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

Péter L. Erdos, I. Miklós, Zoltán Toroczkai

Research output: Contribution to journalArticle

9 Citations (Scopus)

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 languageEnglish
Pages (from-to)481-499
Number of pages19
JournalSIAM Journal on Discrete Mathematics
Volume29
Issue number1
DOIs
Publication statusPublished - 2015

Fingerprint

Balanced Realization
Markov chain
Factorization Theorem
Partition
Decompose
Vertex of a graph
Markov Chain Monte Carlo Algorithms
Degree Sequence
Spectral Gap
Swap
Arbitrary
Graph in graph theory
Conductance
Relaxation Time
Disjoint
Polynomial
Theorem

Keywords

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

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

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.

In: SIAM Journal on Discrete Mathematics, Vol. 29, No. 1, 2015, p. 481-499.

Research output: Contribution to journalArticle

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