The optimal sink and the best source in a Markov chain

TL;DR

This paper demonstrates that in finite irreducible Markov chains, the tail distributions of hitting times can be ordered, revealing a hierarchy of states as sources and sinks with distinct underlying mechanisms.

Contribution

It introduces a novel ordering of states based on hitting time tails and distinguishes the mechanisms for source and sink orderings in Markov chains.

Findings

Hitting time tail distributions can be ordered for typical Markov chains.

States can be ranked as sources or sinks based on their efficiency.

The underlying mechanisms for source and sink orderings are fundamentally different.

Abstract

It is well known that the distributions of hitting times in Markov chains are quite irregular, unless the limit as time tends to infinity is considered. We show that nevertheless for a typical finite irreducible Markov chain and for nondegenerate initial distributions the tails of the distributions of the hitting times for the states of a Markov chain can be ordered, i.e., they do not overlap after a certain finite moment of time. If one considers instead each state of a Markov chain as a source rather than a sink then again the states can generically be ordered according to their efficiency. The mechanisms underlying these two orderings are essentially different though.