Broadcasting in random recursive dags

Simon Briend; Luc Devroye; Gabor Lugosi

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arXiv · 2306.01727

Broadcasting in random recursive dags

Simon Briend·Luc Devroye·Gabor Lugosi

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Abstract

A uniform $k$-{\sc dag} generalizes the uniform random recursive tree by picking $k$ parents uniformly at random from the existing nodes. It starts with $k$ ''roots''. Each of the $k$ roots is assigned a bit. These bits are propagated by a noisy channel. The parents' bits are flipped with probability $p$, and a majority vote is taken. When all nodes have received their bits, the $k$-{\sc dag} is shown without identifying the roots. The goal is to estimate the majority bit among the roots. We identify the threshold for $p$ as a function of $k$ below which the majority rule among all nodes yields an error $c+o(1)$ with $c<1/2$. Above the threshold the majority rule errs with probability $1/2+o(1)$.

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BibTeX RIS

Simon Briend, Luc Devroye, Gabor Lugosi. 2023-06-02. Broadcasting in random recursive dags. https://arxiv.org/abs/2306.01727

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