Robustness, canalyzing functions and systems design

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.