Broadcasting in random recursive dags

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).