  The cut-rank of a set $X$ of vertices in a graph $G$ is defined as the rank
of the $ X \times (V(G)\setminus X)$ matrix over the binary field whose
$(i,j)$-entry is $1$ if the vertex $i$ in $X$ is adjacent to the vertex $j$ in
$V(G)\setminus X$ and $0$ otherwise. We introduce the graph parameter called
the average cut-rank of a graph, defined as the expected value of the cut-rank
of a random set of vertices. We show that this parameter does not increase when
taking vertex-minors of graphs and a class of graphs has bounded average
cut-rank if and only if it has bounded neighborhood diversity. This allows us
to deduce that for each real $\alpha$, the list of induced-subgraph-minimal
graphs having average cut-rank larger than (or at least) $\alpha$ is finite. We
further refine this by providing an upper bound on the size of obstruction and
a lower bound on the number of obstructions for average cut-rank at most (or
smaller than) $\alpha$ for each real $\alpha\ge0$. Finally, we describe
explicitly all graphs of average cut-rank at most $3/2$ and determine up to
$3/2$ all possible values that can be realized as the average cut-rank of some
graph.
