Distributed Pinning Control: Stabilizing Large Boolean Networks Subjected to Perturbations

Stability maintenance in systems refers to the capacity to preserve inherent stability characteristics. In this article, stability maintenance of large boolean networks (BNs) subjected to perturbations is investigated using a distributed pinning control (PC) strategy. The concept of edge removal as a form of perturbation is introduced, and several criteria for achieving global stability are established. Two forms of distributed PCs, one implemented before perturbation occurs and the other after, are introduced. It is noteworthy that the designs of the controllers are solely dependent on the system’s in-neighbors. The proposed method significantly decreases the computational complexity, reducing it from
$O(2^{2|\texttt{V}|})$
to
$O(|\texttt{V}|+ |\texttt{E}| + \kappa \cdot 2^{K})$
, where
$|\texttt{V}|, |\texttt{E}|$
denotes the cardinality of vertices and arcs of the adjacent graph of BN,
$\kappa$
is the number of the pinning nodes, and
$K$
represents the maximum in-degree of the network. In the worst-case scenario, the computational complexity is bounded by
$O(|\texttt{V}|+ |\texttt{E}| + \kappa \cdot 2^{|\texttt{V}|})$
. To validate the effectiveness of the proposed methods, results from multiple gene networks are presented, including a model representing the human rheumatoid arthritis synovial fibroblast, among which only 12 of the 359 nodes are deemed essential.