  The article deals with the description of the statistical behavior of
Gaussian packets on a metric graph. Semiclassical asymptotics of solutions of
the Cauchy problem for the Schr\"{o}dinger equation with initial data
concentrated in the neighborhood of one point on the edge, generates a
classical dynamical system on a graph. In a situation where all times for
packets to pass over edges ("edge travel times") are linearly independent over
the rational numbers, a description of the behavior of such systems is related
to the number-theoretic problem of counting the number of lattice points in an
expanding polyhedron. In this paper we show that for a finite compact graph
packets almost always are distributed evenly. A formula for the leading
coefficient of the asymptotic behavior of the number of packets with an
increasing time is obtained. The article also discusses a situation where the
times of passage over the edges are not linearly independent over the
rationals.
