Global convergence of serial Boolean networks based on algebraic representation

Abstract

This paper analyzes the global convergence of serial Boolean networks (SBNs) via the semi-tensor product of matrices, and presents some new results. Firstly, an algebraic representation is obtained for SBNs, and an algorithm is established for the conversion between the algebraic representations of SBNs and the corresponding Boolean networks. Secondly, the non-equivalence of global convergence between SBNs and the corresponding Boolean networks is revealed, although they have the same fixed points. Thirdly, a necessary and sufficient condition is presented for the global convergence of SBNs. Finally, the obtained results are applied to the evolutionary behaviour analysis of evolutionary networked games with cascading myopic best response adjustment.