Power laws for family sizes in a duplication model

Qian, Luscombe and Gerstein [J. Molecular Biol. 313 (2001) 673--681] introduced a model of the diversification of protein folds in a genome that we may formulate as follows. Consider a multitype Yule process starting with one individual in which there are no deaths and each individual gives birth to a new individual at rate 1. When a new individual is born, it has the same type as its parent with probability $1-r$ and is a new type, different from all previously observed types, with probability $r$. We refer to individuals with the same type as families and provide an approximation to the joint distribution of family sizes when the population size reaches $N$. We also show that if $1\ll S\ll N^{1-r}$, then the number of families of size at least $S$ is approximately $CNS^{-1/(1-r)}$, while if $N^{1-r}\ll S$ the distribution decays more rapidly than any power.