Error Bounds for the Network Scale-Up Method

TL;DR

This paper derives analytical error bounds for the Network Scale-Up Method (NSUM), revealing how network structure and sampling affect estimation accuracy for hidden populations in social networks.

Contribution

It provides the first analytical bounds on NSUM estimation errors, considering adversarial and random network models, and offers improved bounds for specific network types.

Findings

Adversarial networks can cause estimates to be off by a factor of ((n))

Random networks allow for small constant-factor errors with high probability using logarithmic samples

Analytical bounds are validated through extensive numerical experiments

Abstract

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 neighbours 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 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 $\Omega(\sqrt{n})$ 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 Erdos-Renyi 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.