Broadcasting in random recursive dags

• Authors: Simon Briend, Luc P Devroye and Gábor Lugosi.
• ESAIM - Probability and Statistics, Vol. 29, 184 - 203, April 2025.

A uniform k-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-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).