### Abstract

A joint degree matrix (JDM) specifies the number of connections between nodes of given degrees in a graph, for all degree pairs, and uniquely determines the degree sequence of the graph. We consider the space of all balanced realizations of an arbitrary JDM, realizations in which the links between any two fixed-degree groups of nodes are placed as uniformly as possible. We prove that a swap Markov chain Monte Carlo algorithm in the space of all balanced realizations of an arbitrary graphical JDM mixes rapidly, i.e., the relaxation time of the chain is bounded from above by a polynomial in the number of nodes n. To prove fast mixing, we first prove a general factorization theorem similar to the Martin-Randall method for disjoint decompositions (partitions). This theorem can be used to bound from below the spectral gap with the help of fast mixing subchains within every partition and a bound on an auxiliary Markov chain between the partitions. Our proof of the general factorization theorem is direct and uses conductance based methods (Cheeger inequality).

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

Pages (from-to) | 481-499 |

Number of pages | 19 |

Journal | SIAM Journal on Discrete Mathematics |

Volume | 29 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2015 |

### Fingerprint

### Keywords

- Graph sampling
- Joint degree matrix
- Rapidly mixing Markov chains

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*SIAM Journal on Discrete Mathematics*,

*29*(1), 481-499. https://doi.org/10.1137/130929874

**A decomposition based proof for fast mixing of a Markov chain over balanced realizations of a joint degree matrix.** / Erdos, Péter L.; Miklós, I.; Toroczkai, Zoltán.

Research output: Contribution to journal › Article

*SIAM Journal on Discrete Mathematics*, vol. 29, no. 1, pp. 481-499. https://doi.org/10.1137/130929874

}

TY - JOUR

T1 - A decomposition based proof for fast mixing of a Markov chain over balanced realizations of a joint degree matrix

AU - Erdos, Péter L.

AU - Miklós, I.

AU - Toroczkai, Zoltán

PY - 2015

Y1 - 2015

N2 - A joint degree matrix (JDM) specifies the number of connections between nodes of given degrees in a graph, for all degree pairs, and uniquely determines the degree sequence of the graph. We consider the space of all balanced realizations of an arbitrary JDM, realizations in which the links between any two fixed-degree groups of nodes are placed as uniformly as possible. We prove that a swap Markov chain Monte Carlo algorithm in the space of all balanced realizations of an arbitrary graphical JDM mixes rapidly, i.e., the relaxation time of the chain is bounded from above by a polynomial in the number of nodes n. To prove fast mixing, we first prove a general factorization theorem similar to the Martin-Randall method for disjoint decompositions (partitions). This theorem can be used to bound from below the spectral gap with the help of fast mixing subchains within every partition and a bound on an auxiliary Markov chain between the partitions. Our proof of the general factorization theorem is direct and uses conductance based methods (Cheeger inequality).

AB - A joint degree matrix (JDM) specifies the number of connections between nodes of given degrees in a graph, for all degree pairs, and uniquely determines the degree sequence of the graph. We consider the space of all balanced realizations of an arbitrary JDM, realizations in which the links between any two fixed-degree groups of nodes are placed as uniformly as possible. We prove that a swap Markov chain Monte Carlo algorithm in the space of all balanced realizations of an arbitrary graphical JDM mixes rapidly, i.e., the relaxation time of the chain is bounded from above by a polynomial in the number of nodes n. To prove fast mixing, we first prove a general factorization theorem similar to the Martin-Randall method for disjoint decompositions (partitions). This theorem can be used to bound from below the spectral gap with the help of fast mixing subchains within every partition and a bound on an auxiliary Markov chain between the partitions. Our proof of the general factorization theorem is direct and uses conductance based methods (Cheeger inequality).

KW - Graph sampling

KW - Joint degree matrix

KW - Rapidly mixing Markov chains

UR - http://www.scopus.com/inward/record.url?scp=84925351340&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84925351340&partnerID=8YFLogxK

U2 - 10.1137/130929874

DO - 10.1137/130929874

M3 - Article

VL - 29

SP - 481

EP - 499

JO - SIAM Journal on Discrete Mathematics

JF - SIAM Journal on Discrete Mathematics

SN - 0895-4801

IS - 1

ER -