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Algorithms 2016, 9(3), 50; doi:10.3390/a9030050

Utilizing Network Structure to Accelerate Markov Chain Monte Carlo Algorithms

Department of Computer Science, The University of Texas at Dallas, Richardson, TX 75081, USA
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Author to whom correspondence should be addressed.
Academic Editor: Natarajan Meghanathan
Received: 28 April 2016 / Revised: 8 July 2016 / Accepted: 22 July 2016 / Published: 29 July 2016
(This article belongs to the Special Issue Algorithms for Complex Network Analysis)
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Abstract

We consider the problem of estimating the measure of subsets in very large networks. A prime tool for this purpose is the Markov Chain Monte Carlo (MCMC) algorithm. This algorithm, while extremely useful in many cases, still often suffers from the drawback of very slow convergence. We show that in a special, but important case, it is possible to obtain significantly better bounds on the convergence rate. This special case is when the huge state space can be aggregated into a smaller number of clusters, in which the states behave approximately the same way (but their behavior still may not be identical). A Markov chain with this structure is called quasi-lumpable. This property allows the aggregation of states (nodes) into clusters. Our main contribution is a rigorously proved bound on the rate at which the aggregated state distribution approaches its limit in quasi-lumpable Markov chains. We also demonstrate numerically that in certain cases this can indeed lead to a significantly accelerated way of estimating the measure of subsets. The result can be a useful tool in the analysis of complex networks, whenever they have a clustering that aggregates nodes with similar (but not necessarily identical) behavior. View Full-Text
Keywords: Markov chain; convergence rate; lumpable and quasi-lumpable Markov chain Markov chain; convergence rate; lumpable and quasi-lumpable Markov chain
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This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).

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Askarian, A.; Xu, R.; Faragó, A. Utilizing Network Structure to Accelerate Markov Chain Monte Carlo Algorithms. Algorithms 2016, 9, 50.

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