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On nodal domains and higher-order Cheeger inequalities of finite reversible Markov processes

Daneshgar, A ; Sharif University of Technology | 2012

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  1. Type of Document: Article
  2. DOI: 10.1016/j.spa.2012.02.009
  3. Publisher: 2012
  4. Abstract:
  5. Let L be a reversible Markovian generator on a finite set V. Relations between the spectral decomposition of L and subpartitions of the state space V into a given number of components which are optimal with respect to min-max or max-min Dirichlet connectivity criteria are investigated. Links are made with higher-order Cheeger inequalities and with a generic characterization of subpartitions given by the nodal domains of an eigenfunction. These considerations are applied to generators whose positive rates are supported by the edges of a discrete cycle Z N, to obtain a full description of their spectra and of the shapes of their eigenfunctions, as well as an interpretation of the spectrum through a double-covering construction. Also, we prove that for these generators, higher Cheeger inequalities hold, with a universal constant factor 48
  6. Keywords:
  7. Cheeger's inequality ; Dirichlet connectivity spectra ; Markov processes on discrete cycles ; Nodal domains of eigenfunctions ; Optimal partitions of state space ; Principal Dirichlet eigenvalues ; Reversible Markovian generator ; Spectral decomposition ; Cheeger ; Dirichlet ; Eigenvalues ; Markovian ; Nodal domain ; Spectral decomposition ; State space ; Markov processes ; Optimization ; Eigenvalues and eigenfunctions
  8. Source: Stochastic Processes and their Applications ; Volume 122, Issue 4 , April , 2012 , Pages 1748-1776 ; 03044149 (ISSN)
  9. URL: http://www.sciencedirect.com/science/article/pii/S0304414912000312