The Local Convergence of Boolean Networks With Disturbances

This brief is devoted to studying the local convergence of Boolean networks (BNs) with disturbances. On the one hand, the algebraic form of a BN with <inline-formula> <tex-math notation="LaTeX">${n}$ </tex-math></inline-formula> nodes and <inline-formula> <tex-math notation="LaTeX">${n}$ </tex-math></inline-formula> disturbances is obtained by semi-tensor product (STP) of matrices and, based on the algebraic expression, some conditions of the local convergence are presented. On the other hand, by the discrete derivative of Boolean functions at a fixed point, a new matrix with dimension <inline-formula> <tex-math notation="LaTeX">${n\times n}$ </tex-math></inline-formula> (not <inline-formula> <tex-math notation="LaTeX">${2^{n}\times 2^{n}}$ </tex-math></inline-formula>) is constructed to analyze the local convergence and it implies that the computational complexity is dramatically reduced from <inline-formula> <tex-math notation="LaTeX">${\mathcal {O}(2^{2n})}$ </tex-math></inline-formula> to <inline-formula> <tex-math notation="LaTeX">${\mathcal {O}(n^{2})}$ </tex-math></inline-formula> compared with the method of STP. Finally, examples are provided to illustrate the effectiveness of the obtained results.