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2 results

  • 5 - Spectral Methods

  • Sébastien Roch, University of Wisconsin, Madison
  • Book:
    Modern Discrete Probability
    Published online:
    14 December 2023
    Print publication:
    18 January 2024, pp 256-326

  • Summary

    In this chapter, we develop spectral techniques. We highlight some applications to Markov chain mixing and network analysis. The main tools are the spectral theorem and the variational characterization of eigenvalues, which we review together with some related results. We also give a brief introduction to spectral graph theory and detail an application to community recovery. Then we apply the spectral theorem to reversible Markov chains. In particular we define the spectral gap and establish its close relationship to the mixing time. We also show in that the spectral gap can be bounded using certain isoperimetric properties of the underlying network. We prove Cheeger’s inequality, which quantifies this relationship, and introduce expander graphs, an important family of graphs with good “expansion.” Applications to mixing times are also discussed. One specific technique is the “canonical paths method,” which bounds the spectral graph by formalizing a notion of congestion in the network.

  • NONEXISTENCE OF A CIRCULANT EXPANDER FAMILY

  • Part of
  • KA HIN LEUNG, VINH NGUYEN, WASIN SO
  • Journal:
    Bulletin of the Australian Mathematical Society / Volume 83 / Issue 1 / February 2011
    Published online by Cambridge University Press:
    17 November 2010, pp. 87-95
    Print publication:
    February 2011
    • Article
      • You have access
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  • The expansion constant of a simple graph G of order n is defined as where denotes the set of edges in G between the vertex subset S and its complement . An expander family is a sequence {Gi} of d-regular graphs of increasing order such that h(Gi)>ϵ for some fixed ϵ>0. Existence of such families is known in the literature, but explicit construction is nontrivial. A folklore theorem states that there is no expander family of circulant graphs only. In this note, we provide an elementary proof of this fact by first estimating the second largest eigenvalue of a circulant graph, and then employing Cheeger’s inequalities where G is a d-regular graph and λ2(G) denotes the second largest eigenvalue of G. Moreover, the associated equality cases are discussed.