🤖 AI Summary
This work investigates the solvability of root majority bit reconstruction in a random recursive $k$-DAG—constructed from $k$ initial root nodes, where each new node selects $k$ parents uniformly at random and inherits a bit via noisy majority voting (each parent’s bit is flipped independently with probability $p$). We establish the first exact phase transition characterization of noise tolerance for such $k$-DAGs: an explicit critical threshold $p_c(k) = frac{1}{2} - Theta(k^{-1/2})$ is derived. When $p < p_c(k)$, a global majority estimator achieves sub-majority error rate $c + o(1)$ for some $c < 1/2$; when $p > p_c(k)$, the error rate degrades to $1/2 + o(1)$, matching random guessing. This precisely identifies the fundamental information-theoretic limit for reliable broadcast reconstruction on this class of directed acyclic graphs.
📝 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 $1/2+o(1)$.