Interacting Urns on a Finite Directed Graph.

Consider a finite directed graph $G=(V, E)$ and place an urn with balls of two colours: white and black, at each node at time $t=0$. The urns evolve, in discrete time, depending upon a common replacement matrix $R$ and the underlying graph structure. At each time-step, urns reinforce their neighbours according to a fixed replacement matrix $R$. We study asymptotic properties of the fraction of balls of either colour and obtain limit theorems for general replacement matrices. In particular, we show that if the reinforcement is not of what we call {P}\'{o}lya-type, there is always a consensus, almost surely, with Gaussian fluctuations in some regimes. We also prove that for {P}\'{o}lya-type replacements, the fraction of balls of either colour, in each urn, converges almost surely and that this limit is same for every urn. One of the motivations behind studying this model and choosing the replacement matrices comes from opinion dynamics on networks, where opinions are rigid and change slowly with influence from the neighbours.