2009/12/03 by Annabell Berger, Matthias Müller‐Hannemann, Berger, Annabell +1
Mathematics · Medicine · #Alzheimer's disease research and treatments #Data Structures and Algorithms (cs.DS) #Discrete Mathematics (cs.DM) #F.2.2 #FOS: Computer and information sciences #G.2.2 #G.2.3 #Graph theory and applications #Markov Chains and Monte Carlo Methods
paper · pdf · doi:10.48550/arxiv.0912.0685
openalex publication_date 2009/12/03 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
Many applications in network analysis require algorithms to sample uniformly at random from the set of all graphs with a prescribed degree sequence. We present a Markov chain based approach which converges to the uniform distribution of all realizations for both the directed and undirected case. It remains an open challenge whether these Markov chains are rapidly mixing. For the case of directed graphs, 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.