Finding High-Degree Vertices with Inclusive Random Sampling

Abstract

The friendship paradox (FP) is the famous phenomenon that one’s friends typically have more friends than they themselves do. The FP has inspired novel approaches for sampling vertices at random from a network when the goal of the sampling is to find vertices of higher degree. The most famous of these methods involves selecting a vertex at random and then selecting one of its neighbors at random. Another possible method would be to select a random edge from the network and then select one of its endpoints at random, again predicated on the fact that high degree vertices will be overrepresented in the collection of edge endpoints. In this paper we propose a simple tweak to these methods where we consider the degrees of the two vertices involved in the selection process and choose the one with higher degree. We explore the different sampling methods theoretically and establish interesting asymptotic bounds on their performances as a way of highlighting their respective strengths. We also apply the methods experimentally to both synthetic graphs and real-world networks to determine the improvement inclusive sampling offers over exclusive sampling, which version of inclusive sampling is stronger, and what graph characteristics affect these results.