Dynamics of random boolean networks

Boolean networks are models of genetic regulatory networks. S. Kauffman based many of his claims about spontaneous self-organization in complex systems on simulations of randomly constructed Boolean networks. Some of these claims are precise mathematical statements. We analyze these statements using combinatorial methods and show that there is partial agreement with some of Kauffman's conclusions, but in other cases there is disagreement. Our key finding is an algebraic parameter that determines the likelihood of ordered behavior in a random Boolean network. There is a threshold such that when the parameter is less than the threshold, ordered behavior is prevalent, and when it is greater than the threshold, chaotic behavior is highly likely. When the parameter equals the threshold, some forms of ordered behavior persist, but others do not.