Broadcasting on Bounded Degree DAGs

We study the following generalization of the well-known model of broadcasting on trees. Consider an infinite directed acyclic graph (DAG) with a unique source node $X$. Let the collection of nodes at distance $k$ from $X$ be called the $k$th layer. At time zero, the source node is given a bit. At time $k\geq 1$, each node in the $(k-1)$th layer inspects its inputs and sends a bit to its descendants in the $k$th layer. Each bit is flipped with a probability of error $\delta \in \left(0,\frac{1}{2}\right)$ in the process of transmission. The goal is to be able to recover the original bit with probability of error better than $\frac{1}{2}$ from the values of all nodes at an arbitrarily deep layer $k$. Besides its natural broadcast interpretation, the DAG broadcast is a natural model of noisy computation. Some special cases of the model represent information flow in biological networks, and other cases represent noisy finite automata models. We show that there exist DAGs with bounded degree and layers of size $\omega(\log(k))$ that permit recovery provided $\delta$ is sufficiently small and find the critical $\delta$ for the DAGs constructed. Our result demonstrates a doubly-exponential advantage for storing a bit in bounded degree DAGs compared to trees. On the negative side, we show that if the DAG is a two-dimensional regular grid, then recovery is impossible for any $\delta \in \left(0,\frac{1}{2}\right)$ provided all nodes use either AND or XOR for their processing functions.