Approximate Discovery of Random Graphs

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 Erd˝os-R´enyi 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.

• Type: Conference
• Year: 2007
• Where: SAGA
• Authors: Thomas Erlebach, Alexander Hall, Matús Mihalák
• Keywords: Network Discovery, Nε /n, Random Graph, SAGA 2007