Asymptotic Behavior of Conjunctive Boolean Networks Over Weakly Connected Digraphs

Abstract

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 has 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 an 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.

Controllability of Conjunctive Boolean Networks With Application to Gene Regulation

“…First, recall that in the study of orbit-controlling sets, we considered only CBNs whose dependency graphs are strongly connected, because the periodic orbits of CBNs with weakly connected dependency graphs have not yet been fully characterized. Most recently, we have made some progress in this direction in [49], where we have investigated the asymptotic behavior of weakly connected CBNs. We plan to generalize the result of orbit-controllability obtained in this paper to a general weakly connected dependency graph.…”

Attractor detection and enumeration algorithms for Boolean networks

“…Gao et al characterized periodic attractors of a conjunctive BN (i.e., each Boolean function is AND of literals) whose underlying directed graph is strongly connected by establishing bijection between the set of periodic attractors and the set of binary necklaces (i.e., character strings over the binary set , where their all rotations are dealt with as equivalent) of a certain length [79] . Chen et al extended this study to BNs over weakly connected directed graph by applying network reduction [80] . It is to be noted that network reduction methods were utilized also in some other methods mentioned in this section.…”

Further Results on the Controllability of Boolean Control Networks

“…Some applications of STP include the analysis of controllability [6]- [9], observability [6,10]- [13], stability and stabilization [14]- [18], optimal control [19]- [21] and so on. Moreover, other kinds of BNs and BCNs, such as the conjunctive Boolean networks (CBNs) [22]- [26], are recently prevalence. It is no surprise that the research on the BNs and BCNs has become increasingly attractive and challenging.…”