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