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A Phase Transition and Stochastic Domination in Pippenger's Probabilistic Failure Model for Boolean Networks with Unreliable Gates | | Maxim Raginsky
; | | Date: |
4 Nov 2003 | | Subject: | Probability; Combinatorics; Mathematical Physics; Disordered Systems and Neural Networks MSC-class: 82B26; 94C10; 60K10; 05C20; 05C80 | math.PR cond-mat.dis-nn math-ph math.CO math.MP | | Abstract: | 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-ErdH{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 $n oinfty$, 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. | | Source: | arXiv, math.PR/0311045 | | Services: | Forum | Review | PDF | Favorites | |
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