Properties of stochastic Kronecker graphs

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 $α^{{u}\cdot{v}}γ^{(1-{u})\cdot(1-{v})}β^{n-{u}\cdot{v}-(1-{u})\cdot(1-{v})}$ independently of the presence or absence of any other edge, for fixed parameters $0<α,β,γ<1$. 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 $0<α,β,γ<1$. In addition, we analyze the number of subgraphs present in the stochastic Kronecker graph and study the typical neighborhood of any given vertex.

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