Asymptotic Analysis of Equivalences and Core-Structures in Kronecker-Style Graph Models

TL;DR

This paper provides an asymptotic analysis of Kronecker-style graph models, revealing differences between variants and highlighting limitations of empirical studies in capturing true model behaviors.

Contribution

It proves asymptotic equivalences and divergences among Kronecker-style models, clarifying their theoretical properties and practical implications.

Findings

R-MAT variants are asymptotically equivalent but differ from SKG.

Differences among models are observable at small graph sizes.

Asymptotic analysis uncovers unexpected behaviors within models.

Abstract

Growing interest in modeling large, complex networks has spurred significant research into generative graph models. Kronecker-style models (SKG and R-MAT) are often used due to their scalability and ability to mimic key properties of real-world networks. Although a few papers theoretically establish these models' behavior for specific parameters, many claims used to justify their use are supported only empirically. In this work, we prove several results using asymptotic analysis which illustrate that empirical studies may not fully capture the true behavior of the models. Paramount to the widespread adoption of Kronecker-style models was the introduction of a linear-time edge-sampling variant (R-MAT), which existing literature typically treats as interchangeable with SKG. We prove that although several R-MAT formulations are asymptotically equivalent, their behavior diverges from that of SKG. Further, we show these results are observable even at relatively small graph sizes. Second, we consider a case where asymptotic analysis reveals unexpected behavior within a given model.