Star sampling with and without replacement

Star sampling (SS) is a random sampling procedure on a graph wherein each sample consists of a randomly selected vertex (the star center) and its one-hop neighbors (the star endpoints). We consider the use of star sampling to find any member of an arbitrary target set of vertices in a graph, where the figure of merit (cost) is either the expected number of samples (unit cost) or the expected number of star centers plus star endpoints (linear cost) until a vertex in the target set is encountered, either as a star center or as a star point. We analyze this performance measure on three related star sampling paradigms: SS with replacement (SSR), SS without center replacement (SSC), and SS without star replacement (SSS). We derive exact and approximate expressions for the expected unit and linear costs of SSR, SSC, and SSS on Erdos-Renyi (ER) graphs. Our results show there is i) little difference in unit cost, but ii) significant difference in linear cost, across the three paradigms. Although our results are derived for ER graphs, experiments on "real-world" graphs suggest our performance expressions are reasonably accurate for non-ER graphs.