  We study distance properties of a general class of random directed acyclic
graphs (DAGs). In a DAG, many natural notions of distance are possible, for
there exists multiple paths between pairs of nodes. The distance of interest
for circuits is the maximum length of a path between two nodes. We give laws of
large numbers for the typical depth (distance to the root) and the minimum
depth in a random DAG. This completes the study of natural distances in random
DAGs initiated (in the uniform case) by Devroye and Janson (2009+). We also
obtain large deviation bounds for the minimum of a branching random walk with
constant branching, which can be seen as a simplified version of our main
result.
