A generalized Pólya's urn with graph based interactions

Given a finite connected graph G, place a bin at each vertex. Two bins are called a pair if they share an edge of G. At discrete times, a ball is added to each pair of bins. In a pair of bins, one of the bins gets the ball with probability proportional to its current number of balls raised by some fixed power α>0. We characterize the limiting behavior of the proportion of balls in the bins. The proof uses a dynamical approach to relate the proportion of balls to a vector field. Our main result is that the limit set of the proportion of balls is contained in the equilibria set of the vector field. We also prove that if α<1 then there is a single point v=v(G,α) with non-zero entries such that the proportion converges to v almost surely. A special case is when G is regular and α≤1. We show e.g. that if G is non-bipartite then the proportion of balls in the bins converges to the uniform measure almost surely.