Limiting properties of random graph models with vertex and edge weights

TL;DR

This paper reviews and extends results on the asymptotic behavior of heaviest paths in weighted random directed graphs, including convergence properties and applications across various fields.

Contribution

It provides a comprehensive overview, simplifies proof techniques, and extends the analysis of weighted paths in random graphs with both vertex and edge weights.

Findings

Convergence of weighted random graphs to Poisson-based models

Asymptotic analysis of longest and heaviest paths

Extension to graphs with signed edge weights

Abstract

This paper provides an overview of results, concerning longest or heaviest paths, in the area of random directed graphs on the integers along with some extensions. We study first-order asymptotics of heaviest paths allowing weights both on edges and vertices and assuming that weights on edges are signed. We aim at an exposition that summarizes, simplifies, and extends proof ideas. We also study sparse graph asymptotics, showing convergence of the weighted random graphs to a certain weighted graph that can be constructed in terms of Poisson processes. We are motivated by numerous applications, ranging from ecology to parallel computing model. It is the latter set of applications that necessitates the introduction of vertex weights. Finally, we discuss some open problems and research directions.