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The Diameter of a Long-Range Percolation Graph

  • Don Coppersmith
  • David Gamarnik
  • Maxim Sviridenko
Conference paper
Part of the Trends in Mathematics book series (TM)

Abstract

e consider the following long-range percolation model: an undirected graph with the node set {0, 1,... ,N}d, has edges (x, y) selected with probability ≈ ,β/∥x - y∥s if ∥x - y∥ > 1, and with probability 1 if ∥x - y∥ = 1, for some parameters β, s > 0. This model was introduced by Benjamini and Berger [2], who obtained bounds on the diameter of this graph for the one-dimensional case d = 1 and for various values of s, but left cases s = 1,2 open. We show that, with high probability, the diameter of this graph is Θ(log N / log log N) when s = d, and, for some constants 0 < η1 < η2 < 1, it is at most Nη2 when s = 2d, and is at least Nη2 when d = 1, s = 2, β < 1 or when s > 2d. We also provide a simple proof that the diameter is at most log0(1) N with high probability, when d < s < 2d, established previously in [2].

Keywords

Short Path Random Graph Side Length Percolation Model Circle Graph 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Basel AG 2002

Authors and Affiliations

  • Don Coppersmith
    • 1
  • David Gamarnik
    • 1
  • Maxim Sviridenko
    • 1
  1. 1.IBM T.J.Watson Research CenterYorktown HeightsUSA

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