On the number of star samples to find a vertex or edge with given degree in a graph

Abstract

A (single) star sample of a large graph consists of a vertex, its neighbors, and the degrees of each neighbor. Estimators are derived for the expected number of star samples required to find any vertex with a given degree l, and to find any edge with given degrees {k, l}, when the star vertex is selected with replacement, independently and uniformly at random. A modified construction of the classic Erdős-Renyi (ER) graph is given, wherein a random number of edges are placed independently and uniformly at random on the set of vertices. Three results are given: i) although the modified ER graph is a multigraph, the fraction of multiple edges is exponentially small in the edge probability, ii) the random degrees under the classic ER and modified ER constructions have equal means and the variances have a ratio near one for small edge probability, and iii) the probability of a star sample containing a vertex with a target degree is approximately equal to a simple combinatorial quantity under the modified ER construction. This quantity is our primary approximation of the performance under star sampling, requiring knowledge only of certain natural summary statistics for the graph. Numerical results comparing the actual and approximate number of star samples required to find vertices and edges with target degrees are given for both synthetic ER graphs and for real-world graphs from the SNAP repository.