Natural Computing

, Volume 16, Issue 3, pp 427–439 | Cite as

Activity in Boolean networks

  • Abhijin Adiga
  • Hilton Galyean
  • Chris J. Kuhlman
  • Michael Levet
  • Henning S. Mortveit
  • Sichao Wu


In this paper we extend the notion of activity for Boolean networks introduced by Shmulevich and Kauffman (Phys Rev Lett 93(4):48701:1–4, 2004). In contrast to existing theory, we take into account the actual graph structure of the Boolean network. The notion of activity measures the probability that a perturbation in an initial state produces a different successor state than that of the original unperturbed state. It captures the notion of sensitive dependence on initial conditions, and provides a way to rank vertices in terms of how they may impact predictions. We give basic results that aid in the computation of activity and apply this to Boolean networks with threshold functions and nor functions for elementary cellular automata, d-regular trees, square lattices, triangular lattices, and the Erdős–Renyi random graph model. We conclude with some open questions and thoughts on directions for future research related to activity, including long-term activity.


Boolean networks Finite dynamical system Activity Sensitivity Network Sensitive dependence on initial conditions 



We thank our external collaborators and members of the Network Dynamics and Simulation Science Laboratory (NDSSL) for their suggestions and comments. We also thank two anonymous reviewers for valuable comments. This work has been partially supported by DTRA Grant HDTRA1-11-1-0016.


  1. Adiga A, Kuhlman CJ, Mortveit HS, Vullikanti AKS (2013) Sensitivity of diffusion dynamics to network uncertainty. In: Proceedings of the twenty-seventh AAAI conference on artificial intelligence (AAAI-13), July 14–18, 2013. Bellevue, Washington, USA, pp 2–8Google Scholar
  2. Adiga A, Galyean H, Kuhlman CJ, Levet M, Mortveit HS, Wu S (2015) Network structure and activity in Boolean networks. In: Kari J (ed) Cellular automata and discrete complex systems: proceedings of AUTOMATA 2015, Turku, Finland, June 8–10, 2015, Lecture Notes in Computer Science, vol 9099, pp 210–223, doi: 10.1007/978-3-662-47221-7_16
  3. Aldana M, Coppersmith S, Kadanoff LP (2003) Boolean dynamics with random couplings. In: Perspectives and problems in nonlinear science, Springer, pp 23–89Google Scholar
  4. Baetens JM, De Baets B (2010) Phenomenological study of irregular cellular automata based on Lyapunov exponents and Jacobians. Chaos 20:1–15. doi: 10.1063/1.3460362 MathSciNetCrossRefMATHGoogle Scholar
  5. Baetens JM, Van der Weeën P, De Baets B (2012) Effect of asynchronous updating on the stability of cellular automata. Chaos Solitons Fractals 45:383–394. doi: 10.1016/j.chaos.2012.01.002 CrossRefMATHGoogle Scholar
  6. Derrida B, Pomeau Y (1986) Random networks of automata: a simple annealed approximation. Europhys Lett 1:45–49CrossRefGoogle Scholar
  7. Fretter C, Szejka A, Drossel B (2009) Perturbation propagation in random and evolved Boolean networks. N J Phys 11:1–13. doi: 10.1088/1367-2630/11/3/033005 CrossRefGoogle Scholar
  8. Ghanbarnejad F, Klemm K (2012) Impact of individual nodes in Boolean network dynamics. EPL (Europhys Lett) 99(5):58,006CrossRefGoogle Scholar
  9. Goles E, Martinez S (1990) Neural and automata networks: dynamical behaviour and applications. Kluwer Academic Publishers, BerlinCrossRefMATHGoogle Scholar
  10. Kauffman SA (1969) Metabolic stability and epigenesis in randomly constructed genetic nets. J Theor Biol 22:437–467MathSciNetCrossRefGoogle Scholar
  11. Kauffman SA (1993) The origins of order: self-organization and selection in evolution. Oxford University Press, OxfordGoogle Scholar
  12. Kuhlman CJ, Mortveit HS (2014) Attractor stability in nonuniform Boolean networks. Theor Comput Sci 559:20–33. doi: 10.1016/j.tcs.2014.08.010 special volume: Non-uniform Cellular AutomataMathSciNetCrossRefMATHGoogle Scholar
  13. Layne L, Dimitrova E, Matthew M (2012) Nested canalyzing depth and network stability. Bull Math Biol. doi: 10.1007/s11538-011-9692-y
  14. Luo JX, Turner MS (2012) Evolving sensitivity balances Boolean networks. PLoS One 7(e36):010. doi: 10.1371/journal.pone.0036010 Google Scholar
  15. Mortveit HS, Reidys CM (2007) An introduction to sequential dynamical systems. Universitext, Springer. doi: 10.1007/978-0-387-49879-9
  16. Pomerance A, Ott E, Girvan M, Losert W (2009) The effect of network topology on the stability of discrete state models of genetic control. Proc Nat Acad Sci 106(20):8209–8214CrossRefGoogle Scholar
  17. Ribeiro AS, Kauffman SA (2007) Noisy attractors and ergodic sets in models of gene regulatory networks. J Theor Biol 247:743–755MathSciNetCrossRefGoogle Scholar
  18. Robert F (1986) Discrete iterations. A Metric Study. No. 6 in Springer Series in Computational Mathematics, SpringerGoogle Scholar
  19. Serra R, Villani M, Barbieri A, Kauffman S, Colacci A (2010) On the dynamics of random Boolean networks subject to noise: attractors, ergodic sets and cell types. J Theor Biol 265(2):185–193MathSciNetCrossRefGoogle Scholar
  20. Shmulevich I, Kauffman SA (2004) Activities and sensitivities in Boolean network models. Phys Rev Lett 93(4):048701:1–4CrossRefGoogle Scholar
  21. Shmulevich I, Lähdesmäki H, Dougherty ER, Astola J, Zhang W (2003) The role of certain post classes in Boolean network models of genetic networks. Proc Nat Acad Sci 100(19):10,734–10,739CrossRefGoogle Scholar
  22. Xiao Y, Dougherty ER (2007) The impact of function perturbations in Boolean networks. Bioinformatics 23(10):1265–1273CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  • Abhijin Adiga
    • 1
  • Hilton Galyean
    • 2
  • Chris J. Kuhlman
    • 1
  • Michael Levet
    • 4
  • Henning S. Mortveit
    • 1
    • 3
  • Sichao Wu
    • 1
  1. 1.Network Dynamics and Simulation Science Laboratory, VBIVirginia TechBlacksburgUSA
  2. 2.Department of PhysicsVirginia TechBlacksburgUSA
  3. 3.Department of MathematicsVirginia TechBlacksburgUSA
  4. 4.Department of Computer ScienceUniversity of South CarolinaColumbiaUSA

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