  Estimating the size of hard-to-reach populations is an important problem for
many fields. The Network Scale-up Method (NSUM) is a relatively new approach to
estimate the size of these hard-to-reach populations by asking respondents the
question, "How many X's do you know," where X is the population of interest
(e.g. "How many female sex workers do you know?"). The answers to these
questions form Aggregated Relational Data (ARD). The NSUM has been used to
estimate the size of a variety of subpopulations, including female sex workers,
drug users, and even children who have been hospitalized for choking. Within
the Network Scale-up methodology, there are a multitude of estimators for the
size of the hidden population, including direct estimators, maximum likelihood
estimators, and Bayesian estimators. In this article, we first provide an
in-depth analysis of ARD properties and the techniques to collect the data.
Then, we comprehensively review different estimation methods in terms of the
assumptions behind each model, the relationships between the estimators, and
the practical considerations of implementing the methods. Finally, we provide a
summary of the dominant methods and an extensive list of the applications, and
discuss the open problems and potential research directions in this area.
