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oai:drops-oai.dagstuhl.de:8331 2024-03-06T10:42:09Z ddc:004 open_access

Toward a Theory of Markov Influence Systems and their Renormalization Chazelle, Bernard Markov influence systems nonlinear Markov chains dynamical systems renormalization graph sequence parsing Nonlinear Markov chains are probabilistic models commonly used in physics, biology, and the social sciences. In "Markov influence systems" (MIS), the transition probabilities of the chains change as a function of the current state distribution. This work introduces a renormalization framework for analyzing the dynamics of MIS. It comes in two independent parts: first, we generalize the standard classification of Markov chain states to the dynamic case by showing how to "parse" graph sequences. We then use this framework to carry out the bifurcation analysis of a few important MIS families. In particular, we show that irreducible MIS are almost always asymptotically periodic. We also give an example of "hyper-torpid" mixing, where a stationary distribution is reached in super-exponential time, a timescale that cannot be achieved by any Markov chain. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Bernard Chazelle 2018 Is Part Of LIPIcs, Volume 94, 9th Innovations in Theoretical Computer Science Conference (ITCS 2018) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.ITCS.2018.58 urn:nbn:de:0030-drops-83317 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2018.58 eng https://creativecommons.org/licenses/by/3.0/legalcode