Broadcasting On Random Networks

We study a generalization of the problem of broadcasting on trees to the setting of directed acyclic graphs (DAGs). At time 0, a source vertex Chi transmits a uniform bit along binary symmetric channels (BSCs) to a set of vertices called layer 1. Each vertex except Chi has indegree d. At time kappa >= 1, vertices at layer kappa apply d-input Boolean processing functions to their received bits and send out the results to vertices at layer kappa + 1. We say that broadcasting is possible if we can reconstruct Chi with probability of error bounded away from 1/2 using the values of all vertices at an arbitrarily deep layer k. This question is closely related to models of reliable computation and storage, probabilistic cellular automata, and information flow in biological networks.In this work, we analyze randomly constructed DAGs and demonstrate that broadcasting is only possible if the BSC noise level is below a certain (degree and function dependent) critical threshold. Specifically, for every d >= 3, we identify the critical threshold for random DAGs with layers of size Omega(log(kappa)) and majority processing functions. For d = 2, we establish a similar result for the NAND processing function. Furthermore, for odd d >= 3, we prove that the identified thresholds cannot be improved by other processing functions if reconstruction is required from a single vertex. Finally, for any BSC noise level, in quasi-polynomial or randomized polylogarithmic time in the depth, we construct deterministic bounded degree DAGs with layers of size Theta(log(kappa)) that admit reconstruction using lossless expander graphs.