Mean first-passage time calculations: comparison of the deterministic Hill’s algorithm with Monte Carlo simulations

  • M. Torchala
  • P. Chelminiak
  • P. A. Bates
Regular Article


Accurate determination of mean first-passage times (MFPTs) between any two states of a complex network still attracts considerable attention. Appropriate methods should take into account the discrepancy in MFPTs when a random walker moves first from a source to a target and then in the opposite direction. In addition, it is desirable to allow fast evaluation of mean first-passage times when transition probabilities are allowed to vary over time. For our calculations we make use of Hill’s algorithm, which enables the exact calculation of MFPTs. As we show in this work, when given a fixed distance to travel, the calculation of a particular MFPT depends on the choice of source and target points and their relative position on a lattice. We also demonstrate, when the network contains a relatively low number of cycles, that this deterministic technique provides exact results much faster in comparison to the more standard, but computationally demanding, stochastic Monte Carlo simulation method, where only approximate results can be obtained that are highly dependent on the number of walkers. Therefore, our specific implementation of Hill’s algorithm should facilitate efficient and accurate computation of MFPTs on a variety of network topologies of practical interest to the broad scientific community.


Statistical and Nonlinear Physics 

Supplementary material

10051_2012_441_MOESM1_ESM.pdf (454 kb)
Supplementary material, approximately 454 KB.


  1. 1.
    Z. Zhang, S. Gao, Eur. Phys. J. B 80, 209 (2011)ADSCrossRefGoogle Scholar
  2. 2.
    M. Shlesinger, Nature 450, 40 (2007)ADSCrossRefGoogle Scholar
  3. 3.
    A. Einstein, Ann. Phys. 17, 549 (1905)MATHCrossRefGoogle Scholar
  4. 4.
    M. von Smoluchowski, Phys. Z. 16, 318 (1915)Google Scholar
  5. 5.
    M. von Smoluchowski, Phys. Z. 17, 585 (1916)Google Scholar
  6. 6.
    E. Schroedinger, Phys. Z. 16, 289 (1915)Google Scholar
  7. 7.
    M. Kurzynski, P. Chelminiak, J. Statist. Phys. 110, 137 (2003)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    S. Condamin, O. Benichou, V. Tejedor, R. Voituriez, J. Klafter, Nature 450, 77 (2007)ADSCrossRefGoogle Scholar
  9. 9.
    G. Barton, Elements of Green Functions and Propagation: Potentials, Diffusion and Waves (Oxford University Press, 1989)Google Scholar
  10. 10.
    D. Ben-Avraham, S. Havlin, Diffusion and Reactions in Fractals and Disordered Systems (Cambridge University Press, 2000)Google Scholar
  11. 11.
    O. Benichou, C. Chevalier, J. Klafter, B. Meyer, R. Voituriez, Nat. Chem. 2, 472 (2010)CrossRefGoogle Scholar
  12. 12.
    G. Kirchhoff, Poggendorffs Ann. Phys. Chem. 72, 495 (1847)Google Scholar
  13. 13.
    T.L. Hill, J. Theo. Biol. 10, 442 (1966)CrossRefGoogle Scholar
  14. 14.
    T.L. Hill, Proc. Natl. Acad. Sci. USA 85, 2879 (1988)ADSMATHCrossRefGoogle Scholar
  15. 15.
    T.L. Hill, Proc. Natl. Acad. Sci. USA 85, 3271 (1988)ADSMATHCrossRefGoogle Scholar
  16. 16.
    T.L. Hill, Proc. Natl. Acad. Sci. USA 85, 5345 (1988)ADSMATHCrossRefGoogle Scholar
  17. 17.
    T.L. Hill, Free Energy Transduction and Biochemical Cycle Kinetics (Springer-Verlag, 1989)Google Scholar
  18. 18.
    M. Torchala, Numerical simulations of the protein dynamics influence on some biomolecular electron transfer processes, Ph.D. thesis, Adam Mickiewicz University, Poznan, Poland, 2010Google Scholar
  19. 19.
    I. Shmulevich, E.R. Dougherty, W. Zhang, Bioinformatics 18, 1319 (2002)CrossRefGoogle Scholar
  20. 20.
    S.V. Krivov, M. Karplus, Proc. Natl. Acad. Sci. USA 101, 14766 (2004)ADSCrossRefGoogle Scholar
  21. 21.
    M. Kurzynski, The Thermodynamic Machinery of Life (Springer, 2006)Google Scholar
  22. 22.
    E.W. Dijkstra, Numer. Math. 1, 269 (1959)MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    R. Sedgewick, Algorithms in C++. Graph Algorithms (Addison-Wesley, 2002)Google Scholar
  24. 24.
    S. Condamin, V. Tejedor, R. Voituriez, O. Benichou, J. Klafter, Proc. Natl. Acad. Sci. USA 105, 5675 (2008)ADSCrossRefGoogle Scholar
  25. 25.
    R. Wilson, Introduction to Graph Theory (Pearson Education Limited, 1996)Google Scholar
  26. 26.
    R. Albert, A.-L. Barabási, Rev. Mod. Phys. 74, 47 (2002)ADSMATHCrossRefGoogle Scholar
  27. 27.
    D.P. Landau, K. Binder, A guide to Monte Carlo simulations in statistical physics (Cambridge University Press, 2000)Google Scholar
  28. 28.
    M. Torchala, Adv. Cheminfo. 1, 21 (2007)Google Scholar
  29. 29.
    M. Kurzynski, M. Torchala, P. Chelminiak (2011), e-print Scholar

Copyright information

© EDP Sciences, SIF, Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Biomolecular Modelling LaboratoryCancer Research UK London Research InstituteLondonUK
  2. 2.Faculty of PhysicsAdam Mickiewicz UniversityPoznanPoland

Personalised recommendations