The Phase Transition in Random Catalytic Sets

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 systems. Here we study catalytic random networks with respect to the final outcome of products. We show that an analytical treatment of this longstanding problem is possible by mapping the problem onto a set of recurrence equations. The solution of these equations shows 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 one. The order parameter is the number of final products. We are able to further understand the origin of this phase transition as a transition from one set of solutions from a quadratic equation to the other.