Uniform sampling of digraphs with a fixed degree sequence

Abstract

Many applications in network analysis require algorithms to sample uniformly at random from the set of all digraphs with a prescribed degree sequence. We present a Markov chain based approach which converges to the uniform distribution of all realizations. It remains an open challenge whether the Markov chain is rapidly mixing. We also explain in this paper that a popular switching algorithm fails in general to sample uniformly at random because the state graph of the Markov chain decomposes into different isomorphic components. We call degree sequences for which the state graph is strongly connected arc swap sequences. To handle arbitrary degree sequences, we develop two different solutions. The first uses an additional operation (a reorientation of induced directed 3-cycles) which makes the state graph strongly connected, the second selects randomly one of the isomorphic components and samples inside it. Our main contribution is a precise characterization of arc swap sequences, leading to an efficient recognition algorithm. Finally, we point out some interesting consequences for network analysis. © 2010 Springer-Verlag.