In a uniform random recursive k-directed acyclic graph, there is a root, 0, and each node in turn, from 1 to n, chooses k uniform random parents from among the nodes of smaller index. If Sn is the shortest path distance from node n to the root, then we determine the constant σ such that Sn/log n→σ in probability as n→∞. We also show that max 1≤i≤nSi/log n→σ in probability.
L. Devroye’s research was sponsored by NSERC Grant A3456. The research was mostly done at the Institute Mittag-Leffler during the programme Discrete Probability held in 2009.
"Long and short paths in uniform random recursive dags." Ark. Mat. 49 (1) 61 - 77, April 2011. https://doi.org/10.1007/s11512-009-0118-0