Home

# New Classes of Degree Sequences with Fast Mixing Swap Markov Chain Sampling

Abstract

In network modelling of complex systems one is often required to sample random realizations of networks that obey a given set of constraints, usually in the form of graph measures. A much studied class of problems targets uniform sampling of simple graphs with given degree sequence or also with given degree correlations expressed in the form of a Joint Degree Matrix. One approach is to use Markov chains based on edge switches (swaps) that preserve the constraints, are irreducible (ergodic) and fast mixing. In 1999, Kannan, Tetali and Vempala (KTV) proposed a simple swap Markov chain for sampling graphs with given degree sequence, and conjectured that it mixes rapidly (in polynomial time) for arbitrary degree sequences. Although the conjecture is still open, it has been proved for special degree sequences, in particular for those of undirected and directed regular simple graphs, half-regular bipartite graphs, and graphs with certain bounded maximum degrees. Here we prove the fast mixing KTV conjecture for novel, exponentially large classes of irregular degree sequences. Our method is based on a canonical decomposition of degree sequences into split graph degree sequences, a structural theorem for the space of graph realizations and on a factorization theorem for Markov chains. After introducing bipartite ‘splitted’ degree sequences, we also generalize the canonical split graph decomposition for bipartite and directed graphs.

References
Hide All
[1] Barrus, M. D. (2016) On realization graphs of degree sequences. Discrete Math. 339 21462152.
[2] Barrus, M. D. and Donovan, E. (2015) Neighborhood degree lists of graphs. arXiv:1507.08212v1
[3] Barrus, M. D. and West, D. B. (2012) The A 4-structure of a graph. J. Graph Theory 71 159175.
[4] Bassler, K. E., Del Genio, C. I., Erdős, P. L., Miklós, I. and Toroczkai, Z. (2015) Exact sampling of graphs with prescribed degree correlations. New J. Phys. 17 083052
[5] Bezáková, I. (2008) Sampling binary contingency tables. Comput. Sci. Eng. 10 2631.
[6] Bezáková, I., Bhatnagar, N. and Randall, D. (2011) On the Diaconis–Gangolli Markov chain for sampling contingency tables with cell-bounded entries. J. Combin. Optim. 22 457468.
[7] Bezáková, I., Bhatnagar, N. and Vigoda, E. (2007) Sampling binary contingency tables with a greedy start. Random Struct. Alg. 30 168205.
[8] Blitzstein, J. and Diaconis, P. (2011) A sequential importance sampling algorithm for generating random graphs with prescribed degrees. Internet Math. 6 489522.
[9] Brualdi, R. A. and Ryser, H. J. (1992) Combinatorial Matrix Theory, Cambridge University Press.
[10] Cheeger, J. (1970) A lower bound for the smallest eigenvalue of the Laplacian. In Problems in Analysis (Gunning, R. C., ed.), Princeton University Press, pp. 195199.
[11] Chen, Y., Diaconis, P., Holmes, S. P. and Liu, J. S. (2005) Sequential Monte Carlo methods for statistical analysis of tables. J. Amer. Statist. Assoc. 100 (469) 109120.
[12] Chvátal, V. and Hammer, P. L. (1977) Aggregation of inequalities in integer programming. Ann. Discrete Math. 1 145162.
[13] Cooper, C., Dyer, M. and Greenhill, C. (2007) Sampling regular graphs and a peer-to-peer network. Comput. Probab. Comput. 16 557593.
[14] Cooper, C., Dyer, M. and Greenhill, C. (2012) Corrigendum: Sampling regular graphs and a peer-to-peer network. arXiv:1203.6111v1
[15] Cryan, M., Dyer, M., Goldberg, L. A., Jerrum, M. and Martin, R. A. (2006) Rapidly mixing Markov chains for sampling contingency tables with a constant number of rows. SIAM J. Comput. 36 247278.
[16] Cryan, M., Dyer, M. E. and Randall, D. (2010) Approximately counting integral flows and cell-bounded contingency tables. SIAM J. Comput. 39 26832703.
[17] Czabarka, É., Dutle, A., Erdős, P. L. and Miklós, I. (2015) On realizations of a joint degree matrix. Discrete Appl. Math. 181 283288.
[18] Del Genio, C. I., Kim, H., Toroczkai, Z. and Bassler, K. E. (2010) Efficient and exact sampling of simple graphs with given arbitrary degree sequence. PLOS ONE 5 e10012.
[19] Diaconis, P. and Gangolli, A. (1995) Rectangular arrays with fixed margins. In Discrete Probability and Algorithms (Aldous, D. et al., eds), Springer, pp. 1541.
[20] Diaconis, P. and Saloff-Coste, L. (1993) Comparison theorems for reversible Markov chains. Ann. Appl. Probab. 3 696730.
[21] Erdős, P. L., Király, Z. and Miklós, I. (2013) On graphical degree sequences and realizations. Combin. Probab. Comput. 22 366383.
[22] Erdős, P. L., Kiss, Z. S., Miklós, I. and Soukup, L. (2015) Approximate counting of graphical realizations. PLOS ONE 20 e0131300.
[23] Erdős, P. L., Miklós, I. and Toroczkai, Z. (2010) A simple Havel–Hakimi type algorithm to realize graphical degree sequences of directed graphs. Electron. J. Combin. 17 R66.
[24] Erdős, P. L., Miklós, I. and Toroczkai, Z. (2015) A decomposition based proof for fast mixing of a Markov chain over balanced realizations of a joint degree matrix. SIAM J. Discrete Math. 29 481499.
[25] Feder, T., Guetz, A., Mihail, M. and Saberi, A. (2006) A local switch Markov chain on given degree graphs with application in connectivity of peer-to-peer networks. In FOCS '06: 47th Annual IEEE Symposium on Foundations of Computer Science, pp. 69–76.
[26] Földes, S. and Hammer, P. L. Split graphs. In Proc. Eighth Southeastern Conference on Combinatorics, Graph Theory and Computing, Vol. XIX of Congressus Numerantium, Utilitas Mathematica, pp. 311–315.
[27] Greenhill, C. (2011) A polynomial bound on the mixing time of a Markov chain for sampling regular directed graphs. Electron. J. Combin. 16 557593.
[28] Greenhill, C. (2015) The switch Markov chain for sampling irregular graphs. In Proc. 26th ACM–SIAM Symposium on Discrete Algorithms, pp. 1564–1572.
[29] Gross, E., Petrović, S. and Stasi, D. (2017) Goodness-of-fit for log-linear network models: Dynamic Markov bases using hypergraphs. Ann. Inst. Statist. Math. 69 673704.
[30] Hammer, P. L., Peled, U. N. and Sun, X. (1990) Difference graphs. Discrete Appl. Math. 28 3544.
[31] Hammer, P. L. and Simeone, B. (1981) The splittance of a graph. Combinatorica 1 275284.
[32] Kannan, R., Tetali, P. and Vempala, S. (1999) Simple Markov-chain algorithms for generating bipartite graphs and tournaments. Random Struct. Alg. 14 293308.
[33] Kim, H., Del Genio, C. I., Bassler, K. E. and Toroczkai, Z. (2012) Constructing and sampling directed graphs with given degree sequences. New J. Phys. 14 023012.
[34] Kim, H., Toroczkai, Z., Erdős, P. L., Miklós, I. and Székely, L. A. (2009) Degree-based graph construction. J. Phys. A: Math. Theor. 42 392001.
[35] Kleitman, D. J. and Wang, D. L. (1973) Algorithms for constructing graphs and digraphs with given valences and factors. Discrete Math. 6 7988.
[36] LaMar, M. D. (2012) Splits digraphs. Discrete Math. 312 13141325.
[37] Levin, D. A., Peres, Y. and Wilmer, E. L. (2008) Markov Chains and Mixing Times, AMS.
[38] Madras, R. and Randall, D. (2002) Markov chain decomposition for convergence rate analysis. Ann. Appl. Probab. 12 581606.
[39] Martin, R. and Randall, D. (2006) Disjoint decomposition of Markov chains and sampling circuits in Cayley graphs. Combin. Probab. Comput. 15 411448.
[40] Miklós, I., Erdős, P. L. and Soukup, L. (2013) Towards random uniform sampling of bipartite graphs with given degree sequence. Electron. J. Combin. 20 P16.
[41] Online Encyclopedia of Integer Sequences . https://oeis.org/A029894
[42] Petrović, S. (2017) A survey of discrete methods in (algebraic) statistics for networks. In Algebraic and Geometric Methods in Discrete Mathematics (Harrington, H., Omar, M. and Wright, M., eds), Vol. 685 of Contemporary Mathematics, AMS, pp. 260281.
[43] Randall, D. (2006) Rapidly mixing Markov chains with applications in computer science and physics. Comput. Sci. Eng. 8 3041.
[44] Rao, A. R., Jana, R. and Bandyopadhyay, S. (1996) A Markov chain Monte Carlo method for generating random (0,1)-matrices with given marginals. Sankhy=ā: Ind. J. Stat. 58 225370.
[45] Ryser, H. J. (1957) Combinatorial properties of matrices of zeros and ones. Canad. J. Math. 9 371377.
[46] Sinclair, A. (1992) Improved bounds for mixing rates of Markov chains and multicommodity flow. Combin. Probab. Comput. 1 351370.
[47] Slavković, A., Zhu, X. and Petrović, S. (2015) Fibers of multi-way contingency tables given conditionals: Relation to marginals, cell bounds and Markov bases. Ann. Inst. Stat. Math. 67 621648.
[48] Taylor, R. (1981) Constrained switching in graphs. In Combinatorial Mathematics VIII: Proc. Eighth Australian Conference on Combinatorial Mathematics, Vol. 884 of Lecture Notes in Mathematics, Springer, pp. 314336.
[49] Tyshkevich, R. (1980) The canonical decomposition of a graph (in Russian). Doklady Akademii Nauk SSSR 24 677679.
[50] Tyshkevich, R. (2000) Decomposition of graphical sequences and unigraphs. Discrete Math. 220 201238.
[51] Tyshkevich, R., Melnikov, O. and Kotov, V. (1981) On graphs and degree sequences (in Russian). Kibernetika 6 58.
Recommend this journal

Combinatorics, Probability and Computing
• ISSN: 0963-5483
• EISSN: 1469-2163
• URL: /core/journals/combinatorics-probability-and-computing
Who would you like to send this to? *

×

## Full text viewsFull text views reflects the number of PDF downloads, PDFs sent to Google Drive, Dropbox and Kindle and HTML full text views.

Total number of HTML views: 0
Total number of PDF views: 55 *