Asymptotic Behavior of Conjunctive Boolean Networks over Weakly Connected Digraphs

A conjunctive Boolean network (CBN) is a finite state dynamical system, whose variables take values from a binary set, and the value update rule for each variable is a Boolean function consisting only of logic and operations. We investigate the asymptotic behavior of CBNs by computing their periodic orbits. When the underlying digraph is strongly connected, the periodic orbits of the associated CBN have been completely understood, one-to-one corresponding to binary necklaces of a certain length given by the loop number of the graph. We characterize in the paper the periodic orbits of CBNs over arbitrary weakly connected digraphs. We establish, among other things, a new method to investigate their asymptotic behavior. Specifically, we introduce a graphical approach, termed system reduction, which turns the underlying digraph into a special weakly connected digraph, whose strongly connected components are all cycles. We show that the reduced system uniquely determines the asymptotic behavior of the original system. Moreover, we provide a constructive method for computing the periodic orbit of the reduced system, which the system will enter for a given, but arbitrary initial condition.