Properties of stochastic Kronecker graphs

TL;DR

This paper rigorously analyzes the properties of stochastic Kronecker graphs, showing they do not exhibit power law degree distributions and examining their subgraph counts and local neighborhoods.

Contribution

It provides a theoretical proof that stochastic Kronecker graphs lack power law degree distributions for all parameters, complementing previous empirical observations.

Findings

Does not feature power law degree distribution

Analyzes subgraph counts in the graph

Studies the typical neighborhood of vertices

Abstract

The stochastic Kronecker graph model introduced by Leskovec et al. is a random graph with vertex set $\mathbb Z_2^n$, where two vertices $u$ and $v$ are connected with probability $\alpha^{{u}\cdot{v}}\gamma^{(1-{u})\cdot(1-{v})}\beta^{n-{u}\cdot{v}-(1-{u})\cdot(1-{v})}$ independently of the presence or absence of any other edge, for fixed parameters $0<\alpha,\beta,\gamma<1$. They have shown empirically that the degree sequence resembles a power law degree distribution. In this paper we show that the stochastic Kronecker graph a.a.s. does not feature a power law degree distribution for any parameters $0<\alpha,\beta,\gamma<1$. In addition, we analyze the number of subgraphs present in the stochastic Kronecker graph and study the typical neighborhood of any given vertex.