Continuum Cascade Model of Directed Random Graphs: Traveling Wave   Analysis
  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. 
Yoshiaki Itoh, P. L. Krapivsky
math.PR cond-mat.stat-mech q-bio.PE