Uniform random sampling not recommended for large graph size estimation

Abstract

The norm of data size estimation is to use uniform random samples whenever possible. There have been tremendous efforts in obtaining uniform random samples using methods such as Metropolis–Hasting random walk or importance sampling [2]. This paper shows that, on the contrary to the common practice, uniform random sampling should be avoided when PPS (probability proportional to size) sampling is available for large data.To develop intuition of the sampling process, we discuss the sampling and estimation problem in the context of graph. The size is the number of nodes in the graph; uniform random sampling corresponds to uniform random node (RN) sampling; and PPS sampling is approximated by random edge (RE) sampling. In this setting, we show that for large graphs RE sampling outperforms RN sampling with a ratio proportional to the normalized graph degree variance. This result is particularly important in the era of big data, when data are typically large and scale-free [3], resulting in large degree variance.We derive the result by giving the variances of RN and RE estimators. Each step of the derivation is supported and demonstrated by simulation studies assuming power law distributions. Then we use 18 real-world networks to verify the result. Furthermore, we show that the performance of random walk (RW) sampling is data dependent and can be significantly worse than RN and RE. More specifically, RW can estimate online social networks but not Web graphs due to the difference of the graph conductance.