A Framework of Pinning Control for Nonperiodical Stable Behaviors of Large-Scale Asynchronous Boolean Networks

In this article, two pinning control (PC) schemes are proposed to achieve nonperiodical stable behaviors for asynchronous Boolean networks (BNs), from the aspects of state transition digraph (STG) and dependence digraph (DD). First, under the framework of algebraic state-space representation of asynchronous BNs, a nonuniform PC is proposed based on STG and feedback vertex set. The nonuniform pinning nodes (PNs) are determined under the transformation of certain columns of the state transition matrices. Due to the high computational complexity of using STG, a uniform PC is further proposed based on the DD of asynchronous BNs, where PNs are easily found using a feedback arc set (FAS). Compared with the nonuniform PC with computational complexity
$O(n2^{2n})$
(
$n$
is the size of network), the uniform PC has advantages of lower computational complexity
$O(n^{2}+n2^{K})$
(
$K$
is the largest indegree of in-neighbors). Finally, simulations on gene networks with different sizes are given to illustrate the effectiveness of the obtained results that only almost 1%–33% nodes are needed. Especially, as for a network with 321 genes, only two nodes (1%) are needed, which well reflects the core idea of PC approach.