Probability Theory and Related Fields

, Volume 170, Issue 3–4, pp 933–960 | Cite as

Random walk on sparse random digraphs

  • Charles BordenaveEmail author
  • Pietro Caputo
  • Justin Salez


A finite ergodic Markov chain exhibits cutoff if its distance to equilibrium remains close to its initial value over a certain number of iterations and then abruptly drops to near 0 on a much shorter time scale. Originally discovered in the context of card shuffling (Aldous and Diaconis in Am Math Mon 93:333–348, 1986), this remarkable phenomenon is now rigorously established for many reversible chains. Here we consider the non-reversible case of random walks on sparse directed graphs, for which even the equilibrium measure is far from being understood. We work under the configuration model, allowing both the in-degrees and the out-degrees to be freely specified. We establish the cutoff phenomenon, determine its precise window and prove that the cutoff profile approaches a universal shape. We also provide a detailed description of the equilibrium measure.

Mathematics Subject Classification

Primary 05C80 05C81 


  1. 1.
    Addario-Berry, L., Balle, B., Perarnau, G.: Diameter and stationary distribution of random \(r\)-out digraphs. ArXiv e-prints (2015)Google Scholar
  2. 2.
    Aldous, D.: Random walks on finite groups and rapidly mixing Markov chains. In: Seminar on Probability, XVII, volume 986 of Lecture Notes in Math., pp. 243–297. Springer, Berlin (1983)Google Scholar
  3. 3.
    Aldous, D., Diaconis, P.: Shuffling cards and stopping times. Am. Math. Mon. 93, 333–348 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Barral, J.: Moments, continuité, et analyse multifractale des martingales de Mandelbrot. Probab. Theory Relat. Fields 113(4), 535–569 (1999)CrossRefzbMATHGoogle Scholar
  5. 5.
    Barral, J.: Mandelbrot cascades and related topics. In: Feng, D.-J., Lau, K.-S. (eds.) Geometry and Analysis of Fractals. Springer Proceedings in Mathematics & Statistics, vol. 88, pp. 1–45. Springer, Heidelberg (2014)Google Scholar
  6. 6.
    Ben-Hamou, A., Salez, J.: Cutoff for non-backtracking random walks on sparse random graphs. ArXiv e-prints (2015)Google Scholar
  7. 7.
    Benjamini, I., Kozma, G., Wormald, N.: The mixing time of the giant component of a random graph. Random Struct. Algorithms 45(3), 383–407 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Berestycki, N., Lubetzky, E., Peres, Y., Sly, A.: Random walks on the random graph. ArXiv e-prints (2015)Google Scholar
  9. 9.
    Chatterjee, S.: Stein’s method for concentration inequalities. Probab. Theory Related Fields 138(1), 305–321 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Chen, G.-Y., Saloff-Coste, L.: The cutoff phenomenon for ergodic Markov processes. Electron. J. Probab. 13(3), 26–78 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Chen, N., Litvak, N., Olvera-Cravioto, M.: Ranking algorithms on directed configuration networks. ArXiv e-prints, Sept. (2014)Google Scholar
  12. 12.
    Cooper, C.: Random walks, interacting particles, dynamic networks: randomness can be helpful. In: Structural Information and Communication, Complexity, pp. 1–14 (2011)Google Scholar
  13. 13.
    Cooper, C., Frieze, A.: The size of the largest strongly connected component of a random digraph with a given degree sequence. Comb. Probab. Comput. 13(3), 319–337 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Cooper, C., Frieze, A.: The cover time of random regular graphs. SIAM J. Discrete Math. 18(4), 728–740 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Cooper, C., Frieze, A.: The cover time of sparse random graphs. Random Struct. Algorithms 30(1–2), 1–16 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Cooper, C., Frieze, A.: The cover time of the preferential attachment graph. J. Comb. Theory Ser. B 97(2), 269–290 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Cooper, C., Frieze, A.: The cover time of the giant component of a random graph. Random Struct. Algorithms 32(4), 401–439 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Cooper, C., Frieze, A.: Stationary distribution and cover time of random walks on random digraphs. J. Comb. Theory Ser. B 102(2), 329–362 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Cooper, C., Frieze, A.: Vacant sets and vacant nets: component structures induced by a random walk. arXiv preprint arXiv:1404.4403 (2014)
  20. 20.
    Diaconis, P.: The cutoff phenomenon in finite Markov chains. Proc. Natl. Acad. Sci. U. S. A. 93(4), 1659–1664 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Diaconis, P., Graham, R.L., Morrison, J.A.: Asymptotic analysis of a random walk on a hypercube with many dimensions. Random Struct. Algorithms 1(1), 51–72 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Diaconis, P., Shahshahani, M.: Generating a random permutation with random transpositions. Probab. Theory Relat. Fields 57(2), 159–179 (1981)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Ding, J., Lubetzky, E., Peres, Y.: Mixing time of near-critical random graphs. Ann. Probab. 40(3), 979–1008 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Fountoulakis, N., Reed, B.A.: The evolution of the mixing rate of a simple random walk on the giant component of a random graph. Random Struct. Algorithms 33(1), 68–86 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Freedman, D.A.: On tail probabilities for martingales. Ann. Probab. 3, 100–118 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Lacoin, H.: The cutoff profile for the simple-exclusion process on the circle. ArXiv e-prints, Feb. (2015)Google Scholar
  27. 27.
    Levin, D.A., Peres, Y., Wilmer, E.L.: Markov Chains and Mixing Times. American Mathematical Society, Providence (2009)zbMATHGoogle Scholar
  28. 28.
    Liu, Q.: The growth of an entire characteristic function and the tail probabilities of the limit of a tree martingale. In: Chauvin, B., Cohen, S., Rouault, A. (eds.) Trees. Progress in Probability, vol. 40, pp. 51–80. Birkhäuser, Basel (1996)Google Scholar
  29. 29.
    Liu, Q.: Sur une équation fonctionnelle et ses applications: une extension du théorème de Kesten–Stigum concernant des processus de branchement. Adv. Appl. Probab. 29(2), 353–373 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Liu, Q.: On generalized multiplicative cascades. Stoch. Process. Appl. 86(2), 263–286 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Liu, Q.: Asymptotic properties and absolute continuity of laws stable by random weighted mean. Stoch. Process. Appl. 95(1), 83–107 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Lubetzky, E., Peres, Y.: Cutoff on all Ramanujan graphs. ArXiv e-prints (2015)Google Scholar
  33. 33.
    Lubetzky, E., Sly, A.: Cutoff phenomena for random walks on random regular graphs. Duke Math. J. 153(3), 475–510 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    McDiarmid, C.: Concentration. In: Habib, M., McDiarmid, C., Ramirez-Alfonsin, J., Reed, B. (eds.) Probabilistic Methods for Algorithmic Discrete Mathematics. Algorithms and Combinatorics, vol. 16, pp. 195–248. Springer, Berlin (1998)CrossRefGoogle Scholar
  35. 35.
    Nachmias, A., Peres, Y.: Critical random graphs: diameter and mixing time. Ann. Probab. 36(4), 1267–1286 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Peres, Y.: American institute of mathematics (AIM) research workshop “sharp thresholds for mixing times”, Palo Alto (2004).
  37. 37.
    Rösler, U.: A fixed point theorem for distributions. Stoch. Process. Appl. 42(2), 195–214 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Saloff-Coste, L.: Random walks on finite groups. In: Kesten, H. (ed.) Probability on Discrete Structures, pp. 263–346. Springer, Berlin (2004)Google Scholar
  39. 39.
    Villani, C.: Optimal Transport, Volume 338 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer, Berlin (2009)Google Scholar

Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  • Charles Bordenave
    • 1
    Email author
  • Pietro Caputo
    • 2
  • Justin Salez
    • 3
  1. 1.Institut de Mathématiques de Toulouse Centre National de la Recherche Scientifique & Université Toulouse 3ToulouseFrance
  2. 2.Dipartimento di Matematica e FisicaUniversità Roma Tre Largo San Murialdo 1RomaItaly
  3. 3.Laboratoire de Probabilités et Modèles Aléatoires Université Paris DiderotParis CEDEX 13France

Personalised recommendations