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_30: 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.
@cite_36: In network modeling of complex systems one is often required to sample random realizations of networks that obey a given set of constraints, usually in form of graph measures. A much studied class of problems targets uniform sampling of simple graphs with given degree sequence or also with given degree correlations expressed in the form of a joint degree matrix. One approach is to use Markov chains based on edge switches (swaps) that preserve the constraints, are irreducible (ergodic) and fast mixing. In 1999, Kannan, Tet ali and Vempala (KTV) proposed a simple swap Markov chain for sampling graphs with given degree sequence and conjectured that it mixes rapidly (in poly-time) for arbitrary degree sequences. While the conjecture is still open, it was proven for special degree sequences, in particular, for those of undirected and directed regular simple graphs, of half-regular bipartite graphs, and of graphs with certain bounded maximum degrees. Here we prove the fast mixing KTV conjecture for novel, exponentially large classes of irregular degree sequences. Our method is based on a canonical decomposition of degree sequences into split graph degree sequences, a structural theorem for the space of graph realizations and on a factorization theorem for Markov chains. After introducing bipartite splitted degree sequences, we also generalize the canonical split graph decomposition for bipartite and directed graphs.
@cite_6: Abstract A binary operation ∘ is introduced on the set of graphs and is transferred to graphical sequences. The decomposition theorem stating that any graph and any graphical sequence can be uniquely decomposed into indecomposable components with respect to the operation ∘ is proved. An exhaustive description of the structure of unigraphs based on this theorem is given.
@cite_39: A joint degree matrix (JDM) specifies the number of connections between nodes of given degrees in a graph, for all degree pairs, and uniquely determines the degree sequence of the graph. We consider the space of all balanced realizations of an arbitrary JDM, realizations in which the links between any two fixed-degree groups of nodes are placed as uniformly as possible. We prove that a swap Markov chain Monte Carlo algorithm in the space of all balanced realizations of an arbitrary graphical JDM mixes rapidly, i.e., the relaxation time of the chain is bounded from above by a polynomial in the number of nodes @math . To prove fast mixing, we first prove a general factorization theorem similar to the Martin--Randall method for disjoint decompositions (partitions). This theorem can be used to bound from below the spectral gap with the help of fast mixing subchains within every partition and a bound on an auxiliary Markov chain between the partitions. Our proof of the general factorization theorem is direct and ...
@cite_19: In this paper we consider a simple Markov chain for bipartite graphs with given degree sequence on @math vertices. We show that the mixing time of this Markov chain is bounded above by a polynomial in @math in case of half-regular degree sequence. The novelty of our approach lies in the construction of the multicommodity flow in Sinclair's method.
@cite_40: This paper has two parts. In the first part we consider a simple Markov chain for d-regular graphs on n vertices, where d = d(n) may grow with n. We show that the mixing time of this Markov chain is bounded above by a polynomial in n and d. In the second part of the paper, a related Markov chain for d-regular graphs on a varying number of vertices is introduced, for even constant d. This is a model for a certain peer-to-peer network. We prove that the related chain has mixing time which is bounded above by a polynomial in N, the expected number of vertices, provided certain assumptions are met about the rate of arrival and departure of vertices.
@cite_27: 
We conclude this section with two recent papers. Erd o s, Mikl ' o s and Toroczkai @cite_36 showed how to build on the results of @cite_40 @cite_30 @cite_19 using several ingredients including a Markov chain factorisation theorem by the same authors @cite_39 and a certain canonical decomposition of degree sequences due to Tyshkevich @cite_27 @cite_6 . Their approach works by taking degree sequences for which rapid mixing of the switch chain is known, and combining them in order to construct new degree sequences for which they prove that the switch chain is also rapidly mixing. Erd o also considered the directed setting, where Theorem now provides a wider range of directed degree sequences for which rapid mixing is known, extending the foundation of the method used in @cite_36 . This should further enlarge the set of more directed degree sequences for which the directed switch chain can be shown to be rapidly mixing.