Robustness, canalyzing functions and systems design
- 196 Downloads
We study a notion of knockout robustness of a stochastic map (Markov kernel) that describes a system of several input random variables and one output random variable. Robustness requires that the behaviour of the system does not change if one or several of the input variables are knocked out. Gibbs potentials are used to give a mechanistic description of the behaviour of the system after knockouts. Robustness imposes structural constraints on these potentials. We show that robust systems can be described in terms of suitable interaction families of Gibbs potentials, which allows us to address the problem of systems design. Robustness is also characterized by conditional independence constraints on the joint distribution of input and output. The set of all probability distributions corresponding to robust systems can be decomposed into a finite union of components, and we find parametrizations of the components.
KeywordsRobustness Knockouts Conditional independence Markov kernels
This work has been supported by the Volkswagen Foundation and the Santa Fe Institute. Nihat Ay thanks David Krakauer and Jessica Flack for many stimulating discussions on robustness. We thank the reviewers for their detailed remarks which helped us to improve our manuscript.
- Birolini A (2010) Reliability engineering: theory and practice. 6th edn. Springer, BerlinGoogle Scholar
- Drton M, Sturmfels B, Sullivant S (2009) Lectures on algebraic statistics. 1st ed., ser. Oberwolfach Seminars. Birkhäuser, Basel, vol. 39.Google Scholar
- Kauffman S (1993) The origins of order: self-organization and selection in evolution. Oxford University Press, OxfordGoogle Scholar
- Kauffman S, Peterson C, Samuelsson B, Troein C (2003) Random Boolean network models and the yeast transcriptional network. PNAS 100(25):14796–14799Google Scholar
- Kauffman S, Peterson C, Samuelsson B, Troein C (2004) Genetic networks with canalyzing Boolean rules are always stable. PNAS 101(49):17102–17107Google Scholar
- Lauritzen SL (1996) Graphical Models. In: Oxford Statistical Science Series, 1st edn. Oxford University Press, OxfordGoogle Scholar
- Swanson I, Taylor A (2011) Minimal primes of ideals arising from conditional independence statements. J Algebra. Preprint: arXiv:1107.5604v3Google Scholar
- de Visser JAGM, Hermisson J, Wagner GP, Meyers LA, Bagheri-Chaichian H, Blanchard JL, Chao L, Cheverud JM, Elena SF, Fontana W, Gibson G, Hansen TF, Krakauer D, Lewontin RC, Ofria C, Rice SH, von Dassow G, Wagner A, Whitlock MC (2003) Perspective: evolution and detection of genetic robustness. Evolution 57(9):1959–1972Google Scholar