Optimal steering to invariant distributions for networks flows

TL;DR

This paper develops methods for optimally controlling network flows to reach desired invariant distributions using entropic measures, with special cases for reversible priors and connections to Boltzmann distributions.

Contribution

It introduces a novel framework for steering Markov chains to invariant distributions minimizing entropic distance, including reversible cases and thermodynamic analogies.

Findings

Optimal steering strategies derived for finite and infinite horizons.

Reversible priors lead to reversible solutions.

Connections established between network flow control and thermodynamic concepts.

Abstract

We derive novel results on the ergodic theory of irreducible, aperiodic Markov chains. We show how to optimally steer the network flow to a stationary distribution over a finite or infinite time horizon. Optimality is with respect to an entropic distance between distributions on feasible paths. When the prior is reversible, it shown that solutions to this discrete time and space steering problem are reversible as well. A notion of temperature is defined for Boltzmann distributions on networks, and problems analogous to cooling (in this case, for evolutions in discrete space and time) are discussed.