Robustness, Canalyzing Functions and Systems Design

TL;DR

This paper investigates the concept of robustness in systems modeled by Markov kernels, analyzing how robustness constraints affect the system's ability to distinguish input states and describing the structure of robust systems using Gibbs potentials and algebraic methods.

Contribution

It introduces a formal framework for robustness in systems with input and output variables, characterizes the structure of robust systems via Gibbs potentials, and provides algebraic decompositions of the space of robust distributions.

Findings

Robust systems impose specific conditional independence constraints.

The set of all robust distributions decomposes into finitely many algebraic components.

Interaction families of Gibbs potentials effectively describe robust systems.

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.