Distributed Pinning Set Stabilization of Large-Scale Boolean Networks

Abstract

In this article, we design the distributed pinning controllers to globally stabilize a Boolean network (BN), especially a sparsely connected large-scale one, toward a preassigned subset of states through the node-to-node message exchange. Given an appointed set of states, system nodes are partitioned into two disjoint parts, whose states are, respectively, fixed or arbitrary with respect to the given state set. With such node division, three parts of pinned nodes are selected and the state feedback controllers are accordingly designed such that the resulting BN satisfies all three conditions: the information of the arbitrary-state nodes cannot be passed to the others, the subgraph of network structure induced by the fixed-state nodes is acyclic, and the fixed states of these nodes are compatible with the preassigned state set. If the network structure of controlling BN is acyclic, the stabilizing time is revealed to be no more than the diameter of the resulting subgraph plus one. Based on this, we further design the pinning controllers with the constraint of stabilizing time. Noting that the overall procedure runs in an exponentially increasing time complexity with respect to the largest number of functional variables in the dynamics of pinned nodes, the sparsely connected large-scale BNs can be well addressed in a reasonable amount of time. Finally, we demonstrate the applications of our theoretical results in a T-cell large granular lymphocyte (T-LGL) survival signal network with 29 nodes and a T-cell receptor signaling network with 90 nodes.