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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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