## Reactive flows and unproductive cycles for random walks on complex networks

Please always quote using this URN: urn:nbn:de:0297-zib-54239

- We present a comprehensive theory for analysis and understanding of transition events between an initial set A and a target set B for general ergodic finite-state space Markov chains or jump processes, including random walks on networks as they occur, e.g., in Markov State Modelling in molecular dynamics. The theory allows us to decompose the probability flow generated by transition events between the sets A and B into the productive part that directly flows from A to B through reaction pathways and the unproductive part that runs in loops and is supported on cycles of the underlying network. It applies to random walks on directed networks and nonreversible Markov processes and can be seen as an extension of Transition Path Theory. Information on reaction pathways and unproductive cycles results from the stochastic cycle decomposition of the underlying network which also allows to compute their corresponding weight, thus characterizing completely which structure is used how often in transition events. The new theory is illustrated by an application to a Markov State Model resulting from weakly damped Langevin dynamics where the unproductive cycles are associated with periodic orbits of the underlying Hamiltonian dynamics.

Author: | Ralf Banisch, Natasa Djurdjevac Conrad, Christof Schütte |
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Document Type: | ZIB-Report |

Tag: | Complex networks; Markow State Methods; cycle decomposition; molecular transition networks; reactive trajectories; transition path theory |

MSC-Classification: | 05-XX COMBINATORICS (For finite fields, see 11Txx) / 05Cxx Graph theory (For applications of graphs, see 68R10, 81Q30, 81T15, 82B20, 82C20, 90C35, 92E10, 94C15) / 05C81 Random walks on graphs |

90-XX OPERATIONS RESEARCH, MATHEMATICAL PROGRAMMING / 90Bxx Operations research and management science / 90B15 Network models, stochastic | |

Date of first Publication: | 2015/03/27 |

Series (Serial Number): | ZIB-Report (15-19) |

ISSN: | 1438-0064 |

Published in: | Appeared in: The European Physical Journal Special Topics, vol. 224, issue 12 ,(2015) pp. 2369-2387 |

DOI: | https://doi.org/10.1140/epjst/e2015-02417-8 |