Approximate Discovery of Random Graphs

  • Thomas Erlebach
  • Alexander Hall
  • Matúš Mihal’ák
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4665)

Abstract

In the layered-graph query model of network discovery, a query at a node v of an undirected graph G discovers all edges and non-edges whose endpoints have different distance from v. We study the number of queries at randomly selected nodes that are needed for approximate network discovery in Erdős-Rényi random graphs Gn,p. We show that a constant number of queries is sufficient if p is a constant, while Ω(nα) queries are needed if p = nε/n, for arbitrarily small choices of ε = 3 / (6 ·i + 5) with i ∈ ℕ. Note that α> 0 is a constant depending only on ε. Our proof of the latter result yields also a somewhat surprising result on pairwise distances in random graphs which may be of independent interest: We show that for a random graph Gn,p with p = nε/n, for arbitrarily small choices of ε> 0 as above, in any constant cardinality subset of the nodes the pairwise distances are all identical with high probability.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Thomas Erlebach
    • 1
  • Alexander Hall
    • 2
  • Matúš Mihal’ák
    • 3
  1. 1.Department of Computer Science, University of Leicester, University Road, Leicester LE1 7RHUK
  2. 2.Department EECS, UC Berkeley, CA 94720USA
  3. 3.Institute for TCS, ETH Zurich, CH-8092 ZurichSwitzerland

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