Electric network for non-reversible Markov chains

TL;DR

This paper extends the electrical network analogy to non-reversible Markov chains by introducing a voltage multiplier component, enabling computation of key probabilistic quantities through electrical properties despite the loss of classical Rayleigh monotonicity.

Contribution

It introduces a novel electrical network model for non-reversible Markov chains using a voltage multiplier, generalizing classical reversible network results.

Findings

Absorption and escape probabilities can be computed via electrical properties.

Effective resistance remains a key quantity despite non-reversibility.

Classical Rayleigh monotonicity does not hold in the new network model.

Abstract

We give an analogy between non-reversible Markov chains and electric networks much in the flavour of the classical reversible results originating from Kakutani, and later Kem\'eny-Snell-Knapp and Kelly. Non-reversibility is made possible by a voltage multiplier -- a new electronic component. We prove that absorption probabilities, escape probabilities, expected number of jumps over edges and commute times can be computed from electrical properties of the network as in the classical case. The central quantity is still the effective resistance, which we do have in our networks despite the fact that individual parts cannot be replaced by a simple resistor. We rewrite a recent non-reversible result of Gaudilli\`ere-Landim about the Dirichlet and Thomson principles into the electrical language. We also give a few tools that can help in reducing and solving the network. The subtlety of our network is, however, that the classical Rayleigh monotonicity is lost.