New Classes of Degree Sequences with Fast Mixing Swap Markov Chain Sampling

In network modeling of complex systems one is often required to sample random realizations of networks that obey a given set of constraints, often 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, Tetali 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 for the general case, it was proven for special degree sequences, in particular, for those of undirected and directed regular simple graphs (Cooper, Dyer, Greenhill, 2007; Greenhill, 2011), of half-regular bipartite graphs (Miklós, Erdős, Soukup, 2013), and of graphs with certain bounded maximum degrees (Greenhill, 2015). 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 due to Tyshkevich (1984, 2000), a structural theorem for the space of graph realizations by Barrus (2015) and on a factorization theorem for Markov chains (Erdős, Miklós, Toroczkai, 2015). After introducing bipartite splitted degree sequences, we also generalize Tyshkevich’s decomposition for bipartite and directed graphs.