  We study approximations of billiard systems by lattice graphs. It is
demonstrated that under natural assumptions the graph wavefunctions approximate
solutions of the Schroedinger equation with energy rescaled by the billiard
dimension. As an example, we analyze a Sinai billiard with attached leads. The
results illustrate emergence of global structures in large quantum graphs and
offer interesting comparisons with patterns observed in complex networks of a
different nature.
