We endow the set of probability measures on a weighted graph with a Monge--Kantorovich metric, induced by a function defined on the set of vertices. The graph is assumed to have $n$ vertices and so, the boundary of the probability simplex is an affine $(n-2)$--chain. Characterizing the geodesics of minimal length which may intersect the boundary, is a challenge we overcome even when the endpoints of the geodesics dont share the same connected components. It is our hope that this work would be a preamble to the theory of Mean Field Games on graphs.
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