Continuum Cascade Model of Directed Random Graphs: Traveling Wave Analysis

TL;DR

This paper analyzes the asymptotic behavior of the longest directed path and cascade tree size in a class of directed random graphs using traveling wave techniques, revealing their growth characteristics.

Contribution

It introduces a traveling wave analysis approach to study the asymptotic properties of directed paths and cascade trees in a novel class of random graphs.

Findings

Asymptotic behavior of longest path length derived

Growth rate of cascade tree size characterized

Traveling wave techniques effectively applied

Abstract

We study a class of directed random graphs. In these graphs, the interval [0,x] is the vertex set, and from each y\in [0,x], directed links are drawn to points in the interval (y,x] which are chosen uniformly with density one. We analyze the length of the longest directed path starting from the origin. In the large x limit, we employ traveling wave techniques to extract the asymptotic behavior of this quantity. We also study the size of a cascade tree composed of vertices which can be reached via directed paths starting at the origin.