Long and short paths in uniform random recursive dags

TL;DR

This paper analyzes the asymptotic behavior of shortest path lengths in uniform random recursive k-dags, establishing a constant ratio for typical and maximum distances as the graph size grows.

Contribution

It determines the limiting constant for shortest path distances in uniform random recursive k-dags, extending understanding of their structural properties.

Findings

S_n/log(n) converges to a constant    in probability

Maximum shortest path length scaled by log(n) also converges to    in probability

Provides asymptotic analysis of path lengths in recursive directed acyclic graphs

Abstract

In a uniform random recursive k-dag, 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 S_n is the shortest path distance from node n to the root, then we determine the constant \sigma such that S_n/log(n) tends to \sigma in probability as n tends to infinity. We also show that max_{1 \le i \le n} S_i/log(n) tends to \sigma in probability.