The switch Markov chain for sampling irregular graphs (Extended Abstract)

The problem of efficiently sampling from a set of(undirected) graphs with a given degree sequence has many applications. One approach to this problem uses a simple Markov chain, which we call the switch chain, to perform the sampling. The switch chain is known to be rapidly mixing for regular degree sequences. We prove that the switch chain is rapidly mixing for any degree sequence with minimum degree at least 1 and with maximum degree $d_{\max}$ which satisfies $3\leq d_{\max}\leq \frac{1}{4}\, \sqrt{M}$, where $M$ is the sum of the degrees. The mixing time bound obtained is only an order of $n$ larger than that established in the regular case, where $n$ is the number of vertices.

[1]  Prasad Tetali,et al.  Simple Markov-Chain Algorithms for Generating Bipartite Graphs and Tournaments (Extended Abstract) , 1999, SODA.

[2]  Martin E. Dyer,et al.  Sampling regular graphs and a peer-to-peer network , 2005, SODA '05.

[3]  Brendan D. McKay,et al.  Uniform Generation of Random Regular Graphs of Moderate Degree , 1990, J. Algorithms.

[4]  J. Besag,et al.  Generalized Monte Carlo significance tests , 1989 .

[5]  István Miklós,et al.  Approximate Counting of Graphical Realizations , 2015, PloS one.

[6]  Brendan D. McKay,et al.  Subgraphs of Dense Random Graphs with Specified Degrees , 2010, Combinatorics, Probability and Computing.

[7]  A. Sinclair.  Improved Bounds for Mixing Rates of Markov Chains and Multicommodity Flow , 1992, Combinatorics, Probability and Computing.

[8]  Brendan D. McKay,et al.  Asymptotic enumeration by degree sequence of graphs with degreeso(n1/2) , 1991, Comb..

[9]  Amin Saberi,et al.  A Sequential Algorithm for Generating Random Graphs , 2007, Algorithmica.

[10]  Gregory Gutin,et al.  Digraphs - theory, algorithms and applications , 2002 .

[11]  A. Rao,et al.  A Markov chain Monte carol method for generating random (0, 1)-matrices with given marginals , 1996 .

[12]  Pu Gao,et al.  Uniform Generation of Random Regular Graphs , 2015, 2015 IEEE 56th Annual Symposium on Foundations of Computer Science.

[13]  P. Diaconis,et al.  Algebraic algorithms for sampling from conditional distributions , 1998 .

[14]  Michael Drew Lamar.  On uniform sampling simple directed graph realizations of degree sequences , 2009, ArXiv.

[15]  J. Petersen.  Die Theorie der regulären graphs , 1891 .

[16]  Nicholas Wormald,et al.  Enumeration of graphs with a heavy-tailed degree sequence , 2014, 1404.1250.

[17]  Van H. Vu,et al.  Generating Random Regular Graphs , 2003, STOC '03.

[18]  István Miklós,et al.  Towards Random Uniform Sampling of Bipartite Graphs with given Degree Sequence , 2010, Electron. J. Comb..

[19]  H. Ryser.  Combinatorial Properties of Matrices of Zeros and Ones , 1957, Canadian Journal of Mathematics.

[20]  Prasad Tetali,et al.  Simple Markov-chain algorithms for generating bipartite graphs and tournaments , 1997, SODA '97.

[21]  Michael Drew Lamar,et al.  Directed 3-cycle anchored digraphs and their application in the uniform sampling of realizations from a fixed degree sequence , 2011, Proceedings of the 2011 Winter Simulation Conference (WSC).

[22]  Istv'an Mikl'os,et al.  Constructing, sampling and counting graphical realizations of restricted degree sequences , 2013, 1301.7523.

[23]  Mark Jerrum,et al.  Fast Uniform Generation of Regular Graphs , 1990, Theor. Comput. Sci..

[24]  Zoltán Toroczkai,et al.  A Decomposition Based Proof for Fast Mixing of a Markov Chain over Balanced Realizations of a Joint Degree Matrix , 2015, SIAM J. Discret. Math..

[25]  Matthias Müller-Hannemann,et al.  Uniform Sampling of Digraphs with a Fixed Degree Sequence , 2010, WG.

[26]  James Y. Zhao.  Expand and Contract: Sampling graphs with given degrees and other combinatorial families , 2013, ArXiv.

[27]  Brendan D. McKay,et al.  Random dense bipartite graphs and directed graphs with specified degrees , 2009, Random Struct. Algorithms.

[28]  Béla Bollobás,et al.  A Probabilistic Proof of an Asymptotic Formula for the Number of Labelled Regular Graphs , 1980, Eur. J. Comb..

[29]  Eric Vigoda,et al.  A polynomial-time approximation algorithm for the permanent of a matrix with non-negative entries , 2001, STOC '01.

[30]  Michael Drew Lamar.  Algorithms for realizing degree sequences of directed graphs , 2009, ArXiv.

[31]  Alexander I. Barvinok,et al.  The number of graphs and a random graph with a given degree sequence , 2010, Random Struct. Algorithms.

[32]  Nicholas C. Wormald,et al.  Generating Random Regular Graphs Quickly , 1999, Combinatorics, Probability and Computing.

[33]  F. Jotzo,et al.  Double counting and the Paris Agreement rulebook , 2019, Science.

[34]  M. Jerrum.  Counting, Sampling and Integrating: Algorithms and Complexity , 2003 .

[35]  Regina Tyshkevich,et al.  Decomposition of graphical sequences and unigraphs , 2000, Discret. Math..

[36]  Catherine S. Greenhill.  A Polynomial Bound on the Mixing Time of a Markov Chain for Sampling Regular Directed Graphs , 2011, Electron. J. Comb..

[37]  W. T. Tutte.  A Short Proof of the Factor Theorem for Finite Graphs , 1954, Canadian Journal of Mathematics.

[38]  Martin Dyer,et al.  Corrigendum: Sampling regular graphs and a peer-to-peer network , 2012 .

[39]  R. Taylor.  Contrained switchings in graphs , 1981 .