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Articles

Global convergence of serial Boolean networks based on algebraic representation

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Pages 633-647 | Received 25 Nov 2015, Accepted 28 Nov 2016, Published online: 26 Dec 2016
 

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 t+1, denoted by xi(t+1), is selected from the best strategy among the strategies of its neighbours j at time t.

Additional information

Funding

The work was supported by the National Natural Science Foundation of China under [grant number 61374065] and [grant number 61503225]; the Natural Science Fund for Distinguished Young Scholars of Shandong Province under [grant number JQ201613]; the Natural Science Foundation of Shandong Province under [grant number ZR2015FQ003].

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