Phase Transitions in Random Networks with Multiple States

The critical boundaries separating ordered from chaotic behavior in randomly wired S-state networks are calculated. These networks are a natural generalization of random Boolean net and are proposed as an extended approach to genetic regulatory systems, sets of cells in different states or collectives of agents engaged into a set of S possible tasks. An order parameter for the transition is computed and analyzed and a Lyapunov exponent for the sustem is defined. The relevance of these networks to biology, their relationships with standard cellular automata and possible extensions are outlined.