  We introduce the nested canalyzing depth of a function, which measures the
extent to which it retains a nested canalyzing structure. We characterize the
structure of functions with a given depth and compute the expected activities
and sensitivities of the variables. This analysis quantifies how canalyzation
leads to higher stability in Boolean networks. It generalizes the notion of
nested canalyzing functions (NCFs), which are precisely the functions with
maximum depth. NCFs have been proposed as gene regulatory network models, but
their structure is frequently too restrictive and they are extremely sparse. We
find that functions become decreasingly sensitive to input perturbations as the
canalyzing depth increases, but exhibit rapidly diminishing returns in
stability. Additionally, we show that as depth increases, the dynamics of
networks using these functions quickly approach the critical regime, suggesting
that real networks exhibit some degree of canalyzing depth, and that NCFs are
not significantly better than functions of sufficient depth for many
applications of the modeling and reverse engineering of biological networks.
