Waiting for a Bat to Fly By (in Polynomial Time)

  • Authors:

  • Itai Benjamini;Gady Kozma;László Lovász;Dan Romik;Gábor Tardos

  • Affiliations:

  • Faculty of Mathematics and Computer Science, The Weizmann Institute of Science, POB 26, Rehovot 76100, Israel (e-mail: itai.benjamini@weizmann.ac.il);Institute for Advanced Study, 1 Einstein Drive, Princeton, New Jersey 08540, USA (e-mail: gady@ias.edu);Microsoft Research, One Microsoft Way, Redmond, WA 98052, USA (e-mail: lovasz@microsoft.com);Department of Statistics, 367 Evans Hall, University of California, Berkeley, CA 94720-3860, USA (e-mail: romik@stat.berkeley.edu);Rényi Institute, Hungarian Academy of Sciences, Pf. 127, H-1354 Budapest, Hungary (e-mail: tardos@renyi.hu)

  • Venue:
  • Combinatorics, Probability and Computing
  • Year:
  • 2006

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Abstract

We observe returns of a simple random walk on a finite graph to a fixed node, and would like to infer properties of the graph, in particular properties of the spectrum of the transition matrix. This is not possible in general, but at least the set of eigenvalues can be recovered under fairly general conditions, e.g., when the graph has a node-transitive automorphism group. The main result is that by observing polynomially many returns, it is possible to estimate the spectral gap of such a graph up to a constant factor.