Hidden Order of Boolean Networks

Abstract—It is a common belief that the order of a Boolean network is mainly determined by its attractors, including fixed points and cycles. Using semi-tensor product (STP) of matrices and the algebraic state-space representation (ASSR) of Boolean networks, this paper reveals that in addition to this explicit order, there is certain implicit or hidden order, which is determined by the fixed points and limit cycles of their dual networks. The structure and certain properties of dual networks are investigated. Instead of a trajectory, which describes the evolution of a state, hidden order provides a global picture to describe the evolution of the overall network. It is our conjecture that the order of networks is mainly determined by hidden attractors via their corresponding hidden orders. Finally, the hidden order of kvalued logical networks is also explored. The previously obtained results about Boolean networks are extended to the k-valued case.

[1]  Daizhan Cheng,et al.  Input-State Approach to Boolean Networks , 2009, IEEE Transactions on Neural Networks.

[2]  D. Cheng,et al.  An Introduction to Semi-Tensor Product of Matrices and Its Applications , 2012 .

[3]  Anna Gerber,et al.  At Home In The Universe , 2016 .

[4]  Daizhan Cheng,et al.  Canonical Form of Boolean Networks , 2019, 2019 Chinese Control Conference (CCC).

[5]  W. Feng,et al.  observability ? , 2013 .

[6]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.

[7]  Stuart A. Kauffman,et al.  ORIGINS OF ORDER , 2019, Origins of Order.

[8]  J. Urry.  Complexity , 2006, Interpreting Art.

[9]  G. Winskel.  What Is Discrete Mathematics , 2007 .

[10]  Daizhan Cheng,et al.  State–Space Analysis of Boolean Networks , 2010, IEEE Transactions on Neural Networks.

[11]  S. Kauffman.  At Home in the Universe: The Search for the Laws of Self-Organization and Complexity , 1995 .

[12]  Daizhan Cheng,et al.  Invariant Subspace Approach to Boolean (Control) Networks , 2021, ArXiv.

[13]  Ettore Fornasini,et al.  Observability, Reconstructibility and State Observers of Boolean Control Networks , 2013, IEEE Transactions on Automatic Control.

[14]  D. Cheng,et al.  Analysis and control of Boolean networks: A semi-tensor product approach , 2010, 2009 7th Asian Control Conference.