Interacting Urn Schemes on Finite Ancestral Directed Acyclic Graphs

We study interacting finite-color urn schemes on directed acyclic graphs, allowing the graph to be infinite. Each urn evolves through reinforcements driven by colors drawn from its in-neighboring urns via edge-dependent reinforcement matrices. Assuming that every vertex has only finitely many ancestors, we prove almost sure convergence of urn proportions and show that the limiting configuration is determined by vertices with no ancestors or self-loops. Under additional balance and irreducibility assumptions on reinforcement matrices, we also obtain second-order asymptotic results in all regimes of appropriately defined parameters.