  The $k$-sample $\mathbb{G}(k,W)$ from a graphon $W:[0,1]^2\to [0,1]$ is the
random graph on $\{1,\dots,k\}$, where we sample $x_1,\dots,x_k\in [0,1]$
uniformly at random and make each pair $\{i,j\}\subseteq \{1,\dots,k\}$ an edge
with probability $W(x_i,x_j)$, with all these choices being mutually
independent. Let the random variable $X_k(W)$ be the number of edges in
$\mathbb{G}(k,W)$.
  Vera T. S\'os asked in 2012 whether two graphons $U,W$ are necessarily weakly
isomorphic if the random variables $X_k(U)$ and $X_k(W)$ have the same
distribution for every integer $k\ge 2$. This question when one of the graphons
$W$ is a constant function was answered positively by Endre Cs\'oka and
independently by Jacob Fox, Tomasz {\L}uczak and Vera T. S\'os. Here we
investigate the question when $W$ is a 2-step graphon and prove that the answer
is positive for a 3-dimensional family of such graphons.
  We also present some related results.
