We study Pippenger's model of Boolean networks with unreliable gates. In this
model, the conditional probability that a particular gate fails, given the
failure status of any subset of gates preceding it in the network, is bounded
from above by some $\epsilon$. We show that if we pick a Boolean network with
$n$ gates at random according to the Barak-Erd\H{o}s model of a random acyclic
digraph, such that the expected edge density is $c n^{-1}\log n$, and if
$\epsilon$ is equal to a certain function of the size of the largest reflexive,
transitive closure of a vertex (with respect to a particular realization of the
random digraph), then Pippenger's model exhibits a phase transition at $c=1$.
Namely, with probability $1-o(1)$ as $n\to\infty$, we have the following: for
$0 \le c \le 1$, the minimum of the probability that no gate has failed, taken
over all probability distributions of gate failures consistent with Pippenger's
model, is equal to $o(1)$, whereas for $c >1$ it is equal to
$\exp(-\frac{c}{e(c-1)}) + o(1)$. We also indicate how a more refined analysis
of Pippenger's model, e.g., for the purpose of estimating probabilities of
monotone events, can be carried out using the machinery of stochastic
domination.