On the log-normality of the degree distribution in large homogeneous binary multiplicative attribute graph models

TL;DR

This paper rigorously proves that in large homogeneous binary multiplicative attribute graph models, the degree distribution converges to a log-normal distribution, providing precise convergence rates and approximations.

Contribution

It offers a rigorous convergence result to log-normality under weaker conditions and derives explicit rates of convergence for the degree distribution in MAG models.

Findings

Degree distribution converges to log-normal in large MAGs.

Provides Berry-Esseen type convergence rate estimate.

Develops log-normal approximations for degree distribution.

Abstract

The muliplicative attribute graph (MAG) model was introduced by Kim and Leskovec as a mathematically tractable model for networks where network structure is believed to be shaped by features or attributes associated with individual nodes. For large homogeneous binary MAGs, they argued through approximation arguments that the "tail of [the] degree distribution follows a log-normal distribution" as the number of nodes becomes unboundedly large and the number of attributes scales logarithmically with the number of nodes. Under the same limiting regime, we revisit the asymptotic behavior of the degree distribution: Under weaker conditions we obtain a precise convergence result to log-normality, develop from it reasoned log-normal approximations to the degree distribution and derive various rates of convergence. In particular, we show that a certain transformation of the node degree converges in distribution to a log-normal distribution, and give its convergence rate in the form of a Berry-Esseen type estimate.