Graph transformation method for calculating waiting times in Markov chains.

We describe an exact approach for calculating transition probabilities and waiting times in finite-state discrete-time Markov processes. All the states and the rules for transitions between them must be known in advance. We can then calculate averages over a given ensemble of paths for both additive and multiplicative properties in a nonstochastic and noniterative fashion. In particular, we can calculate the mean first-passage time between arbitrary groups of stationary points for discrete path sampling databases, and hence extract phenomenological rate constants. We present a number of examples to demonstrate the efficiency and robustness of this approach.

[1]  Goldhirsch,et al.  Relation between the classical resistance of inhomogeneous networks and diffusion. , 1987, Physical review. B, Condensed matter.

[2]  I. Goldhirsch,et al.  Distribution functions for random walk processes on networks: An analytic method , 1987 .

[3]  Mikhail J. Atallah,et al.  Algorithms and Theory of Computation Handbook , 2009, Chapman & Hall/CRC Applied Algorithms and Data Structures series.

[4]  L. Mirny,et al.  Diffusion in correlated random potentials, with applications to DNA. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[5]  N. Kampen,et al.  Stochastic processes in physics and chemistry , 1981 .

[6]  Balakrishnan,et al.  Analytic calculation of the diffusion coefficient for random walks on strips of finite width: Dependence on size and nature of boundaries. , 1993, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[7]  Webb,et al.  Universal scaling of nonlocal and local resistance fluctuations in small wires. , 1990, Physical review. B, Condensed matter.

[8]  W. H. Weinberg,et al.  Theoretical foundations of dynamical Monte Carlo simulations , 1991 .

[9]  R. Berry,et al.  Solid‐Liquid Phase Behavior in Microclusters , 2007 .

[10]  Joseph Klafter,et al.  On mean residence and first passage times in finite one-dimensional systems , 1998 .

[11]  R. K. Shyamasundar,et al.  Introduction to algorithms , 1996 .

[12]  Dibyendu Mukherjee,et al.  Kinetic Monte Carlo simulation of the effect of coalescence energy release on the size and shape evolution of nanoparticles grown as an aerosol , 2003 .

[13]  David J. Wales,et al.  Kinetic analysis of discrete path sampling stationary point databases , 2006, cond-mat/0604165.

[14]  Sidney Redner,et al.  Scaling of the first-passage time and the survival probability on exact and quasi-exact self-similar structures , 1989 .

[15]  M. Slemrod,et al.  DYNAMICS OF FIRST ORDER PHASE TRANSITIONS , 1984 .

[16]  David J. Wales,et al.  The free energy landscape and dynamics of met-enkephalin , 2003 .

[17]  Nico M. van Dijk.  Queueing networks and product forms - a systems approach , 1993, Wiley-Interscience series in systems and optimization.

[18]  N. Meyers,et al.  H = W. , 1964, Proceedings of the National Academy of Sciences of the United States of America.

[19]  Robert E. Tarjan,et al.  Fibonacci heaps and their uses in improved network optimization algorithms , 1987, JACM.

[20]  M. B. Priestley.  Probability: An Introduction , 1961 .

[21]  David A. Reed,et al.  Surface diffusivity and the time correlation of concentration fluctuations , 1981 .

[22]  31a-PS-52 Multifractals of Normalized First Passage Time in Sierpinski Gasket , 1997, cond-mat/9804146.

[23]  I. Goldhirsch,et al.  Biased diffusion on random networks: mean first passage time and DC conductivity , 1985 .

[24]  Gunter Bolch,et al.  Queueing Networks and Markov Chains , 2005 .

[25]  A. B. Bortz,et al.  A new algorithm for Monte Carlo simulation of Ising spin systems , 1975 .

[26]  W. Ebeling.  Stochastic Processes in Physics and Chemistry , 1995 .

[27]  Landauer,et al.  Diffusive traversal time: Effective area in magnetically induced interference. , 1987, Physical review. B, Condensed matter.

[28]  Mean first-passage and residence times of random walks on asymmetric disordered chains , 2003, cond-mat/0302408.

[29]  Pandey,et al.  Conductivity exponent for stirred superconductor-insulator mixtures. , 1987, Physical review. A, General physics.

[30]  Paul C. Bressloff,et al.  A `sum-over-paths' approach to diffusion on trees , 1996 .

[31]  G. Grimmett,et al.  Probability and random processes , 2002 .

[32]  R. Tao.  COMMENT: Studies of the spectral dimension for branched Koch curves , 1987 .

[33]  Weblog Wikipedia,et al.  In Wikipedia the Free Encyclopedia , 2005 .

[34]  Julius Jellinek,et al.  Energy Landscapes: With Applications to Clusters, Biomolecules and Glasses , 2005 .

[35]  David J. Wales,et al.  Energy landscapes of model glasses. II. Results for constant pressure , 2003 .

[36]  G. Chartrand.  Introductory Graph Theory , 1984 .

[37]  David Stirzaker,et al.  One Thousand Exercises in Probability , 2001 .

[38]  P. Gács,et al.  Algorithms , 1992 .

[39]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[40]  Dan Suciu,et al.  Journal of the ACM , 2006 .

[41]  Murthy,et al.  Validity of the mean-field approximation for diffusion on a random comb. , 1996, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[42]  Novotny.  Monte Carlo algorithms with absorbing Markov chains: Fast local algorithms for slow dynamics. , 1995, Physical review letters.

[43]  Joseph W. Haus,et al.  Diffusion in regular and disordered lattices , 1987 .

[44]  Guy Pujolle,et al.  Introduction to queueing networks , 1987 .

[45]  Thomas H. Cormen,et al.  Introduction to algorithms [2nd ed.] , 2001 .

[46]  Wilkinson,et al.  Transport and dispersion in random networks with percolation disorder. , 1988, Physical review. A, General physics.

[47]  T. Ala‐Nissila,et al.  Exact and efficient discrete random walk method for time-dependent two-dimensional environments. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[48]  Goldhirsch,et al.  Biased random walk on networks. , 1987, Physical review. A, General physics.

[49]  J. Doye,et al.  Characterizing the network topology of the energy landscapes of atomic clusters. , 2004, The Journal of chemical physics.

[50]  Kurt Binder,et al.  Dynamics of First Order Phase Transitions , 1975 .

[51]  M. Klein,et al.  Coarse-grain molecular dynamics simulations of diblock copolymer surfactants interacting with a lipid bilayer , 2004 .

[52]  David J Wales,et al.  Folding of the GB1 hairpin peptide from discrete path sampling. , 2004, The Journal of chemical physics.

[53]  M. Raykin.  First-passage probability of a random walk on a disordered one-dimensional lattice , 1993 .

[54]  V. Balakrishnan,et al.  Diffusion coefficient for random walks on strips with spatially inhomogeneous boundaries , 1993 .

[55]  Jean-Claude Latombe,et al.  Stochastic roadmap simulation for the study of ligand-protein interactions , 2002, ECCB.

[56]  G. Dienes,et al.  Radiation Effects in Solids , 1953 .

[57]  Goldhirsch,et al.  First-passage-time distribution in a random random walk. , 1990, Physical review. A, Atomic, molecular, and optical physics.

[58]  Thomas W. Reps,et al.  An Incremental Algorithm for a Generalization of the Shortest-Path Problem , 1996, J. Algorithms.

[59]  S. Redner,et al.  Random walk in a random multiplicative environment , 1989 .

[60]  L. Mirny,et al.  Kinetics of protein-DNA interaction: facilitated target location in sequence-dependent potential. , 2004, Biophysical journal.

[61]  E. Schöll,et al.  Kinetic Monte Carlo simulation of formation of microstructures in liquid droplets , 2004 .

[62]  R. Whetten,et al.  Statistical thermodynamics of the cluster solid-liquid transition. , 1990, Physical review letters.

[63]  Sei‐Yong Kim,et al.  CRYSTALLINITY AND AVERAGE GRAIN SIZE OF FILMS GROWN BY CHEMICAL VAPOR DEPOSITION , 1995 .

[64]  I. Goldhirsch,et al.  The building blocks of random walks , 1989 .

[65]  D. Wales,et al.  A doubly nudged elastic band method for finding transition states. , 2004, The Journal of chemical physics.

[66]  C. Broeck,et al.  Transport properties on a random comb , 1995 .

[67]  M. G. Bulmer,et al.  Principles of Statistics. , 1969 .

[68]  Havlin,et al.  Mean first-passage time on loopless aggregates. , 1989, Physical review. A, General physics.

[69]  MULTIFRACTAL MEASURES CHARACTERIZED BY THE ITERATIVE MAP WITH TWO CONTROL PARAMETERS , 2000, cond-mat/0009074.

[70]  Goldhirsch,et al.  Analytic method for calculating properties of random walks on networks. , 1986, Physical review. A, General physics.

[71]  Vijay S Pande,et al.  Using path sampling to build better Markovian state models: predicting the folding rate and mechanism of a tryptophan zipper beta hairpin. , 2004, The Journal of chemical physics.

[72]  Thomas W. Reps,et al.  On the Computational Complexity of Dynamic Graph Problems , 1996, Theor. Comput. Sci..

[73]  You-yan Liu,et al.  Matrix method for random walks on lattices , 1995 .

[74]  V. Pereyra,et al.  Collective surface diffusion: n-fold way kinetic Monte Carlo simulation , 1998 .

[75]  Jean-Claude Latombe,et al.  Stochastic roadmap simulation: an efficient representation and algorithm for analyzing molecular motion , 2002, RECOMB '02.

[76]  D. Wales.  Discrete path sampling , 2002 .

[77]  Murthy,et al.  Mean first-passage time of random walks on a random lattice. , 1989, Physical review. A, General physics.

[78]  A. Heuer.  Energy Landscapes. Applications to Clusters, Biomolecules and Glasses. By David J. Wales. , 2005 .