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 Erdos-Renyi 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.