Two Directed Non-planar Random Networks and Their Scaling Limits

We study two directed non-planar random graphs, each of which has a dependence structure. We prove that each of these models, under diffusive scaling, converges to the Brownian web. To obtain this, we first obtain a Markovian renewal structure of the paths of the graph and then study the coalescence time of any two paths. Finally, we show that the condition required by Coletti and Valle (Ann Inst Henri Poincare Prob Stat 50:899-919, 2014) in their study of the diffusive scaling limit of a generalized Howard model of drainage can be relaxed.