Complex System Dynamics Modeled by a Group-Based Boolean Network

Abstract

Complex systems require unconventional analytical methods such as automaton and Boolean network models. This report presents a new kind of stochastic asynchronous Boolean network model with an updating function using combinatorial rules. In this model, nodes can be reactivated by two active nodes based on set operations within a power set. The characteristic dynamics of two variations of this model are presented, one constructed on power sets using four set operations and the other on Boolean group structures. These models both exhibit an emergent dynamic behavior that includes an initial decay, a persistent regime, and a final decay, characteristic to complex systems which have long lifetimes on stationary states followed by catastrophic behavior. The power set model also naturally gives rise to critical threshold dynamics which are characteristic to the Hill equation, an empirical description of enzyme activation. The group structure model leads naturally to a classification of microstates based on their internal structure (signature). While automorphic microstates must necessarily have the same signature, the converse has only been checked experimentally. Proving this conjecture could be the subject of future work, together with connections to quantum computing and dynamics on hypercubes which are representations of system microstates.