Abstract
One of the simplest methods of generating a random graph with a given degree sequence is provided by the Monte Carlo Markov Chain method using switches. The switch Markov chain converges to the uniform distribution, but generally the rate of convergence is not known. After a number of results concerning various degree sequences, rapid mixing was established for so-called P-stable degree sequences (including that of directed graphs), which covers every previously known rapidly mixing region of degree sequences.
 In this paper we give a non-trivial family of degree sequences that are not P-stable and the switch Markov chain is still rapidly mixing on them. This family has an intimate connection to Tyshkevich-decompositions and strong stability as well.
Highlights
An important problem in network science is to sample graphs with a given degree sequence uniformly
In this paper we study a Markov Chain Monte Carlo (MCMC) approach to this problem
In this paper we study the so-called switch Markov chain
Summary
An important problem in network science is to sample graphs with a given degree sequence (almost) uniformly. Is the switch Markov chain rapidly mixing on the realizations of all graphic degree sequences?. There is a long line of results where the rapid mixing of the switch Markov chain is proven for certain degree sequences, see [2, 19, 13, 6, 7, 12] The switch Markov chain is rapidly mixing on sets of unconstrained, bipartite, and directed degree sequences that are P -stable (see Definition 8.3). Before presenting our main results, let us get familiar with two interesting properties of h0(n)
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