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 G n,p. We show that a constant number of queries is sufficient if p is a constant, while Ω(n α) queries are needed if p = n ε/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 G n,p with p = n ε/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.
Work partially supported by European Commission - Fet Open project DELIS IST-001907 Dynamically Evolving Large Scale Information Systems, for which funding in Switzerland is provided by SBF grant 03.0378-1.
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Erlebach, T., Hall, A., Mihal’ák, M. (2007). Approximate Discovery of Random Graphs. In: Hromkovič, J., Královič, R., Nunkesser, M., Widmayer, P. (eds) Stochastic Algorithms: Foundations and Applications. SAGA 2007. Lecture Notes in Computer Science, vol 4665. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-74871-7_8
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