This is a little section designed to explain the model dynamics in detail, and discuss the underlying dynamical system for historical reason. Originally, the model was introduced as a model to study the origin of life. We imagine a prebiotic pond in early earth. Say the pond has S chemical species. Each species can catalyse any other species by probability p. And the catalysis can be represented by a graph edge. An edge from chemical i to chemical j implies j catalyses creation of i. We assume that this pond is a well contained pond where the system is allowed to reach equilibrium of the catalytic reactions involved with all species.

Let x_i be the relative population of ith chemical species. The system evolves by the following dynamical system/p>

$$\dot{x_i}= \sum_{j=1}^S C_{ij}x_j-x_i \sum_{j,k=1}^S C_{jk}x_k$$

where C is the adjacency matrix of the system. We can show the following:

$$ \vec{x}(t) \xrightarrow{ t \rightarrow \infty } \vec{X}$$

Here, X is the eigenvector of the adjacency matrix C with the largest eigenvalue

$$CX=\lambda_1 X$$

Thus, the dynamics of the catalytic model is actually a non linear model whose solution is simply the perron frobenius eigenvector of the adjacency matrix.