Gamma invariance and Dirichlet proportion for random exchange processes on graphs

We investigate invariant measures on random weighted directed graphs representing random exchange processes.More precisely, we consider measures consisting of independent random variables on each vertex of the graph, and we assume that their expectations are not decomposable according to the graph structure.When in addition such a measure is invariant under a Markov dynamic on the graph, we show that the random variables constituting the considered measure are necessarily gamma-distributed and that the proportions from each vertex along its graph-neighbors are Dirichlet distributed.We illustrate and discuss our results on several examples, including the related original setting of Muratov and Zuyev (2017).