  We study a class of directed random graphs. In these graphs, the interval
[0,x] is the vertex set, and from each y\in [0,x], directed links are drawn to
points in the interval (y,x] which are chosen uniformly with density one. We
analyze the length of the longest directed path starting from the origin. In
the large x limit, we employ traveling wave techniques to extract the
asymptotic behavior of this quantity. We also study the size of a cascade tree
composed of vertices which can be reached via directed paths starting at the
origin.
