We consider generalizations of the classical Polya urn problem: Given finitely many
bins each containing one ball, suppose that additional balls arrive one at a time. For each new ball,
with probability p, create a new bin and place the ball in that
bin; with probability 1–p, place the ball in an existing
bin, such that the probability that the ball is placed in a bin is proportional to
mg
... [Show full abstract] m^\gamma , where m is the number of balls in that bin. For
p=0, the number of bins is fixed and finite,
and the behavior of the process depends on whether is greater than, equal to, or less than 1.
We survey the known results and give new proofs for all three cases. We then consider the case
p>0. When =1, this is equivalent to the so-called
preferential attachment scheme which leads to power law
distribution for bin sizes. When >1, we prove that a single bin dominates, i.e., as
the number of balls goes to infinity, the probability that any new ball either goes into that bin or
creates a new bin converges to 1. When p > 0 and < 1, we show that under the assumption that
certain limits exist, the fraction of bins having m balls shrinks
exponentially as a function of m. We then discuss further
generalizations and pose several open problems.