Theory in Biosciences

, Volume 133, Issue 2, pp 63–78 | Cite as

Robustness, canalyzing functions and systems design

  • Johannes Rauh
  • Nihat Ay
Original Paper


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.


Robustness 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.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Max Planck Institute for Mathematics in the SciencesLeipzigGermany
  2. 2.Santa Fe InstituteSanta FeUSA

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