2007 Current Developments in Mathematical Biology  
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
more » ... 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. Theorem 4. There is a constant r such that for all When λ ≤ 1, r = 1, and when λ > 1, r < 1. Corollary 1. The expected number of α log n-ineffective gates in a random Boolean network is asymptotic to rn. A stronger result is Corollary 2. The number of α log n-ineffective gates in almost all Boolean networks is asymptotic to rn. That is, there is a function ε(n) such that ε(n) → 0 and, letting the random variable X n be the number of α log n-ineffective gates in a random Boolean network with n gates, lim n→∞ pr(|X n − rn| ≤ nε(n)) = 1. Proof. By the previous corollary, E(X n ) = rn + nε(n),
doi:10.1142/9789812706799_0002 fatcat:6eatjzjz4bevloewtffzmrqazu