### 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 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.

Original language | English |
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Title of host publication | Proceedings of the 4th Symposium on Stochastic Algorithms, Foundations, and Applications (SAGA) |

Pages | 82-92 |

Number of pages | 11 |

DOIs | |

Publication status | Published - 2007 |

Externally published | Yes |

## Cite this

Erlebach, T., Hall, A., & Mihalák, M. (2007). Approximate Discovery of Random Graphs. In

*Proceedings of the 4th Symposium on Stochastic Algorithms, Foundations, and Applications (SAGA)*(pp. 82-92) https://doi.org/10.1007/978-3-540-74871-7_8