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A joint degree matrix (JDM) specifies the number of connections between nodesof given degrees in a graph, for all degree pairs and uniquely determines thedegree sequence of the graph. We consider the space of all balancedrealizations of an arbitrary JDM, realizations in which the links between anytwo degree groups are placed as uniformly as possible. We prove that a swapMarkov Chain Monte Carlo (MCMC) algorithm in the space of all balancedrealizations of an {\em arbitrary} graphical JDM mixes rapidly, i.e., therelaxation time of the chain is bounded from above by a polynomial in thenumber of nodes $n$. To prove fast mixing, we first prove a generalfactorization theorem similar to the Martin-Randall method for disjointdecompositions (partitions). This theorem can be used to bound from below thespectral gap with the help of fast mixing subchains within every partition anda bound on an auxiliary Markov chain between the partitions. Our proof of thegeneral factorization theorem is direct and uses conductance based methods(Cheeger inequality).

A decomposition based proof for fast mixing of a Markov chain over balanced realizations of a joint degree matrix