Uniform Sampling of Negative Edge Weights in Shortest Path Networks

TL;DR

This paper introduces a maximum entropy model for sampling edge weights, including negative values, in shortest path networks without negative cycles, using an MCMC method with proven convergence and polynomial mixing time.

Contribution

It proposes a novel MCMC sampling algorithm for negative edge weights in shortest path networks, with a dynamic Johnson's algorithm implementation and empirical performance analysis.

Findings

MCMC process converges to the desired distribution.

Mixing time is polynomial on cycle graphs.

Empirical evaluation shows effective performance of the sampling algorithm.

Abstract

We consider a maximum entropy edge weight model for shortest path networks that allows for negative weights. Given a graph $G$ and possible weights $\mathcal{W}$ typically consisting of positive and negative values, the model selects edge weights $w \in \mathcal{W}^m$ uniformly at random from all weights that do not introduce a negative cycle. We propose an MCMC process and show that (i) it converges to the required distribution and (ii) that the mixing time on the cycle graph is polynomial. We then engineer an implementation of the process using a dynamic version of Johnson's algorithm in connection with a bidirectional Dijkstra search. We empirically study the performance characteristics of an implementation of the novel sampling algorithm as well as the output produced by the model.