An In-Depth Analysis of Stochastic Kronecker Graphs

TL;DR

This paper provides a detailed analysis of the stochastic Kronecker graph model, revealing its limitations in generating realistic degree distributions and core structures, and proposes an enhanced version with improved properties.

Contribution

It offers a rigorous mathematical analysis of SKG, demonstrates its limitations, and introduces an enhanced model with noise that produces more realistic graph properties.

Findings

SKG cannot generate power-law or lognormal degree distributions

Adding noise to SKG results in lognormal distributions

Graphs from SKG often have high isolated vertex counts

Abstract

Graph analysis is playing an increasingly important role in science and industry. Due to numerous limitations in sharing real-world graphs, models for generating massive graphs are critical for developing better algorithms. In this paper, we analyze the stochastic Kronecker graph model (SKG), which is the foundation of the Graph500 supercomputer benchmark due to its favorable properties and easy parallelization. Our goal is to provide a deeper understanding of the parameters and properties of this model so that its functionality as a benchmark is increased. We develop a rigorous mathematical analysis that shows this model cannot generate a power-law distribution or even a lognormal distribution. However, we formalize an enhanced version of the SKG model that uses random noise for smoothing. We prove both in theory and in practice that this enhancement leads to a lognormal distribution. Additionally, we provide a precise analysis of isolated vertices, showing that the graphs that are produced by SKG might be quite different than intended. For example, between 50% and 75% of the vertices in the Graph500 benchmarks will be isolated. Finally, we show that this model tends to produce extremely small core numbers (compared to most social networks and other real graphs) for common parameter choices.