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Issue No.04 - July-Aug. (2013 vol.10)
pp: 927-938
Chao Luo , Fac. of Electron. Inf. & Electr. Eng., Dalian Univ. of Technol., Dalian, China
Xingyuan Wang , Fac. of Electron. Inf. & Electr. Eng., Dalian Univ. of Technol., Dalian, China
ABSTRACT
In this paper, dynamics of asynchronous multiple-valued networks (AMVNs) are investigated based on linear representation. By semitensor product of matrices, we convert AMVNs into the discrete-time linear representation. A general formula to calculate all of network transition matrices of a specific AMVN is achieved. A necessary and sufficient algebraic criterion to determine whether a given state belongs to loose attractors of length s is proposed. Formulas for the numbers of attractors in AMVNs are provided. Finally, algorithms are presented to detect all of the attractors and basins. Examples are shown to demonstrate the feasibility of the proposed scheme.
INDEX TERMS
network theory (graphs), genetics, matrix algebra, attractors, algebraic representation, asynchronous multiple-valued network dynamics, semitensor product, AMVN, discrete-time linear representation, network transition matrices, loose attractors, basins, Vectors, Computational biology, Bioinformatics, Biological system modeling, Heuristic algorithms, Data structures, Boolean functions, dynamics, Multiple-valued networks, asynchronous stochastic update, algebraic representation
CITATION
Chao Luo, Xingyuan Wang, "Algebraic Representation of Asynchronous Multiple-Valued Networks and Its Dynamics", IEEE/ACM Transactions on Computational Biology and Bioinformatics, vol.10, no. 4, pp. 927-938, July-Aug. 2013, doi:10.1109/TCBB.2013.112
REFERENCES
[1] S.A. Kauffman, "Metabolic Stability and Epigenesis in Randomly Constructed Genetic Nets," J. Theoretical Biology, vol. 22, pp. 437-467, 1969.
[2] S. Hashemikhabir, E.S. Ayaz, Y. Kavurucu, T. Can, and T. Kahveci, "Large-Scale Signaling Network Reconstruction," IEEE/ACM Trans. Computational Biology and Bioinformatics, vol. 9, no. 6, pp. 1696-1708, Nov. 2012.
[3] B. Drossel, T. Mihaljev, and F. Greil, "Number and Length of Attractors in a Critical Kauffman Model with Connectivity One," Physical Rev. Letters, vol. 94, p. 088701, 2005.
[4] E. Dubrova and M. Teslenko, "A SAT-Based Algorithm for Finding Attractors in Synchronous Boolean Networks," IEEE/ACM Trans. Computational Biology and Bioinformatics, vol. 8, no. 5, pp. 1393-1399, Sept. 2011.
[5] Z.X. Mai and H.Y. Liu, "Boolean Network-Based Analysis of the Apoptosis Network: Irreversible Apoptosis and Stable Surviving," J. Theoretical Biology, vol. 259, pp. 760-769, 2009.
[6] J. Heidel, J. Maloney, J. Farrow, and J. Rogers, "Finding Cycles in Synchronous Boolean Networks with Applications to Biochemical Systems," Int'l J. Bifurcation Chaos, vol. 13, pp. 535-552, 2003.
[7] M. Aldana, S. Coppersmith, and L.P. Kadanoff, "Boolean Dynamics with Random Couplings," Perspectives and Problems in Nonlinear Science, pp. 23-89, Springer, 2003.
[8] E. Dubrova, "Random Multiple-Valued Networks: Theory and Application," Proc. Int'l Symp. Multiple-Valued Logic, pp. 27-33, 2006.
[9] A. Adamatzky, "On Dynamically Non-Trivial Three-Valued Logics: Oscillatory and Bifurcatory Species," Chaos, Solitons & Fractals, vol. 18, pp. 917-936, 2003.
[10] J.J. Hopfield, "Neural Networks and Physical Systems with Emergent Collective Computational Abilities," Proc. the Nat'l Academy of Sciences USA, vol. 79, pp. 2554-2558, 1982.
[11] F. Dellaert and R. Beer, "Toward an Evolvable Model of Development for Autonomous Agent Synthesis," Proc. Fourth Int'l Workshop the Synthesis and Simulation of Living Systems (Alife 4), pp. 246-257, 1994.
[12] I. Harvey and T. Bossomaier, "Time out of Joint: Attractors in Asynchronous Random Boolean Networks," Proc. Fourth European Conf. Artificial Life, pp. 67-75, 1997.
[13] E.A. Di Paolo, "Rhythmic and Non-Rhythmic Attractors in Asynchronous Random Boolean Networks," BioSystems, vol. 59, pp. 185-195, 2001.
[14] A. Garg, I. Xenarios, L. Mendoza, and G. De Micheli, "An Efficient Method for Dynamic Analysis of Gene Regulatory Networks and In Silico Gene Perturbation Experiments," Proc. 11th Ann. Int'l Conf. Research in Computational Molecular Biology, pp. 62-76, 2007.
[15] M. Bertrand and T. Christof, "Critical Values in Asynchronous Random Boolean Networks," Advances in Artificial Life, vol. 2801, pp. 367-376, 2003.
[16] A. Shreim, A. Berdahl, F. Greil, J. Davidsen, and M. Paczuski, "Attractor and Basin Entropies of Random Boolean Networks under Asynchronous Stochastic Update," Physical Rev. E, vol. 82, article 035102, 2010.
[17] T. Skodawessely and K. Klemm, "Finding Attractors in Asynchronous Boolean Dynamics," Advances in Complex Systems, vol. 14, pp. 439-449, 2011.
[18] J. Hallinan and J. Wiles, "Asynchronous Dynamics of an Artificial Genetic Regulatory Network," Proc. Ninth Int'l Conf. the Simulation and Synthesis of Living Systems, pp. 399-403, 2004.
[19] F. Greil and B. Drossel, "Dynamics of Critical Kauffman Networks under Asynchronous Stochastic Update," Physical Rev. Letters, vol. 95, article 048701, 2005.
[20] X.T. Deng, H.M. Geng, and M.T. Matache, "Dynamics of Asynchronous Random Boolean Networks with Asynchrony Generated by Stochastic Processes," Biosystems, vol. 88, pp. 16-34, 2007.
[21] F. Greil, B. Drossel, and J. Sattler, "Critical Kauffman Networks under Deterministic Asynchronous Update," New J. Physics, vol. 9, article 373, 2007.
[22] L.J. Steggles, "Abstracting Asynchronous Multi-Valued Networks: An Initial Investigation," arXiv, preprint, arXiv:1108.3433, 2011.
[23] E.M. Clarke, O. Grumberg, and D.E. Long, "Model Checking and Abstractions," ACM Trans. Programming Languages and Systems, vol. 16, pp. 1512-1542, 1994.
[24] R. Banks and L.J. Steggles, "A High-Level Petri Net Framework for Multi-Valued Genetic Regulatory Networks," J. Integrative Bioinformatics, vol. 4, article 60, 2007.
[25] A. Mishchenko and R. Brayton, "Simplification of Non-Deterministic Multi-Valued Networks," Proc. IEEE/ACM Int'l Conf. Computer-Aided Design (ICCAD '02), vol. 2, pp. 557-562, 2002.
[26] E. Dubrova, M. Teslenko, and M. Liu, "Finding Attractors in Synchronous Multiple-Valued Networks Using SAT-based Bounded Model Checking," Proc. IEEE 40th Int'l Symp. Multiple-Valued Logic, pp. 144-149, 2010.
[27] A. Garg, L. Mendoza, I. Xenarios, and G. De Micheli, "Modeling of Multiple Valued Gene Regulatory Networks," Proc. IEEE 29th Ann. Int'l Conf. Eng. Medicine and Biology Soc., pp. 1398-1404, 2007.
[28] A. Naldi, D. Thieffry, and C. Chaouiya, "Decision Diagrams for the Representation and Analysis of Logical Models of Genetic Networks," Proc. Int'l Conf. Computational Methods in Systems Biology, pp. 233-247, 2007.
[29] D. Cheng, H. Qi, and Z. Li, Analysis and Control of Boolean Networks: A Semi-Tensor Product Approach. Springer, 2011.
[30] D. Cheng and H. Qi, "Controllability and Observability of Boolean Control Networks," Automatica, vol. 45, pp. 1659-1667, 2009.
[31] F.F. Li, J.T. Sun, and Q. Wu, "Observability of Boolean Control Networks with State Time Delays," IEEE Trans. Neural Networks, vol. 22, no. 6, pp. 948-954, June 2011.
[32] F.F. Li and J.T. Sun, "Controllability of Probabilistic Boolean Control Networks," Automatica, vol. 47, pp. 2765-2771, 2011.
[33] C. Gershenson, "Classification of Random Boolean Networks," Proc. Eighth Int'l Conf. Artificial Life (Artificial Life '02), pp. 1-8, 2002.
[34] C. Gershenson, J. Broekaert, and D. Aerts, "Contextual Random Boolean Networks," Proc. Seventh European Conf. Advances in Artificial Life, pp. 615-624, 2003.
[35] C. Farrow, J. Heidel, J. Maloney, and J. Rogers, "Scalar Equations for Synchronous Boolean Networks with Biological Applications," IEEE Trans. Neural Networks, vol. 15, no. 2, pp. 348-354, Mar. 2004.
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