  Let $\mathfrak{g}$ be an untwisted affine Lie algebra with associated Weyl
group $W_a$. To any level 0 weight $\gamma$ we associate a weighted graph
$\Gamma_\gamma$ that encodes the orbit of $\gamma$ under the action $W_a$. We
show that the graph $\Gamma_\gamma$ encodes the periodic orientation of certain
subsets of alcoves in $W_a$ and therefore can be interpreted as an automaton
determining the reduced expressions in these subsets. Then, by using some
relevant quotients of the homology ring of affine Grassmannians, we show that
$\Gamma_\gamma$ is positively multiplicative. This allows us in particular to
compute the structure constants of the homology rings using elementary linear
algebra on multiplicative graphs. In another direction, the positivity of
$\Gamma_\gamma$ yields the key ingredients to study a large class of central
random walks on alcoves.
