Are We There Yet? When to Stop a Markov Chain while Generating Random Graphs
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.
Keywordsgraph generation Markov chain Monte Carlo independent samples
Unable to display preview. Download preview PDF.
- 3.Holme, P., Zhao, J.: Exploring the assortativity-clustering space of a network’s degree sequence. Phys. Rev. E 75, 046111 (2007)Google Scholar
- 7.Gkantsidis, C., Mihail, M., Zegura, E.W.: The Markov chain simulation method for generating connected power law random graphs. In: ALENEX, pp. 16–25 (2003)Google Scholar
- 8.Stanton, I., Pinar, A.: Constructing and sampling graphs with a prescribed joint degree distribution using Markov chains. ACM Journal of Experimental Algorithmics (to appear)Google Scholar
- 11.Adams, S.: Dilbert: Random number generator (2001), http://search.dilbert.com/comic/RandomNumberGenerator
- 12.Sokal, A.: Monte Carlo methods in statistical mechanics: Foundations and new algorithms (1996)Google Scholar
- 13.Ray, J., Pinar, A., Seshadhri, C.: Are we there yet? when to stop a markov chain while generating random graphs. CoRR abs/1202.3473 (2012)Google Scholar
- 14.Raftery, A., Lewis, S.M.: Implementing MCMC. In: Gilks, W.R., Richardson, S., Spiegelhalter, D.J. (eds.) Markov Chain Monte Carlo in Practice, pp. 115–130. Chapman and Hall (1996)Google Scholar
- 15.Raftery, A.E., Lewis, S.M.: How many iterations in the Gibbs sampler? In: Bernardo, J.M., Berger, J.O., Dawid, A.P., Smith, A.F.M. (eds.) Bayesian Statistics, vol. 4, pp. 765–766. Oxford University Press (1992)Google Scholar
- 18.Newman, M.E.J.: Finding community structure in networks using the eigenvectors of matrices. Phys. Rev. E 74, 036104 (2006)Google Scholar
- 20.Newman, M.E.J.: Prof. M. E. J. Newman’s collection of graphs at University of Michigan, http://www-personal.umich.edu/~mejn/netdata/
- 21.Stanford Network Analysis Platform Collection of Graphs: The Epinions social network from the Stanford Network Analysis Platform collection, http://snap.stanford.edu/data/soc-Epinions1.html