Robustness, Canalyzing Functions and Systems Design

Abstract: We study a notion of robustness of a 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. If the system is required to be robust against too many knockouts, then the output variable cannot distinguish reliably between input states and must be independent of the input. We study how many input states the output variable can distinguish as a function of the required level of robustness. Gibbs potentials allow a mechanistic description of the behaviour of the system after knockouts. Robustness imposes structural constraints on these potentials. We show that interaction families of Gibbs potentials allow to describe robust systems. Given a distribution of the input random variables and the Markov kernel describing the system, we obtain a joint probability distribution. Robustness implies a number of conditional independence statements for this joint distribution. 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. The decomposition corresponds to a primary decomposition of the conditional independence ideal and can be derived from more general results about generalized binomial edge ideals.