  We analyze composed quantum systems consisting of $k$ subsystems, each
described by states in the $n$-dimensional Hilbert space. Interaction between
subsystems can be represented by a graph, with vertices corresponding to
individual subsystems and edges denoting a generic interaction, modeled by
random unitary matrices of order $n^2$. The global evolution operator is
represented by a unitary matrix of size $N=n^k$. We investigate statistical
properties of such matrices and show that they display spectral properties
characteristic to Haar random unitary matrices provided the corresponding graph
is connected. Thus basing on random unitary matrices of a small size $n^2$ one
can construct a fair approximation of large random unitary matrices of size
$n^{k}$. Graph--structured random unitary matrices investigated here allow one
to define the corresponding structured ensembles of random pure states.
