A Phase Transition and Stochastic Domination in Pippenger's Probabilistic Failure Model for Boolean Networks with Unreliable Gates

Details A Phase Transition and Stochastic Domination in Pippenger's Probabilistic Failure Model for Boolean Networks with Unreliable Gates A Phase Transition and Stochastic Domination in Pippenger's Probabilistic Failure Model for Boolean Networks with Unreliable Gates

2003-11-04

arxiv.org/abs/math/0311045v2

Mathematics

Probability

20 pages, 1 eps figure; made some cosmetic changes, corrected a few errors

Scientific paper

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.

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