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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.Copyright © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 46,614–634, 2015

A generalized Pólya's urn with graph based interactions