We consider a directed graph on the integers with a directed edge from vertex i to j present with probability n-1, whenever i < j, independently of all other edges. Moreover, to each edge (i, j) we assign weight n-1(j - i). We show that the closure of vertex 0 in such a weighted random graph converges in distribution to the Poisson-weighted infinite tree as n → ∞. In addition, we derive limit theorems for the length of the longest path in the subgraph of the Poisson-weighted infinite tree which has all vertices at weighted distance of at most ρ from the root.
