Minimum-Weight Path in a Sparse Erd\H{o}s--R\'{e}nyi Graph with Signed Weights

TL;DR

This paper analyzes the distribution of near-minimum weight paths in a sparse Erdős–Rényi graph with signed weights, revealing a Poisson process limit influenced by branching random walk theory.

Contribution

It extends previous work on minimum-weight paths to include signed weights, providing a probabilistic limit description involving branching random walk martingales.

Findings

Rescaled weight and hopcount pairs converge to a Poisson point process.

The limiting process is characterized by the product of independent Biggins martingale limits.

Generalizes prior results from non-negative to signed weights.

Abstract

We consider a sparse Erd\H{o}s--R\'{e}nyi graph $\mathcal{G}(n,\lambda/n)$ where each edge is independently assigned a random signed weight. For two uniformly chosen vertices, we study the joint distribution of the total weights and hopcounts (number of edges) of the near-minimum weight paths connecting them. Under certain conditions on the weight distribution, which ensure in particular that these paths are typically of positive weight, we prove that the point process formed by the rescaled pairs of total weight and hopcount, converges weakly to a Poisson point process with a random intensity. This random intensity is characterized by the product of two independent copies of the Biggins martingale limit of certain branching random walk. This result generalizes the work of Daly, Schulte, and Shneer (arXiv:2308.12149) from non-negative to signed weights.