Activity in Boolean networks
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.
KeywordsBoolean 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.
- 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
- 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
- Aldana M, Coppersmith S, Kadanoff LP (2003) Boolean dynamics with random couplings. In: Perspectives and problems in nonlinear science, Springer, pp 23–89Google Scholar
- Kauffman SA (1993) The origins of order: self-organization and selection in evolution. Oxford University Press, OxfordGoogle Scholar
- Layne L, Dimitrova E, Matthew M (2012) Nested canalyzing depth and network stability. Bull Math Biol. doi: 10.1007/s11538-011-9692-y
- Mortveit HS, Reidys CM (2007) An introduction to sequential dynamical systems. Universitext, Springer. doi: 10.1007/978-0-387-49879-9
- Robert F (1986) Discrete iterations. A Metric Study. No. 6 in Springer Series in Computational Mathematics, SpringerGoogle Scholar