Journal of Theoretical Probability

, Volume 32, Issue 2, pp 684–701 | Cite as

Improved Mixing Rates of Directed Cycles by Added Connection

  • Balázs GerencsérEmail author
  • Julien M. Hendrickx


We investigate the mixing rate of a Markov chain where a combination of long distance edges and non-reversibility is introduced. As a first step, we focus here on the following graphs: starting from the cycle graph, we select random nodes and add all edges connecting them. We prove a square-factor improvement of the mixing rate compared to the reversible version of the Markov chain.


Mixing rate Random graphs Non-reversibility 

Mathematics Subject Classification (2010)

60J10 05C80 


  1. 1.
    Addario-Berry, L., Lei, T.: The mixing time of the Newman–Watts small world. In: Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms, SIAM, 18 Jan 2012, pp. 1661–1668 (2012)Google Scholar
  2. 2.
    Boyd, S., Diaconis, P., Parrilo, P., Xiao, L.: Fastest mixing Markov chain on graphs with symmetries. SIAM J. Optim. 20, 792–819 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Boyd, S., Diaconis, P., Xiao, L.: Fastest mixing Markov chain on a graph. SIAM Rev. 46, 667–689 (2004). (electronic)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Diaconis, P., Holmes, S., Neal, R.M.: Analysis of a nonreversible Markov chain sampler. Ann. Appl. Probab. 10, 726–752 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Durrett, R.: Random Graph Dynamics. Cambridge University Press, Cambridge (2006)CrossRefzbMATHGoogle Scholar
  6. 6.
    Gerencsér, B.: Mixing times of Markov chains on a cycle with additional long range connections. arXiv:1401.1692 (2014)
  7. 7.
    Jerrum, M.: Mathematical foundations of the Markov chain Monte Carlo method. In: Habib, M., McDiarmid, C., Ramirez-Alfonsin, J., Reed, B. (eds.) Probabilistic Methods for Algorithmic Discrete Mathematics, vol. 16 of Algorithms and Combinatorics, pp. 116–165. Springer, Berlin (1998)CrossRefGoogle Scholar
  8. 8.
    Krivelevich, M., Reichman, D., Samotij, W.: Smoothed analysis on connected graphs. SIAM J. Discrete Math. 29, 1654–1669 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Levin, D., Peres, Y., Wilmer, E.: Markov Chains and Mixing Times. American Mathematical Society, Providence (2009)zbMATHGoogle Scholar
  10. 10.
    Lovász, L., Vempala, S.: Hit-and-run from a corner. SIAM J. Comput. 35, 985–1005 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Lovász, L., Vempala, S.: Simulated annealing in convex bodies and an \({O}^*(n^4)\) volume algorithm. J. Comput. Syst. Sci. 72, 392–417 (2006)CrossRefzbMATHGoogle Scholar
  12. 12.
    Montenegro, R., Tetali, P.: Mathematical aspects of mixing times in Markov chains, Foundations and Trends®. Theor. Comput. Sci. 1, 237–354 (2006)zbMATHGoogle Scholar
  13. 13.
    Nedić, A., Olshevsky, A.: Distributed optimization over time-varying directed graphs. IEEE Trans. Automat. Contr. 60, 601–615 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Nedić, A., Ozdaglar, A., Parrilo, P.A.: Constrained consensus and optimization in multi-agent networks. IEEE Trans. Automat. Contr. 55, 922–938 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Newman, M., Moore, C., Watts, D.: Mean-field solution of the small-world network model. Phys. Rev. Lett. 84, 3201–3204 (2000)CrossRefGoogle Scholar
  16. 16.
    Olshevsky, A., Tsitsiklis, J.N.: Convergence speed in distributed consensus and averaging. SIAM J. Control Optim. 48, 33–55 (2009)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.MTA Alfréd Rényi Institute of MathematicsBudapestHungary
  2. 2.ICTEAM InstituteUniversité catholique de LouvainLouvain-la-NeuveBelgium

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