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.
Categories: math.CO