Fair and Biased Random Walks on Undirected Graphs and Related Entropies



The entropy rates of Markov chains (random walks) defined on connected undirected graphs are well studied in many surveys. We study the entropy rates related to the first-passage time probability distributions of fair random walks, their relative (Kullback–Leibler) entropies, and the entropy related to two biased random walks – with the random absorption of walkers and the shortest paths random walks. We show that uncertainty of first-passage times quantified by the entropy rates characterizes the connectedness of the graph. The relative entropy derived for the biased random walks estimates the level of uncertainty between connectivity and connectedness – the local and global properties of nodes in the graph.


Entropy of graphs First-passage times Random walks on graphs 


  1. 1.
    Aldous, D.J.: Lower bounds for covering times for reversible Markov chains and random walks on graphs. J. Theoret. Probab. 2, 91 (1989)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Ben-Israel, A., Greville, Th.N.E.: Generalized Inverses: Theory and Applications, 2nd edn. Springer, New York (2003)MATHGoogle Scholar
  3. 3.
    Blanchard, Ph., Volchenkov, D.: An algorithm generating scale free graphs. Phys. A Stat. Mech. Appl. 315, 677 (2002)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Blanchard, Ph., Volchenkov, D.: Intelligibility and first passage times in complex urban networks. Proc. R. Soc. A 464, 2153–2167 (2008). doi:10.1098/rspa.2007.0329MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Blanchard, Ph., Volchenkov, D.: Mathematical Analysis of Urban Spatial Networks, Series: Understanding Complex Systems, ISBN: 978-3-540-87828-5. Springer, New York (2008)Google Scholar
  6. 6.
    Boccaletti, S., Latora, V., Moreno, Y., Chavez, M., Hwang, D.-U.: Complex networks: structure and dynamics. Phys. Rep. 424, 175 (2006)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Campbell, S.L., Meyer, C.D.: Generalized Inverses of Linear transformations. Dover Publications, New York (1979)MATHGoogle Scholar
  8. 8.
    Cao, X., Handy, S.L., Mokhtarian, P.L.: The influences of the built environment and residential self-selection on pedestrian behavior: evidence from Austin, TX. Transportation 33(1), 1–20 (2006)CrossRefGoogle Scholar
  9. 9.
    Chown, M.: Equation can spot a failing neighbourhood. New Scientist 2628, 4 (2007)Google Scholar
  10. 10.
    Coppersmith, D., Tetali, P., Winkler, P.: Collisions among random walks on a graph. SIAM J. Discrete Math. 6(3), 363–374 (1993)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Cormen, T.H., Leiserson, C.E., Rivest, R.L., Stein, C.: Introduction to Algorithms, 2nd edn. MIT Press and McGraw-Hill, New York (2001). ISBN 0-262-03293-7. Chapter 21: Data structures for Disjoint Sets, pp. 498–524MATHGoogle Scholar
  12. 12.
    Cover, T.M., Thomas, J.A.: Elements of Information Theory. Wiley, New York (1991)MATHCrossRefGoogle Scholar
  13. 13.
    Dantzig, G.B., Fulkerson, R., Johnson, S.M.: Solution of a large-scale. traveling salesman problem. Oper. Res. 2, 393–410 (1954)Google Scholar
  14. 14.
    Dehmer, M.: Information processing in complex networks: Graph entropy and information functionals. Appl. Math. Comput. 201(1-2), 82–94 (2008)MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Dehmer, M., Emmert-Streib, F.: Structural information content of networks: Graph entropy based on local vertex functionals. Comput. Biol. Chem. 32(2), 131–138 (2008)MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Drazin, M.P.: Pseudo-inverses in associative rings and semigroups. Am. Math. Month. 65, 506–514 (1958)MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Faskunger, J.: The Influence of the Built Environment on Physical Activity. The Special Report of Swedish National Institute of Public Health, ISBN 978-91-7257-494-6, October 2007Google Scholar
  18. 18.
    Floriani, E., Volchenkov, D., Lima, R.: A system close to a threshold of instability. J. Phys. A Math. Gen. 36, 4771-4783 (2003)MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Fyhn, M., Hafting, T., Treves, A., Moser, M.-B., Moser, E.I.: Hippocampal remapping and grid realignment in entorhinal cortex. Nature 446, 190–194 (2007)CrossRefGoogle Scholar
  20. 20.
    Gomez-Gardenes, J., Latora, V.: Entropy rate of diffusion processes on complex networks. To appear in Rapid Communications, Phys, Rev. E (2009), E-print arXiv:0712.0278v1 [cond-mat.stat-mech] (2007)Google Scholar
  21. 21.
    Hafting, T., Fyhn, M., Molden, S., Moser, M.-B., Moser, E.I.: Microstructure of a spatial map in the entorhinal cortex. Nature 436, 801–806 (2005)CrossRefGoogle Scholar
  22. 22.
    Hillier, B.: The Common Language of Space: A Way of Looking at the Social, Economic and Environmental Functioning of Cities on a Common Basis. Bartlett School of Graduate Studies, London (2004)Google Scholar
  23. 23.
    Hillier, B., Hanson, J.: The Social Logic of Space. Cambridge University Press, Cambridge (1984)CrossRefGoogle Scholar
  24. 24.
    Jiang, B., Claramunt, C.: Topological analysis of urban street networks. Environ. Plan. B Plan. Des. 31, 151–162 (2004)CrossRefGoogle Scholar
  25. 25.
    Kac, M.: On the notion of recurrence in discrete stochastic processes. Bull. Am. Math. Soc. 53, 1002–1010 (1947). [Reprinted in Kac’s Probability, Number Theory, and Statistical Physics: Selected Papers, pp. 231–239]Google Scholar
  26. 26.
    Kullback, S.: Information Theory and Statistics. Wiley, New York (1959)MATHGoogle Scholar
  27. 27.
    Ledoux, C.-N.: L’Architecture considérée sous le rapport de l’art, des mœurs et de la législation (1804)Google Scholar
  28. 28.
    Lovász, L.: Random Walks On Graphs: A Survey. Bolyai Society Mathematical Studies 2: Combinatorics, Paul Erdös is Eighty, Keszthely (Hungary), p. 1–46 (1993)Google Scholar
  29. 29.
    McNaughton, B.L., Battaglia, F.P., Jensen, O., Moser, M.-B., Moser, E.I.: Path integration and the neural basis of the cognitive map. Nature (Rev. Neurosci.) 7, 663–678 (2006)Google Scholar
  30. 30.
    Meyer, C.D.: The role of the group generalized inverse in the theory of finite Markov chains. SIAM Rev. 17, 443-464 (1975)MathSciNetMATHCrossRefGoogle Scholar
  31. 31.
    Ortega-Andeane, P., Jiménez-Rosas, E., Mercado-Doméenech, S., Estrada-Rodrýguez, C.: Space syntax as a determinant of spatial orientation perception. Int. J. Psychol. 40(1), 11–18 (2005)CrossRefGoogle Scholar
  32. 32.
    Parzen, E.: On estimation of a probability density function and mode. Ann. Math. Stat. 33, 1065–1076 (1962)MathSciNetMATHCrossRefGoogle Scholar
  33. 33.
    Does the Built Environment Influence Physical Activity? Examining The Evidence, Committee on Physical Activity, Health, Transportation, and Land Use, Transportation Research Board, Institute Of Medicine,TRB Special Report 282, Washington, D.C. (2005)Google Scholar
  34. 34.
    Sargolini, F., Fyhn, M., Hafting, T., McNaughton, B.L., Witter, M.P., Moser, M.-B., Moser, E.I.: Conjunctive representation of position, direction, and velocity in entorhinal cortex. Science 312(5774), 758–762 (2006)CrossRefGoogle Scholar
  35. 35.
    Shlesinger, M.F.: First encounters. Nature 450(1), 40–41 (2007)CrossRefGoogle Scholar
  36. 36.
    Volchenkov, D.: Random Walks and Flights over Connected Graphs and Complex Networks. Communications in Nonlinear Science and Numerical Simulation, (2010)Google Scholar
  37. 37.
    Volchenkov, D., Blanchard, Ph.: Nonlinear diffusion through large complex networks with regular subgraphs. J. Stat. Phys. 127(4), 677–697 (2007)MathSciNetMATHCrossRefGoogle Scholar
  38. 38.
    Volchenkov, D., Blanchard, Ph.: Random walks along the streets and canals in compact cities: Spectral analysis, dynamical modularity, information, and statistical mechanics. Phys. Rev. E 75, 026104 (2007)CrossRefGoogle Scholar
  39. 39.
    Wasserman, L.: All of Statistics: A Concise Course in Statistical Inference. Springer Texts in Statistics (2005)Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Bielefeld – Bonn Stochastic Research Center (BiBoS)University of BielefeldBielefeldGermany
  2. 2.The Center of Excellence Cognitive Interaction Technology (CITEC)University of BielefeldBielefeldGermany

Personalised recommendations