We analyze a distributed variation on the P´olya urn process in which a network of tiny artifacts manages the individual urns. Neighboring urns interact by repeatedly adding the same colored ball based on previous random choices. We discover that the process rapidly converges to a definitive random ratio between the colors in every urn and that the rate of convergence of the process at a given node depends on the global topology of the network. In particular, the same ratio appears for the case of complete communication graphs. Surprisingly, this effortless random process supports useful applications, such as clustering and pseudocoordinate computation. We present preliminary numerical studies that validate our theoretical predictions.