A random Boolean network (RBN) is a system of n binary-state nodes with k input connections to each node and a binary-valued response function describing the dependence of the state of a node on its input nodes. One can think of the state of a node representing active or inactive status, and the response function of a node as its regulatory mechanism. Randomness appears in this network in two ways: the configuration of inputs for each node is chosen randomly, and the regulating response function for each node is chosen randomly from the set of all binary-valued functions with a $p$ bias towards active states.

Random Boolean networks were originally developed by Kauffman (1969) as a model for genetic regulatory networks. We will show that one can use the threshold contact process (with threshold 1) to approximate the dynamics of an RBN on a fixed underlying random graph. In the limit as the number of nodes in the network goes to infinity, there is a phase transition in its behavior. We identify the phase transition curve in terms of the parameters of the model, and show that the system either settles into a steady state fairly quickly, or exhibits stochastic behaviour for an exponentially long period of time.