Abstract
Markov chains are convenient means of generating realizations of networks with a given (joint or otherwise) degree distribution, since they simply require a procedure for rewiring edges. The major challenge is to find the right number of steps to run such a chain, so that we generate truly independent samples. Theoretical bounds for mixing times of these Markov chains are too large to be practically useful. Practitioners have no useful guide for choosing the length, and tend to pick numbers fairly arbitrarily. We give a principled mathematical argument showing that it suffices for the length to be proportional to the number of desired number of edges. We also prescribe a method for choosing this proportionality constant. We run a series of experiments showing that the distributions of common graph properties converge in this time, providing empirical evidence for our claims.
This work was funded by the Applied Mathematics Program at the U.S. Department of Energy and performed at Sandia National Laboratories, a multiprogram laboratory operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the United States Department of Energy’s National Nuclear Security Administration under contract DE-AC04-94AL85000.
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Ray, J., Pinar, A., Seshadhri, C. (2012). Are We There Yet? When to Stop a Markov Chain while Generating Random Graphs. In: Bonato, A., Janssen, J. (eds) Algorithms and Models for the Web Graph. WAW 2012. Lecture Notes in Computer Science, vol 7323. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-30541-2_12
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DOI: https://doi.org/10.1007/978-3-642-30541-2_12
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