Correlation between noise resilience and output distribution of Boolean circuits
I have found some hints on my question in Moshe Looks, Competent Program Evolution, p. 31:

Most local perturbations of large random formulae have no effect – 96% of them for
arity five, and 91% for arity ten (the drop is a corollary of the increase in the percentage of
unique behaviors in a fixed size sample as the arity increase). This is a consequence of such
formulae typically being highly redundant – e.g., a large formula that computes a tautology
will likely remain a tautology when randomly perturbed (similarly a large subformula which
computes a tautology, etc.).

More resources would be highly appreciated if you know of any.