Synchronization Analysis of Master-Slave Probabilistic Boolean Networks

In this paper, we analyze the synchronization problem of master-slave probabilistic Boolean networks (PBNs). The master Boolean network (BN) is a deterministic BN, while the slave BN is determined by a series of possible logical functions with certain probability at each discrete time point. In this paper, we firstly define the synchronization of master-slave PBNs with probability one, and then we investigate synchronization with probability one. By resorting to new approach called semi-tensor product (STP), the master-slave PBNs are expressed in equivalent algebraic forms. Based on the algebraic form, some necessary and sufficient criteria are derived to guarantee synchronization with probability one. Further, we study the synchronization of master-slave PBNs in probability. Synchronization in probability implies that for any initial states, the master BN can be synchronized by the slave BN with certain probability, while synchronization with probability one implies that master BN can be synchronized by the slave BN with probability one. Based on the equivalent algebraic form, some efficient conditions are derived to guarantee synchronization in probability. Finally, several numerical examples are presented to show the effectiveness of the main results.

[1]  Z. Jane Wang,et al.  Dynamic analysis of Probabilistic Boolean Network for fMRI study in Parkinson's Disease , 2008, 2008 30th Annual International Conference of the IEEE Engineering in Medicine and Biology Society.

[2]  Jinde Cao,et al.  Synchronization Control for Nonlinear Stochastic Dynamical Networks: Pinning Impulsive Strategy , 2012, IEEE Transactions on Neural Networks and Learning Systems.

[3]  Huijun Gao,et al.  Distributed Robust Synchronization of Dynamical Networks With Stochastic Coupling , 2014, IEEE Transactions on Circuits and Systems I: Regular Papers.

[4]  Jinde Cao,et al.  Synchronization in output-coupled temporal Boolean networks , 2014, Scientific Reports.

[5]  D H Zanette,et al.  Synchronization of Kauffman networks. , 2000, Physical review. E, Statistical, nonlinear, and soft matter physics.

[6]  Fangfei Li,et al.  Minimum energy control and optimal-satisfactory control of Boolean control network , 2013 .

[7]  Daizhan Cheng,et al.  Controllability and observability of Boolean control networks , 2009, Autom..

[8]  Attila Szolnoki,et al.  Optimal interdependence between networks for the evolution of cooperation , 2013, Scientific Reports.

[9]  Richard M. Murray,et al.  Consensus problems in networks of agents with switching topology and time-delays , 2004, IEEE Transactions on Automatic Control.

[10]  B. Drossel,et al.  Number and length of attractors in a critical Kauffman model with connectivity one. , 2004, Physical review letters.

[11]  Jianquan Lu,et al.  Some necessary and sufficient conditions for the output controllability of temporal Boolean control networks , 2014 .

[12]  Yang Liu,et al.  Controllability of probabilistic Boolean control networks based on transition probability matrices , 2015, Autom..

[13]  Koichi Kobayashi,et al.  An integer programming approach to optimal control problems in context-sensitive probabilistic Boolean networks , 2011, Autom..

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

[15]  Michael Margaliot,et al.  Controllability of Boolean control networks via the Perron-Frobenius theory , 2012, Autom..

[16]  Edward R. Dougherty,et al.  Probabilistic Boolean networks: a rule-based uncertainty model for gene regulatory networks , 2002, Bioinform..

[17]  Jinde Cao,et al.  Synchronization of master-slave Boolean networks with impulsive effects: Necessary and sufficient criteria , 2014, Neurocomputing.

[18]  Matjaz Perc,et al.  Spreading of cooperative behaviour across interdependent groups , 2013, Scientific Reports.

[19]  Carsten Peterson,et al.  Random Boolean network models and the yeast transcriptional network , 2003, Proceedings of the National Academy of Sciences of the United States of America.

[20]  Stuart A. Kauffman,et al.  ORIGINS OF ORDER IN EVOLUTION: SELF-ORGANIZATION AND SELECTION , 1992 .

[21]  Stuart A. Kauffman,et al.  The origins of order , 1993 .

[22]  Luo Chao,et al.  Controllability of Boolean control networks under asynchronous stochastic update with time delay , 2016 .

[23]  Rainer Breitling,et al.  What is Systems Biology? , 2010, Front. Physiology.

[24]  T. Ideker,et al.  A new approach to decoding life: systems biology. , 2001, Annual review of genomics and human genetics.

[25]  Kwang-Hyun Cho,et al.  Discovery of a kernel for controlling biomolecular regulatory networks , 2013, Scientific Reports.

[26]  Yuzhen Wang,et al.  Output feedback stabilization control design for Boolean control networks , 2013, Autom..

[27]  Tianguang Chu,et al.  Synchronization of Boolean networks with time delays , 2012, Appl. Math. Comput..

[28]  John Maloney,et al.  Scalar equations for synchronous Boolean networks with biological applications , 2004, IEEE Transactions on Neural Networks.

[29]  Fangfei Li.  Synchronization of coupled large-scale Boolean networks. , 2014, Chaos.

[30]  Daniel W. C. Ho,et al.  A new framework for consensus for discrete-time directed networks of multi-agents with distributed delays , 2012, Int. J. Control.

[31]  Jürgen Kurths,et al.  Cluster explosive synchronization in complex networks. , 2013, Physical review letters.

[32]  Jinde Cao,et al.  Synchronization in an Array of Output-Coupled Boolean Networks With Time Delay , 2014, IEEE Transactions on Neural Networks and Learning Systems.

[33]  Hong Liu,et al.  Controllability of time-delayed Boolean multiplex control networks under asynchronous stochastic update , 2014, Scientific Reports.

[34]  Tianguang Chu,et al.  Complete Synchronization of Boolean Networks , 2012, IEEE Transactions on Neural Networks and Learning Systems.

[35]  Michael K. Ng,et al.  An approximation method for solving the steady-state probability distribution of probabilistic Boolean networks , 2007, Bioinform..

[36]  Daizhan Cheng,et al.  Input-state incidence matrix of Boolean control networks and its applications , 2010, Syst. Control. Lett..

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

[38]  Daizhan Cheng,et al.  A Linear Representation of Dynamics of Boolean Networks , 2010, IEEE Transactions on Automatic Control.

[39]  Jinde Cao,et al.  Adaptive synchronization of uncertain dynamical networks with delayed coupling , 2008 .

[40]  B. Samuelsson,et al.  Superpolynomial growth in the number of attractors in Kauffman networks. , 2003, Physical review letters.

[41]  Jürgen Kurths,et al.  Networks from Flows - From Dynamics to Topology , 2014, Scientific Reports.

[42]  S. Kauffman.  Metabolic stability and epigenesis in randomly constructed genetic nets. , 1969, Journal of theoretical biology.

[43]  Yao-Chen Hung,et al.  Stochastic coupling of two random Boolean networks , 2005 .