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Toward a Theory of Markov Influence Systems and their Renormalization

Author(s): Chazelle, Bernard

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Abstract: 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.
Publication Date: 2018
Citation: Chazelle, Bernard. "Toward a Theory of Markov Influence Systems and their Renormalization." 9th Innovations in Theoretical Computer Science Conference (ITCS) (2018): pp. 58:1-58:18. doi:10.4230/LIPIcs.ITCS.2018.58
DOI: 10.4230/LIPIcs.ITCS.2018.58
ISSN: 1868-8969
Pages: 58:1 - 58:18
Type of Material: Conference Article
Journal/Proceeding Title: 9th Innovations in Theoretical Computer Science Conference (ITCS)
Version: Final published version. This is an open access article.

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