Function perturbation of mix-valued logical networks with impacts on limit sets

In this paper, function perturbation of mix-valued logical networks is first proposed and investigated via semi-tensor product (STP) of matrices. Motivated by the concept of one-bit perturbation in Boolean networks, the definition of general perturbation in mix-valued logical networks is presented and the algebraic expression of the perturbed networks is given by STP. The impacts of function perturbation on fixed points and limit cycles are discussed by analyzing the changes of transition matrix in algebraic form. In addition to identifying one perturbation in mix-valued logical networks, a new way to identify multi-perturbation is given. This new method can be used in producing or removing fixed points and thus exerting effects on limit cycles. All of the theoretical results also hold for Boolean and k-valued logical networks. Finally, the results of perturbation identification are applied to the WNT5A gene network, which shows broad prospects of application.