The phase transition in random catalytic sets

TL;DR

This paper analyzes catalytic networks and demonstrates a phase transition from low to high diversity in product outcomes, depending on initial conditions and rule density, using nonlinear recurrence equations.

Contribution

It introduces an analytical framework for understanding phase transitions in random catalytic networks through nonlinear recurrence equations.

Findings

Existence of a phase transition in product diversity.

Final diversity depends on initial products and rule density.

Transition characterized by a change in solutions of quadratic equations.

Abstract

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.