Reversible Boolean networks II. Phase transitions, oscillations, and localstructures

We continue our consideration of a class of models describing the reversibl e dynamics of N Boolean variables, each with K inputs. We investigate in de tail the behavior of the Hamming distance as well as of the distribution of orbit lengths as N and K are varied. We present numerical evidence for a p hase transition in the behavior of the Hamming distance at a critical value K-c approximate to 1.62 and also an analytic theory that yields the exact bounds 1.5 < K-c < 2. We also discuss the large oscillations that we observ e in the Hamming distance for K < K-c as a function of time as well as in t he distribution of cycle lengths as a function of cycle length for moderate K both greater than and less than K,. We propose that local structures, or subsets of spins whose dynamics are not fully coupled to the other spins i n the system, play a crucial role in generating these oscillations. The sim plest of these structures are linear chains, called linkages, and rings, ca lled circuits. We discuss the properties of the linkages in some detail, an d sketch the properties of circuits. We argue that the observed oscillation phenomena can be largely understood in terms of these local structures. (C ) 2001 Published by Elsevier Science B.V.