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.
Acknowledgements
The authors would like to thank the Editor and the anonymous referee for their constructive comments which improved the quality of this paper.
Notes
No potential conflict of interest was reported by the authors.
1 The so-called cascading myopic best response adjustment is to update each player’s strategy by choosing the best strategy among its neighbours, that is, the strategy of player i at time , denoted by , is selected from the best strategy among the strategies of its neighbours j at time t.