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

Part of: Discrete mathematics in relation to computer science Graph theory

Published online by Cambridge University Press:  02 November 2017

PÉTER L. ERDŐS ,
ISTVÁN MIKLÓS  and
ZOLTÁN TOROCZKAI
PÉTER L. ERDŐS
Affiliation:
MTA A. Rényi Institute of Mathematics, Reáltanoda utca 13-15 Budapest, 1053Hungary (e-mail: erdos.peter@renyi.mta.hu, miklos.istvan@renyi.mta.hu)
ISTVÁN MIKLÓS
Affiliation:
MTA SZTAKI, Lágymányosi út 11, Budapest, 1111 Hungary MTA A. Rényi Institute of Mathematics, Reáltanoda utca 13-15 Budapest, 1053Hungary (e-mail: erdos.peter@renyi.mta.hu, miklos.istvan@renyi.mta.hu)
ZOLTÁN TOROCZKAI
Affiliation:
Department of Physics and Interdisciplinary Center for Network Science and Applications, University of Notre Dame, Notre Dame, IN, 46556, USA (e-mail: toro@nd.edu)
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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.

MSC classification

Primary: 05C07: Vertex degrees
Secondary: 05C30: Enumeration in graph theory 05C81: Random walks on graphs 68R10: Graph theory (including graph drawing)
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 27 , Issue 2 , March 2018 , pp. 186 - 207
Copyright
Copyright © Cambridge University Press 2017 

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References

[1] Barrus, M. D. (2016) On realization graphs of degree sequences. Discrete Math. 339 2146–2152.Google Scholar
[2] Barrus, M. D. and Donovan, E. (2015) Neighborhood degree lists of graphs. arXiv:1507.08212v1 Google Scholar
[3] Barrus, M. D. and West, D. B. (2012) The A 4-structure of a graph. J. Graph Theory 71 159–175.CrossRefGoogle Scholar
[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 CrossRefGoogle Scholar
[5] Bezáková, I. (2008) Sampling binary contingency tables. Comput. Sci. Eng. 10 26–31.Google Scholar
[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. 22 457–468.Google Scholar
[7] Bezáková, I., Bhatnagar, N. and Vigoda, E. (2007) Sampling binary contingency tables with a greedy start. Random Struct. Alg. 30 168–205.Google Scholar
[8] Blitzstein, J. and Diaconis, P. (2011) A sequential importance sampling algorithm for generating random graphs with prescribed degrees. Internet Math. 6 489–522.Google Scholar
[9] Brualdi, R. A. and Ryser, H. J. (1992) Combinatorial Matrix Theory, Cambridge University Press.Google Scholar
[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. 195–199.Google Scholar
[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) 109–120.Google Scholar
[12] Chvátal, V. and Hammer, P. L. (1977) Aggregation of inequalities in integer programming. Ann. Discrete Math. 1 145–162.Google Scholar
[13] Cooper, C., Dyer, M. and Greenhill, C. (2007) Sampling regular graphs and a peer-to-peer network. Comput. Probab. Comput. 16 557–593.Google Scholar
[14] Cooper, C., Dyer, M. and Greenhill, C. (2012) Corrigendum: Sampling regular graphs and a peer-to-peer network. arXiv:1203.6111v1 Google Scholar
[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. 36 247–278.Google Scholar
[16] Cryan, M., Dyer, M. E. and Randall, D. (2010) Approximately counting integral flows and cell-bounded contingency tables. SIAM J. Comput. 39 2683–2703.CrossRefGoogle Scholar
[17] Czabarka, É., Dutle, A., Erdős, P. L. and Miklós, I. (2015) On realizations of a joint degree matrix. Discrete Appl. Math. 181 283–288.CrossRefGoogle Scholar
[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.Google Scholar
[19] Diaconis, P. and Gangolli, A. (1995) Rectangular arrays with fixed margins. In Discrete Probability and Algorithms (Aldous, D. et al., eds), Springer, pp. 15–41.Google Scholar
[20] Diaconis, P. and Saloff-Coste, L. (1993) Comparison theorems for reversible Markov chains. Ann. Appl. Probab. 3 696–730.Google Scholar
[21] Erdős, P. L., Király, Z. and Miklós, I. (2013) On graphical degree sequences and realizations. Combin. Probab. Comput. 22 366–383.Google Scholar
[22] Erdős, P. L., Kiss, Z. S., Miklós, I. and Soukup, L. (2015) Approximate counting of graphical realizations. PLOS ONE 20 e0131300.Google Scholar
[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.Google Scholar
[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. 29 481–499.CrossRefGoogle Scholar
[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.Google Scholar
[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.Google Scholar
[27] Greenhill, C. (2011) A polynomial bound on the mixing time of a Markov chain for sampling regular directed graphs. Electron. J. Combin. 16 557–593.Google Scholar
[28] Greenhill, C. (2015) The switch Markov chain for sampling irregular graphs. In Proc. 26th ACM–SIAM Symposium on Discrete Algorithms, pp. 1564–1572.Google Scholar
[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. 69 673–704.Google Scholar
[30] Hammer, P. L., Peled, U. N. and Sun, X. (1990) Difference graphs. Discrete Appl. Math. 28 35–44.Google Scholar
[31] Hammer, P. L. and Simeone, B. (1981) The splittance of a graph. Combinatorica 1 275–284.Google Scholar
[32] Kannan, R., Tetali, P. and Vempala, S. (1999) Simple Markov-chain algorithms for generating bipartite graphs and tournaments. Random Struct. Alg. 14 293–308.Google Scholar
[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.Google Scholar
[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.Google Scholar
[35] Kleitman, D. J. and Wang, D. L. (1973) Algorithms for constructing graphs and digraphs with given valences and factors. Discrete Math. 6 79–88.Google Scholar
[36] LaMar, M. D. (2012) Splits digraphs. Discrete Math. 312 1314–1325.Google Scholar
[37] Levin, D. A., Peres, Y. and Wilmer, E. L. (2008) Markov Chains and Mixing Times, AMS.Google Scholar
[38] Madras, R. and Randall, D. (2002) Markov chain decomposition for convergence rate analysis. Ann. Appl. Probab. 12 581–606.Google Scholar
[39] Martin, R. and Randall, D. (2006) Disjoint decomposition of Markov chains and sampling circuits in Cayley graphs. Combin. Probab. Comput. 15 411–448.CrossRefGoogle Scholar
[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.Google Scholar
[41] Online Encyclopedia of Integer Sequences . https://oeis.org/A029894 Google Scholar
[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. 260–281.Google Scholar
[43] Randall, D. (2006) Rapidly mixing Markov chains with applications in computer science and physics. Comput. Sci. Eng. 8 30–41.Google Scholar
[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. 58 225–370.Google Scholar
[45] Ryser, H. J. (1957) Combinatorial properties of matrices of zeros and ones. Canad. J. Math. 9 371–377.Google Scholar
[46] Sinclair, A. (1992) Improved bounds for mixing rates of Markov chains and multicommodity flow. Combin. Probab. Comput. 1 351–370.Google Scholar
[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. 67 621–648.Google Scholar
[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. 314–336.Google Scholar
[49] Tyshkevich, R. (1980) The canonical decomposition of a graph (in Russian). Doklady Akademii Nauk SSSR 24 677–679.Google Scholar
[50] Tyshkevich, R. (2000) Decomposition of graphical sequences and unigraphs. Discrete Math. 220 201–238.Google Scholar
[51] Tyshkevich, R., Melnikov, O. and Kotov, V. (1981) On graphs and degree sequences (in Russian). Kibernetika 6 5–8.Google Scholar