2017
DOI: 10.1007/s00332-016-9355-0
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A Graph-Algorithmic Approach for the Study of Metastability in Markov Chains

Abstract: Large continuous-time Markov chains with exponentially small transition rates arise in modeling complex systems in physics, chemistry and biology. We propose a constructive graphalgorithmic approach to determine the sequence of critical timescales at which the qualitative behavior of a given Markov chain changes, and give an effective description of the dynamics on each of them. This approach is valid for both time-reversible and time-irreversible Markov processes, with or without symmetry. Central to this app… Show more

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Cited by 6 publications
(8 citation statements)
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“…64 Further details of the discrete path sampling 14,15 (DPS) methodology for the construction and analysis of kinetic networks can be found in recent reviews. 160,162,163 Markovian dynamics on a kinetic network are described by the linear master equation, [2][3][4][5]…”
Section: Master Equation Dynamicsmentioning
confidence: 99%
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“…64 Further details of the discrete path sampling 14,15 (DPS) methodology for the construction and analysis of kinetic networks can be found in recent reviews. 160,162,163 Markovian dynamics on a kinetic network are described by the linear master equation, [2][3][4][5]…”
Section: Master Equation Dynamicsmentioning
confidence: 99%
“…2 Discrete-time Markov chains 3 (DTMCs) are commonly estimated from trajectory data on a continuous potential energy landscape in the Markov State Model (MSM) framework. [4][5][6][7][8] In a complementary approach, continuous-time Markov chains 1 (CTMCs) can be mapped from a potential energy landscape by geometry optimization 9 of local stationary points in the discrete path sampling (DPS) framework. [10][11][12][13] CTMCs with a countably-infinite state space 14,15 are widely used to represent the number of each species in population dynamics [16][17][18] processes such as chemical and biochemical reaction cycles, [19][20][21][22][23][24] and can be transformed to finite Markov chains with negligible error by truncating the state space.…”
Section: Introductionmentioning
confidence: 99%
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“…[10][11][12][13][14][15][16][17][18][19][20][21][22] Standard linear algebra methods to compute MFPTs encounter numerical issues for metastable Markov chains because the separation of slow and fast timescales in the system dynamics leads to severe ill-conditioning. [23][24][25][26][27][28] Since rare events are ubiquitous in realistic models of stochastic dynamical processes, [29][30][31][32][33][34][35][36][37][38][39][40][41][42][43] more numerically stable algorithms are often required for the analysis of Markov chain dynamics. Graph transformation [44][45][46][47][48][49] (GT) provides an exact method to compute the A ← B MFPT that retains numerical precision even for strongly metastable Markov chains.…”
Section: Introductionmentioning
confidence: 99%