Skip to main content
Log in

Random walk on sparse random digraphs

  • Published:
Probability Theory and Related Fields Aims and scope Submit manuscript

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1

Similar content being viewed by others

Notes

  1. Using an arbitrary deterministic ordering of the set of all tails to break ties.

References

  1. Addario-Berry, L., Balle, B., Perarnau, G.: Diameter and stationary distribution of random \(r\)-out digraphs. ArXiv e-prints (2015)

  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)

  3. Aldous, D., Diaconis, P.: Shuffling cards and stopping times. Am. Math. Mon. 93, 333–348 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  4. Barral, J.: Moments, continuité, et analyse multifractale des martingales de Mandelbrot. Probab. Theory Relat. Fields 113(4), 535–569 (1999)

    Article  MATH  Google Scholar 

  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. Ben-Hamou, A., Salez, J.: Cutoff for non-backtracking random walks on sparse random graphs. ArXiv e-prints (2015)

  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)

    Article  MathSciNet  MATH  Google Scholar 

  8. Berestycki, N., Lubetzky, E., Peres, Y., Sly, A.: Random walks on the random graph. ArXiv e-prints (2015)

  9. Chatterjee, S.: Stein’s method for concentration inequalities. Probab. Theory Related Fields 138(1), 305–321 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  10. Chen, G.-Y., Saloff-Coste, L.: The cutoff phenomenon for ergodic Markov processes. Electron. J. Probab. 13(3), 26–78 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  11. Chen, N., Litvak, N., Olvera-Cravioto, M.: Ranking algorithms on directed configuration networks. ArXiv e-prints, Sept. (2014)

  12. Cooper, C.: Random walks, interacting particles, dynamic networks: randomness can be helpful. In: Structural Information and Communication, Complexity, pp. 1–14 (2011)

  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)

    Article  MathSciNet  MATH  Google Scholar 

  14. Cooper, C., Frieze, A.: The cover time of random regular graphs. SIAM J. Discrete Math. 18(4), 728–740 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  15. Cooper, C., Frieze, A.: The cover time of sparse random graphs. Random Struct. Algorithms 30(1–2), 1–16 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  16. Cooper, C., Frieze, A.: The cover time of the preferential attachment graph. J. Comb. Theory Ser. B 97(2), 269–290 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  17. Cooper, C., Frieze, A.: The cover time of the giant component of a random graph. Random Struct. Algorithms 32(4), 401–439 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  19. Cooper, C., Frieze, A.: Vacant sets and vacant nets: component structures induced by a random walk. arXiv preprint arXiv:1404.4403 (2014)

  20. Diaconis, P.: The cutoff phenomenon in finite Markov chains. Proc. Natl. Acad. Sci. U. S. A. 93(4), 1659–1664 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  22. Diaconis, P., Shahshahani, M.: Generating a random permutation with random transpositions. Probab. Theory Relat. Fields 57(2), 159–179 (1981)

    MathSciNet  MATH  Google Scholar 

  23. Ding, J., Lubetzky, E., Peres, Y.: Mixing time of near-critical random graphs. Ann. Probab. 40(3), 979–1008 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  25. Freedman, D.A.: On tail probabilities for martingales. Ann. Probab. 3, 100–118 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  26. Lacoin, H.: The cutoff profile for the simple-exclusion process on the circle. ArXiv e-prints, Feb. (2015)

  27. Levin, D.A., Peres, Y., Wilmer, E.L.: Markov Chains and Mixing Times. American Mathematical Society, Providence (2009)

    MATH  Google Scholar 

  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)

  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)

    Article  MathSciNet  MATH  Google Scholar 

  30. Liu, Q.: On generalized multiplicative cascades. Stoch. Process. Appl. 86(2), 263–286 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  31. Liu, Q.: Asymptotic properties and absolute continuity of laws stable by random weighted mean. Stoch. Process. Appl. 95(1), 83–107 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  32. Lubetzky, E., Peres, Y.: Cutoff on all Ramanujan graphs. ArXiv e-prints (2015)

  33. Lubetzky, E., Sly, A.: Cutoff phenomena for random walks on random regular graphs. Duke Math. J. 153(3), 475–510 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Chapter  Google Scholar 

  35. Nachmias, A., Peres, Y.: Critical random graphs: diameter and mixing time. Ann. Probab. 36(4), 1267–1286 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  36. Peres, Y.: American institute of mathematics (AIM) research workshop “sharp thresholds for mixing times”, Palo Alto (2004). http://www.aimath.org/WWN/mixingtimes

  37. Rösler, U.: A fixed point theorem for distributions. Stoch. Process. Appl. 42(2), 195–214 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  38. Saloff-Coste, L.: Random walks on finite groups. In: Kesten, H. (ed.) Probability on Discrete Structures, pp. 263–346. Springer, Berlin (2004)

  39. Villani, C.: Optimal Transport, Volume 338 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer, Berlin (2009)

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Charles Bordenave.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bordenave, C., Caputo, P. & Salez, J. Random walk on sparse random digraphs. Probab. Theory Relat. Fields 170, 933–960 (2018). https://doi.org/10.1007/s00440-017-0796-7

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00440-017-0796-7

Mathematics Subject Classification

Navigation