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$k$-有向无环图通过从现有节点中均匀随机选取$k$个父节点，推广了均匀随机递归树。该图始于$k$个“根节点”，每个根节点被赋予一个比特，这些比特通过噪声信道传播。父节点的比特以概率$p$发生翻转，并通过多数投票规则决定。当所有节点接收到其比特后，展示该$k$-有向无环图但不标识根节点。目标是对根节点中的多数比特进行估计。我们确定了作为$k$函数的阈值$p$：当$p$低于该阈值时，所有节点的多数投票规则产生的误差为$c+o(1)$，其中$c<1/2$；当$p$高于该阈值时，多数投票规则以$1/2+o(1)$的概率产生误差。