Stable states of perturbed Markov chains

DOI10.4230/LIPICS.MFCS.2016.18zbMATH Open1398.68214arXiv1508.05299OpenAlexW2963669989MaRDI QIDQ4608577

Publication date: 21 March 2018

Abstract: Given an infinitesimal perturbation of a discrete-time finite Markov chain, we seek the states that are stable despite the perturbation, extit{i.e.} the states whose weights in the stationary distributions can be bounded away from

0

as the noise fades away. Chemists, economists, and computer scientists have been studying irreducible perturbations built with exponential maps. Under these assumptions, Young proved the existence of and computed the stable states in cubic time. We fully drop these assumptions, generalize Young's technique, and show that stability is decidable as long as

f i n O (g)

is. Furthermore, if the perturbation maps (and their multiplications) satisfy

f i n O (g)

or

g i n O (f)

, we prove the existence of and compute the stable states and the metastable dynamics at all time scales where some states vanish. Conversely, if the big-

O

assumption does not hold, we build a perturbation with these maps and no stable state. Our algorithm also runs in cubic time despite the general assumptions and the additional work. Proving the correctness of the algorithm relies on new or rephrased results in Markov chain theory, and on algebraic abstractions thereof.

Full work available at URL: https://arxiv.org/abs/1508.05299

zbMATH Keywords

shortest path metastability evolution tropical cubic time SCC

Mathematics Subject Classification ID

Analysis of algorithms and problem complexity (68Q25) Markov chains (discrete-time Markov processes on discrete state spaces) (60J10) Probability in computer science (algorithm analysis, random structures, phase transitions, etc.) (68Q87)