On Set Size Distribution Estimation and the Characterization of Large   Networks via Sampling
  In this work we study the set size distribution estimation problem, where elements are randomly sampled from a collection of non-overlapping sets and we seek to recover the original set size distribution from the samples. This problem has applications to capacity planning, network theory, among other areas. Examples of real-world applications include characterizing in-degree distributions in large graphs and uncovering TCP/IP flow size distributions on the Internet. We demonstrate that it is hard to estimate the original set size distribution. The recoverability of original set size distributions presents a sharp threshold with respect to the fraction of elements that remain in the sets. If this fraction remains below a threshold, typically half of the elements in power-law and heavier-than-exponential-tailed distributions, then the original set size distribution is unrecoverable. We also discuss practical implications of our findings. 
Fabricio Murai and Bruno Ribeiro and Don Towsley and Pinghui Wang
math.ST cs.IT cs.SI math.IT stat.TH