Observability and Detectability of Large-Scale Boolean Control Networks

  • Kuize ZhangEmail author
  • Lijun Zhang
  • Lihua Xie
Part of the Communications and Control Engineering book series (CCE)


In Chaps.  4 and  5, we investigated how to verify different notions of observability and detectability for Boolean control networks (BCNs), and also studied how to determine the initial state (current state) of a BCN according to a particular notion of observability (detectability). In addition, we proved that the problems of verifying these notions are all NP-hard in the number of nodes. Hence, these problems are generally intractable. Actually, in general, for a BCN with more than 30 nodes, one cannot obtain whether it is observable or detectable in a reasonable amount of time by using a personal computer (PC). Hence BCNs with more than 30 nodes can be regarded as large-scale.


  1. Akutsu T et al (1998) A system for identifying genetic networks from gene expression patterns produced by gene disruptions and overexpressions. Genome Inform Work Genome Inform 9:151–160Google Scholar
  2. Fauré A et al (2006) Dynamical analysis of a generic Boolean model for the control of the mammalian cell cycle. Bioinformatics 22(14):e124CrossRefGoogle Scholar
  3. Fornasini E, Valcher ME (2013) Observability, reconstructibility and state observers of Boolean control networks. IEEE Trans Autom Control 58(6):1390–1401MathSciNetCrossRefGoogle Scholar
  4. Ishii H, Tempo R, Bai EW (2012) A web aggregation approach for distributed randomized pagerank algorithms. IEEE Trans Autom Control 57(11):2703–2717MathSciNetCrossRefGoogle Scholar
  5. Klamt S et al (2006) A methodology for the structural and functional analysis of signaling and regulatory networks. BMC Bioinform 7(56):1–26Google Scholar
  6. Laschov D, Margaliot M (2013) Minimum-time control of Boolean networks. SIAM J Control Optim 51(4):2869–2892MathSciNetCrossRefGoogle Scholar
  7. Laschov D, Margaliot M, Even G (2013) Observability of Boolean networks: a graphtheoretic approach. Automatica 49(8):2351–2362MathSciNetCrossRefGoogle Scholar
  8. Li R, Yang M, Chu T (2015) Controllability and observability of Boolean networks arising from biology. Chaos: Interdiscip J Nonlinear Sci 25(2):023104–15MathSciNetCrossRefGoogle Scholar
  9. Liu Y-Y, Slotine J-J, Barabási A-L (2013) Observability of complex systems. Proc Natl Acad Sci 110(7):2460–2465MathSciNetCrossRefGoogle Scholar
  10. Louati A, Aufaure MA, Lechevallier Y (2013) Graph aggregation: application to social networks. In: Guan R et al (eds) Advances in theory and applications of high dimensional and symbolic data analysis, vol RNTI-E-25, Hermann, pp 157–177Google Scholar
  11. Veliz-Cuba A et al (2014) Steady state analysis of Boolean molecular network models via model reduction and computational algebra. BMC Bioinform 15(1):221Google Scholar
  12. Zhang K, Johansson KH (2018) Efficient observability verification for large-scale Boolean control networks. In: 2018 37th Chinese control conference (CCC), pp 560–567Google Scholar
  13. Zhang K, Zhang L, Xie L (2015) Invertibility and nonsingularity of Boolean control networks. Automatica 60:155–164MathSciNetCrossRefGoogle Scholar
  14. Zhang R et al (2008) Network model of survival signaling in large granular lymphocyte leukemia. Proc Natl Acad Sci USA 105:16308–13CrossRefGoogle Scholar
  15. Zhao Q (2005) A remark on scalar equations for synchronous boolean networks with biological applications by CF arrow, J Heidel, J Maloney, J Rogers. IEEE Trans Neural Netw 16(6):1715–1716Google Scholar
  16. Zhao Y, Ghosh BK, Cheng D (2016) Control of large-scale Boolean networks via network aggregation. IEEE Trans Neural Netw Learn Syst 27(7):1527–1536MathSciNetCrossRefGoogle Scholar
  17. Zhao Y, Kim J, Filippone M (2013) Aggregation algorithm towards large-scale Boolean network analysis. IEEE Trans Autom Control 58(8):1976–1985MathSciNetCrossRefGoogle Scholar

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© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.School of Electrical Engineering and Computer ScienceKTH Royal Institute of TechnologyStockholmSweden
  2. 2.School of Marine Science and TechnologyNorthwestern Polytechnical UniversityXi’anChina
  3. 3.School of Electrical and Electronic EngineeringNanyang Technological UniversitySingaporeSingapore

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