Graph search via star sampling with and without replacement

TL;DR

This paper analyzes star sampling strategies on graphs to efficiently find target vertices, deriving cost expressions and comparing sampling paradigms on random and real-world graphs.

Contribution

It provides exact and approximate cost formulas for star sampling with different replacement schemes on Erdős-Rényi graphs and evaluates their performance on real-world networks.

Findings

SSC/SSS outperform SSR in low-density ER graphs under unit cost.

SSS outperforms SSR/SSC under linear cost in low- to moderate-density ER graphs.

Cost approximations are accurate for ER graphs; SSR and SSC are reliable for real-world graphs.

Abstract

Star sampling (SS) is a random sampling procedure on a graph wherein each sample consists of a randomly selected vertex (the star center) and its (one-hop) neighbors (the star points). We consider the use of SS to find any member of a target set of vertices in a graph, where the figure of merit (cost) is either the expected number of samples (unit cost) or the expected number of star centers plus star points (linear cost) until a vertex in the target set is encountered, either as a star center or as a star point. We analyze these two performance measures on three related star sampling paradigms: SS with replacement (SSR), SS without center replacement (SSC), and SS without star replacement (SSS). Exact and approximate expressions are derived for the expected unit and linear costs of SSR, SSC, and SSS on Erd\H{o}s-R\'{e}nyi (ER) random graphs. The approximations are seen to be accurate. SSC/SSS are notably better than SSR under unit cost for low-density ER graphs, while SSS is notably better than SSR/SSC under linear cost for low- to moderate-density ER graphs. Simulations on twelve "real-world" graphs shows the cost approximations to be of variable quality: the SSR and SSC approximations are uniformly accurate, while the SSS approximation, derived for an ER graph, is of variable accuracy.