  Time- and state-discrete dynamical systems are frequently used to model
molecular networks. This paper provides a collection of mathematical and
computational tools for the study of robustness in Boolean network models. The
focus is on networks governed by $k$-canalizing functions, a recently
introduced class of Boolean functions that contains the well-studied class of
nested canalizing functions. The activities and sensitivity of a function
quantify the impact of input changes on the function output. This paper
generalizes the latter concept to $c$-sensitivity and provides formulas for the
activities and $c$-sensitivity of general $k$-canalizing functions as well as
canalizing functions with more precisely defined structure. A popular measure
for the robustness of a network, the Derrida value, can be expressed as a
weighted sum of the $c$-sensitivities of the governing canalizing functions,
and can also be calculated for a stochastic extension of Boolean networks.
These findings provide a computationally efficient way to obtain Derrida values
of Boolean networks, deterministic or stochastic, that does not involve
simulation.
