Longest path distance in random circuits

TL;DR

This paper investigates the maximum path length in random directed acyclic graphs (DAGs), providing laws of large numbers for typical and minimal depths, and extending previous work on distances in such graphs.

Contribution

It offers new laws of large numbers for the typical and minimal depths in random DAGs, completing prior research on natural distances in these graphs.

Findings

Established laws of large numbers for typical depth in random DAGs

Derived large deviation bounds for the minimum depth

Extended understanding of distance properties in random DAGs

Abstract

We study distance properties of a general class of random directed acyclic graphs (DAGs). In a DAG, many natural notions of distance are possible, for there exists multiple paths between pairs of nodes. The distance of interest for circuits is the maximum length of a path between two nodes. We give laws of large numbers for the typical depth (distance to the root) and the minimum depth in a random DAG. This completes the study of natural distances in random DAGs initiated (in the uniform case) by Devroye and Janson (2009+). We also obtain large deviation bounds for the minimum of a branching random walk with constant branching, which can be seen as a simplified version of our main result.