Boolean networks

Several lines of development from the cybernetics movement (notably in immunology, genetics and management science) led in the 1960s to a study of random Boolean networks—notably by Stuart Kauffman and Crayton Walker. Such systems are like cellular automata on networks, except for the fact that when they are set up each node has a rule that is randomly chosen from all 2^{2s} possible ones with s inputs. With s = 2 class 2 behavior (see Chapter 6) tends to dominate. But for s > 2, the behavior one sees quickly approaches what is typical for a random mapping in which the network representing the evolution of the 2^{m} states of the m underlying nodes is itself connected essentially randomly (see page 963). (Attempts were made in the 1980s to study phase transitions as a function of s in analogy to ones in percolation and spin glasses.) Note that in almost all work on random Boolean networks averages are in effect taken over possible configurations, making it impossible to see anything like the kind of complex behavior that I discuss in cellular automata and many other systems in this book.