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A Linear Representation of Dynamics of Boolean Networks

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2 Author(s)
Daizhan Cheng ; Academy of Mathematics and Systems Sciences, Chinese Academy of Sciences, Key Laboratory of Systems and Control, Beijing, China ; Hongsheng Qi

A new matrix product, called semi-tensor product of matrices, is reviewed. Using it, a matrix expression of logic is proposed, where a logical variable is expressed as a vector, a logical function is expressed as a multiple linear mapping. Under this framework, a Boolean network equation is converted into an equivalent algebraic form as a conventional discrete-time linear system. Analyzing the transition matrix of the linear system, formulas are obtained to show a) the number of fixed points; b) the numbers of cycles of different lengths; c) transient period, for all points to enter the set of attractors; and d) basin of each attractor. The corresponding algorithms are developed and used to some examples.

Published in:

IEEE Transactions on Automatic Control  (Volume:55 ,  Issue: 10 )