Positively multiplicative graphs are graphs whose adjacency matrix can be embedded in a matrix algebra admitting a distinguished basis labelled by its vertices with nonnegative structure constants. It is easy to get such graphs from the group algebra of the character algebra of a finite group. Other simple examples are obtained from classical bases of symmetric functions. More subtlety, it is also possible to define numerous multiplicative graphs from the affine Grassmannian associated to an affine Weyl group. These graphs are then related to interesting probabilistic models (random walks in alcoves, TASEP etc.) and the related positively multiplicative graphs give finite automatons recognizing reduced expressions of affine Grassmannian elements. The talk will consist in an introduction to these notions and problems. This is a work in collaboration with J. Guilhot (IDP Tours) and P. Tarrago (LPSM Paris).