  The notion of (auto) catalytic networks has become a cornerstone in
understanding the possibility of a sudden dramatic increase of diversity in
biological evolution as well as in the evolution of social and economical
systems. Here we study catalytic random networks with respect to the final
outcome diversity of products. We show that an analytical treatment of this
longstanding problem is possible by mapping the problem onto a set of
non-linear recurrence equations. The solution of these equations show a crucial
dependence of the final number of products on the initial number of products
and the density of catalytic production rules. For a fixed density of rules we
can demonstrate the existence of a phase transition from a practically
unpopulated regime to a fully populated and diverse one. The order parameter is
the number of final products. We are able to further understand the origin of
this phase transition as a crossover from one set of solutions from a quadratic
equation to the other.
