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Graph transformation and shortest paths algorithms for finite Markov chains.

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The graph transformation (GT) algorithm accurately calculates mean first-passage times in Markov chains. This study generalizes GT for path properties and analyzes challenging metastable chains, offering robust methods for dynamical process analysis.

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Area of Science:

  • Computational Chemistry and Physics
  • Statistical Mechanics
  • Dynamical Systems Theory

Background:

  • Finite Markov chains are fundamental for modeling dynamical processes.
  • Calculating first-passage times and path properties is crucial but challenging, especially for metastable systems.
  • Existing methods like kinetic path sampling can struggle with fat-tailed distributions common in realistic processes.

Purpose of the Study:

  • To provide an overview of the graph transformation (GT) algorithm's formulations.
  • To generalize the GT formalism for calculating various path properties beyond mean first-passage time.
  • To analyze first-passage path ensembles in challenging, metastable Markov chains and compare GT with other methods.

Main Methods:

  • Iterative and block formulations of the graph transformation (GT) algorithm.
  • Generalization of GT to sum of contributions from individual transitions, including path action.
  • Comparison with kinetic path sampling and the recursive enumeration algorithm (REA) for metastable Markov chains.

Main Results:

  • The graph transformation (GT) algorithm robustly computes mean first-passage properties.
  • Highest-probability paths may not be representative in metastable Markov chains.
  • Modified REA using net productive fluxes enables decomposition of reactive flux and analysis of competing dynamics.

Conclusions:

  • The generalized GT formalism offers a numerically stable approach for analyzing path properties in Markov chains.
  • Metastability significantly impacts the relevance of dominant first-passage paths.
  • Analyzing transition flux paths provides quantitative insights into complex dynamics, even in challenging metastable regimes.