The switch Markov chain for sampling irregular graphs and digraphs

Abstract

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 d max which satisfies 3 ≤ d max ≤ 1 3 √ M , where M is the sum of the degrees. The mixing time bound obtained is only a factor n larger than that established in the regular case, where n is the number of vertices. Our result covers a wide range of degree sequences, including power-law density-bounded graphs with parameter γ > 5/2 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 1 4 √ m, where m 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 m 2 larger than that established in the regular case.Here the running time of the sampling algorithm must be (deterministically) polynomially bounded but the output need not be exactly uniform: however, the user can specify how far from the uniform distribution the samples may be. Other approaches to the problem of sampling graphs (or directed graphs) are discussed in Section 1.1.The switch chain is a natural and well-studied Markov chain for sampling from a set of graphs with a given degree sequence. Each move of the switch chain selects two distinct edges uniformly at random and attempts to replace these edges by a perfect matching of the four endvertices, chosen uniformly at random. The proposed move is rejected if the four endvertices are not distinct or if a multiple edge would be formed. We call each such move a switch. The precise definitions of the transitions for the switch chain for undirected and directed graphs are given at the start of Sections 2 and 3, respectively.Ryser [34] used switches to study the structure of 0-1 matrices. Markov chains based on switches have been introduced by Besag and Clifford [5] for 0-1 matrices (bipartite graphs), Diaconis and Sturmfels [9] for contingency tables and Rao, Jana and Bandyopadhyay [33] for directed graphs.The switch chain is aperiodic and its transition matrix is symmetric. It is well-known that the switch chain is irreducible for any (undirected) degree sequence: see [32,37]. Irreducibility for the directed chain is not guaranteed, see Rao et al. [33]. However, Berger and Müller-Hanneman [4] and LaMar [24,26] gave characterisations of directed degree sequences for which the switch chain is irreducible. In particular, the switch chain is irreducible for regular directed graphs (see for example Greenhill [15, Lemma 2.2]).In order for the switch chain to b...

Rapid Mixing of the Switch Markov Chain for Strongly Stable Degree Sequences and 2-Class Joint Degree Matrices

“…Thus, for the flow transformation we use embedding arguments similar to those in Feder et al [20]. 4 Using the aforementioned approach we obtain the following two main results: 1) We show rapid mixing of the switch chain for strongly stable families of degree sequences (Theorem 4), thus providing a rather short proof that unifies and extends the results in [7,24] (Corollaries 5 and 6).…”

“…The graphs G, G ′ ∈ G(d) are switch adjacent if G can be obtained from G ′ with positive probability in one transition of this chain and vice versa. It is well-known that the switch chain is irreducible, aperiodic and symmetric (e.g., [25] and references therein), and, thus, has uniform stationary distribution over G(d). Furthermore, is it a matter of simple counting that P(G, G ′ ) −1 ≤ 6n 4 for all switch adjacent G, G ′ ∈ G(d), and the maximum in-and out-degrees of the state space graph of the switch chain are bounded by n 4 .…”

“…Several variants of the problem have been studied, including sampling undirected, bipartite, connected, or directed graphs. In particular, there is an extensive line of work on Markov Chain Monte Carlo (MCMC) approaches, see, e.g., [29,30,20,7,25,34,19,18,9,6]. In such an approach one studies a random walk on the space of all graphical realizations.…”

“…Our Contribution. The proofs of the results in [34,24,19,18] for the analysis of the switch Markov chain in undirected (or bipartite) graphs are all using conceptually similar ideas to the ones introduced by Cooper, Dyer and Greenhill [7] for the analysis of the switch chain for regular undirected graphs, and are based on the multicommodity flow method of Sinclair [39]. The individual parts of this template can become quite technical and require long proofs.…”

IndeCut evaluates performance of network motif discovery algorithms

“…Through experiments on smal"test graphs" where samples spaces can be empirically enumerated by producing all possible graphs in the space, it has been shown that the same sampling strategy can have very different performance outcomes in terms of uniform and independent sampling [3] depending on graph topology. For example, while d-regular graphs rarely pose a problem, small graphs with highly irregular or "uneven" degree sequences frequently cause difficulty [3,24,25]. This creates a severe concern for the accurate performance of network motif discovery algorithms on real biological networks, which often contain large source hubs ("master regulators") and/or target hubs (heavily regulated nodes) [26,27].…”