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Population size estimates for hidden and hard-to-reach populations areparticularly important when members are known to suffer from disproportionhealth issues or to pose health risks to the larger ambient population in whichthey are embedded. Efforts to derive size estimates are often frustrated by arange of factors that preclude conventional survey strategies, including socialstigma associated with group membership or members' involvement in illegalactivities.  This paper extends prior research on the problem of network population sizeestimation, building on established survey/sampling methodologies commonly usedwith hard-to-reach groups. Three novel one-step, network-based population sizeestimators are presented, to be used in the context of uniform random sampling,respondent-driven sampling, and when networks exhibit significant clusteringeffects. Provably sufficient conditions for the consistency of these estimators(in large configuration networks) are given. Simulation experiments across awide range of synthetic network topologies validate the performance of theestimators, which are seen to perform well on a real-world location-basedsocial networking data set with significant clustering. Finally, the proposedschemes are extended to allow them to be used in settings where participantanonymity is required. Systematic experiments show favorable tradeoffs betweenanonymity guarantees and estimator performance.  Taken together, we demonstrate that reasonable population estimates can bederived from anonymous respondent driven samples of 250-750 individuals, withinambient populations of 5,000-40,000. The method thus represents a novel andcost-effective means for health planners and those agencies concerned withhealth and disease surveillance to estimate the size of hidden populations.Limitations and future work are discussed in the concluding section.

One-step Estimation of Networked Population Size: Respondent-Driven Capture-Recapture with Anonymity