Preferential growth: Exact solution of the time-dependent distributions

We consider a preferential growth model where particles are added one by one to the system consisting of clusters of particles. A new particle can either form a new cluster (with probability $q)$ or join an already existing cluster with a probability proportional to the size thereof. We calculate exactly the probability ${\mathcal{P}}_i\mathrm{}(k,t)\mathrm{}$ that the size of the $i\mathrm{th}$ cluster at time t is k. We analyze the asymptotics, the scaling properties of the size distribution and of the mean size, as well as the relation of our system to recent network models.