Broadcasting on Random Directed Acyclic Graphs

We study a generalization of the well-known model of broadcasting on trees to the case of directed acyclic graphs (DAGs). At time $0$, a source vertex $X$ sends out a uniform bit along binary symmetric channels to a set of vertices called layer $1$. Each vertex except $X$ is assumed to have indegree $d$. At time $k\geq1$, vertices at layer $k$ apply $d$-input Boolean processing functions to their received bits and send out the results to vertices at layer $k+1$. We say that broadcasting is possible if we can reconstruct $X$ with probability of error bounded away from $1/2$ using knowledge of all vertices at an arbitrarily deep layer $k$. This question is also related to models of reliable computation and storage, and information flow in biological networks. In this paper, we study randomly constructed DAGs, for which we show that broadcasting is only possible if the noise level is below a certain (degree and function dependent) critical threshold. For $d\geq3$, and random DAGs with layer sizes $\Omega(\log k)$ and majority processing functions, we identify the critical threshold. For $d=2$, we establish a similar result for NAND processing functions. We also prove a partial converse for odd $d\geq3$ illustrating that the identified thresholds are impossible to improve by selecting different processing functions if the decoder is restricted to using a single vertex. Finally, for any noise level, we construct explicit DAGs (using expander graphs) with bounded degree and layer sizes $\Theta(\log k)$ admitting reconstruction. In particular, we show that such DAGs can be generated in deterministic quasi-polynomial time or randomized polylogarithmic time in the depth. These results portray a doubly-exponential advantage for storing a bit in bounded degree DAGs compared to trees, where $d=1$ but layer sizes need to grow exponentially with depth in order for broadcasting to be possible.