Let $G$ be a random graph with $n$ nodes, with the nodes numbered. $G$ induces a distribution in the set of all graphs of $n$ nodes and we can identify this distribution with $G$.
Given the nodes $i,j$, we consider the Bernoulli random variable $a_{ij}$ which is $1$ when the link $i,j$ exists in $G$. Do the distributions of $a_{ij}, \forall i,j$ uniquely determine the distribution of $G$?