Epidemiologists and social scientists have used the Network Scale-Up Method (NSUM) for over thirty years to estimate the size of a hidden sub-population within a social network. This method involves querying a subset of network nodes about the number of their neighbors belonging to the hidden sub-population. In general, NSUM assumes that the social network topology and the hidden sub-population distribution are well-behaved; hence, the NSUM estimate is close to the actual value. However, bounds on NSUM estimation errors have not been analytically proven. This paper provides analytical bounds on the error incurred by the two most popular NSUM estimators. These bounds assume that the queried nodes accurately provide their degree and the number of neighbors belonging to the hidden sub-population. Our key findings are twofold. First, we show that when an adversary designs the network and places the hidden sub-population, then the estimate can be a factor of Ω(√n) off from the real value (in a network with n nodes). Second, we also prove error bounds when the underlying network is randomly generated, showing that a small constant factor can be achieved with high probability using samples of logarithmic size O(log n). We present improved analytical bounds for Erdős-Rényi and Scale-Free networks. Our theoretical analysis is supported by an extensive set of numerical experiments designed to determine the effect of the sample size on the accuracy of the estimates in both synthetic and real networks.

Indirect surveys, in which respondents provide information about other people they know, have been proposed for estimating (nowcasting) the size of a hidden population where privacy is important or the hidden population is hard to reach. Examples include estimating casualties in an earthquake, conditions among female sex workers, and the prevalence of drug use and infectious diseases. The Network Scale-up Method (NSUM) is the classical approach to developing estimates from indirect surveys, but it was designed for one-shot surveys. Further, it requires certain assumptions and asking for or estimating the number of individuals in each respondent's network. In recent years, surveys have been increasingly deployed online and can collect data continuously (e.g., COVID-19 surveys on Facebook during much of the pandemic). Conventional NSUM can be applied to these scenarios by analyzing the data independently at each point in time, but this misses the opportunity of leveraging the temporal dimension. We propose to use the responses from indirect surveys collected over time and develop analytical tools (i) to prove that indirect surveys can provide better estimates for the trends of the hidden population over time, as compared to direct surveys and (ii) to identify appropriate temporal aggregations to improve the estimates. We demonstrate through extensive simulations that our approach outperforms traditional NSUM and direct surveying methods. We also empirically demonstrate the superiority of our approach on a real indirect survey dataset of COVID-19 cases.