Degree Distribution Identifiability of Stochastic Kronecker Graphs

TL;DR

This paper investigates the limitations of stochastic Kronecker graph models in replicating real-world degree distributions, introduces methods to distinguish between models, and proposes algorithms to reduce degree distribution oscillations.

Contribution

The paper introduces statistical and computational notions of identifiability for SKG models, and presents algorithms to mitigate degree distribution oscillations by mixing seeds of coprime dimensions.

Findings

SKG models cannot generate power-law or lognormal degree distributions.

Models in different identifiability classes can be distinguished by graph features like isolated vertices.

Using seeds of coprime dimensions reduces degree oscillations in generated graphs.

Abstract

Large-scale analysis of the distributions of the network graphs observed in naturally-occurring phenomena has revealed that the degrees of such graphs follow a power-law or lognormal distribution. Seshadhri, Pinar, and Kolda (J. ACM, 2013) proved that stochastic Kronecker graph (SKG) models cannot generate graphs with degree distribution that follows a power-law or lognormal distribution. As a result, variants of the SKG model have been proposed to generate graphs which approximately follow degree distributions, without any significant oscillations. However, all existing solutions either require significant additional parameterization or have no provable guarantees on the degree distribution. -- In this work, we present statistical and computational identifiability notions which imply the separation of SKG models. Specifically, we prove that SKG models in different identifiability classes can be separated by the existence of isolated vertices and connected components in their corresponding generated graphs. This could explain the large (i.e., $>50\%$) fraction of isolated vertices in some popular graph generation benchmarks. -- We present and analyze an efficient algorithm that can get rid of oscillations in the degree distribution by mixing seeds of relative prime dimensions. For an initial $2\times 1$ and $2\times 2$ seed, a crucial subroutine of this algorithm solves a degree-2 and degree-4 optimization problem in the variables of the initial seed, respectively. We generalize this approach to solving optimization problems for $m\times n$ seeds, for any $m, n\in\mathbb{N}$. -- The use of $3\times 3$ seeds alone cannot get rid of significant oscillations. We prove that such seeds result in degree distribution that is bounded above by an exponential tail and thus cannot result in a power-law or lognormal.