On the time to identify the nodes in a random graph

Abstract

The sampling of networks is an important problem at the frontier of statistical network analysis, and the identification of population members of a network is an important step in the sampling process. In this work, we study the random time τ to identify the nodes in an Erdős-Rényi random graph through egocentric sampling We derive the exact distribution of τ and give an exact formula for computing the mean time Eτ as a function of the size of the network. We explore how Eτ varies with the size of the network, the probability of edges, and network sparsity. We establish the scaling of τ with network size in both sparse and dense random graphs, highlighting special cases that demonstrate sub-linear scaling of τ with the size of the network. All theoretical results are non-asymptotic. Lastly, we discuss possible extensions to classes of random graphs beyond Erdős-Rényi random graphs.