# Probability, preferential attachment, “rich get richer”

Imagine you have $N$ empty bins. At every timestep $t$ you throw a ball to a randomly chosen bin ($t$ is therefore also the total number of balls in this system). Probability that a ball falls into a bin $i$ is proportional to the number of balls already in the bin $k_i$ , plus some initial conditions. So the bins get "wider" as more balls falls into them. I want to find distribution of the balls $P_t: t \rightarrow \infty$

$P_t \propto k_i + A$ , where $k_i$ is the number of the balls in the bin $i$ and $A$ is some constant (initial conditions)

$P_{0}(k_i) = {1 \over{N}}$

so I think that

$P_t(k_i) = {{k_{i-1}+A}\over{NA+\sum_jk_j}}$

How to find the distribution for $t \rightarrow \infty$ ?

Related:

Preferential attachment

Rich get richer

Posted also here:

http://mathoverflow.net/questions/86799/probability-preferential-attachment-rich-get-richer

-
Crossposted on MO. Please do not do that. – Did Jan 27 '12 at 7:44
This looks exactly like the Pólya urn model, whose limit distribution is known to be multivariate Dirichlet. – Did Jan 27 '12 at 7:45
OK, I promise not to do that again. Could you please explain your comment? I am not an expert at this stuff. – Jakub M. Jan 27 '12 at 7:50
Did you read the link? – Did Jan 27 '12 at 8:15
You are right, I was confused. en.wikipedia.org/wiki/Multivariate_Pólya_distribution is then an answer. Thanks a lot! – Jakub M. Jan 27 '12 at 9:03