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 $1/2+o(1)$.

翻译：均匀 $k$-{\sc dag} 通过从现有节点中均匀随机选择 $k$ 个父节点，推广了均匀随机递归树。它起始于 $k$ 个“根节点”。每个根节点被分配一个比特。这些比特通过一个有噪信道传播。父节点的比特以概率 $p$ 发生翻转，并采取多数表决。当所有节点都接收到它们的比特后，$k$-{\sc dag} 被展示出来，但不标识根节点。目标是估计根节点中的多数比特。我们确定了 $p$ 作为 $k$ 的函数的阈值，在该阈值以下，所有节点间的多数表决规则产生的误差为 $c+o(1)$，其中 $c<1/2$。在该阈值以上，多数表决规则出错的概率为 $1/2+o(1)$。