Random walks on graphs with interval weights and precise marginals

TL;DR

This paper introduces a novel model of random walks on graphs with interval weights, connecting it to reversible imprecise Markov chains, and addresses computational challenges with a local optimization algorithm.

Contribution

It is the first to model reversible imprecise Markov chains with non-convex probability models and proposes an efficient local optimization method for this complex setting.

Findings

The local optimization algorithm efficiently approximates global solutions.

The model extends the theory of imprecise Markov chains to reversible cases.

Numerical tests demonstrate practical applicability and computational feasibility.

Abstract

We propose a model of random walks on weighted graphs where the weights are interval valued, and connect it to reversible imprecise Markov chains. While the theory of imprecise Markov chains is now well established, this is a first attempt to model reversible chains. In contrast with the existing theory, the probability models that have to be considered are now non-convex. This presents a difficulty in computational sense, since convexity is critical for the existence of efficient optimization algorithms used in the existing models. The second part of the paper therefore addresses the computational issues of the model. The goal is finding sets of weights which maximize or minimize expectations corresponding to multiple steps transition probabilities. In particular, we present a local optimization algorithm and numerically test its efficiency. We show that its application allows finding close approximations of the globally best solutions in reasonable time.