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From observed transitions to hidden paths in Markov networks

Alexander M. Maier1, Udo Seifert1, and Jann van der Meer1,2

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Phys. Rev. Research 7, 033067 – Published 17 July, 2025

DOI: https://doi.org/10.1103/p4k1-1dvb

Abstract

The number of observable degrees of freedom is typically limited in experiments. Here we consider discrete Markov networks in which an observer has access to a few visible transitions and the waiting times between these transitions. Focusing on the underlying structure of a discrete network, we present methods to infer local and global properties of the network from observed data. First, we derive bounds on the microscopic entropy production along the hidden paths between two visible transitions, which complement extant bounds on mean entropy production and affinities of hidden cycles. Second, we demonstrate how the operationally accessible data encodes information about the topology of shortest hidden paths, which can be used to identify potential clusters of states or exclude their existence. Finally, we outline a systematic way to combine the inferred data, resulting in an algorithm that finds the candidates for a minimal graph of the underlying network, i.e., a graph that is part of the original one and compatible with the observations. Our results highlight the interplay between thermodynamic methods, waiting-time distributions, and topological aspects like network structure, which can be expected to provide novel insights in other setups of coarse graining as well.

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Outline

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Phys. Rev. Research 7, 033067– Published 17 July, 2025
  • Received 16 April 2025
  • Accepted 24 June 2025
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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