### 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.

Original language | English |
---|---|

Pages (from-to) | 186-207 |

Number of pages | 22 |

Journal | Combinatorics Probability and Computing |

Volume | 27 |

Issue number | 2 |

DOIs | |

Publication status | Published - Mar 1 2018 |

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### ASJC Scopus subject areas

- Theoretical Computer Science
- Statistics and Probability
- Computational Theory and Mathematics
- Applied Mathematics

### Cite this

*Combinatorics Probability and Computing*,

*27*(2), 186-207. https://doi.org/10.1017/S0963548317000499

**New Classes of Degree Sequences with Fast Mixing Swap Markov Chain Sampling.** / Erdås, Péter L.; Miklós, I.; Toroczkai, Zoltán.

Research output: Contribution to journal › Article

*Combinatorics Probability and Computing*, vol. 27, no. 2, pp. 186-207. https://doi.org/10.1017/S0963548317000499

}

TY - JOUR

T1 - New Classes of Degree Sequences with Fast Mixing Swap Markov Chain Sampling

AU - Erdås, Péter L.

AU - Miklós, I.

AU - Toroczkai, Zoltán

PY - 2018/3/1

Y1 - 2018/3/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85033398118&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85033398118&partnerID=8YFLogxK

U2 - 10.1017/S0963548317000499

DO - 10.1017/S0963548317000499

M3 - Article

AN - SCOPUS:85033398118

VL - 27

SP - 186

EP - 207

JO - Combinatorics Probability and Computing

JF - Combinatorics Probability and Computing

SN - 0963-5483

IS - 2

ER -