence should be on log(1/ε). However, this algorithm is valid only when d max = O(M 1/4−τ ) for some τ > 0 and no longer has the advantage of simplicity, and so it is surpassed by the fast, precisely uniform sampling algorithm of Arman, Gao and Wormald [3], described in (II) above. (Other authors, such as Chen et al [8], have used sequential importance sampling to sample bipartite graphs with given degrees, but without rigorous analysis.…”

“…, d) is regular, the expected number of trials before a simple configuration is sampled is exp Gao and Wormald [17] built on earlier work of McKay and Wormald [24] to give a fast algorithm for exactly uniform sampling of d-regular graphs. Using a recent improvement of Arman, Gao and Wormald [3], a uniformly random d-regular graph on n vertices can be generated in expected runtime O(dn + d 4 ) whenever d = o(n 1/2 ), and a random graph with degree sequence d can be generated in runtime O(M) whenever d 4 max = O(M). It is likely that this approach could be adapted to the problem of sampling hypergraphs uniformly.…”