Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
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)
Boyd, S., Diaconis, P., Parrilo, P., Xiao, L.: Fastest mixing Markov chain on graphs with symmetries. SIAM J. Optim. 20, 792–819 (2009)
Boyd, S., Diaconis, P., Xiao, L.: Fastest mixing Markov chain on a graph. SIAM Rev. 46, 667–689 (2004). (electronic)
Diaconis, P., Holmes, S., Neal, R.M.: Analysis of a nonreversible Markov chain sampler. Ann. Appl. Probab. 10, 726–752 (2000)
Durrett, R.: Random Graph Dynamics. Cambridge University Press, Cambridge (2006)
Gerencsér, B.: Mixing times of Markov chains on a cycle with additional long range connections. arXiv:1401.1692 (2014)
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)
Krivelevich, M., Reichman, D., Samotij, W.: Smoothed analysis on connected graphs. SIAM J. Discrete Math. 29, 1654–1669 (2015)
Levin, D., Peres, Y., Wilmer, E.: Markov Chains and Mixing Times. American Mathematical Society, Providence (2009)
Lovász, L., Vempala, S.: Hit-and-run from a corner. SIAM J. Comput. 35, 985–1005 (2006)
Lovász, L., Vempala, S.: Simulated annealing in convex bodies and an \({O}^*(n^4)\) volume algorithm. J. Comput. Syst. Sci. 72, 392–417 (2006)
Montenegro, R., Tetali, P.: Mathematical aspects of mixing times in Markov chains, Foundations and Trends®. Theor. Comput. Sci. 1, 237–354 (2006)
Nedić, A., Olshevsky, A.: Distributed optimization over time-varying directed graphs. IEEE Trans. Automat. Contr. 60, 601–615 (2015)
Nedić, A., Ozdaglar, A., Parrilo, P.A.: Constrained consensus and optimization in multi-agent networks. IEEE Trans. Automat. Contr. 55, 922–938 (2010)
Newman, M., Moore, C., Watts, D.: Mean-field solution of the small-world network model. Phys. Rev. Lett. 84, 3201–3204 (2000)
Olshevsky, A., Tsitsiklis, J.N.: Convergence speed in distributed consensus and averaging. SIAM J. Control Optim. 48, 33–55 (2009)
Author information
Authors and Affiliations
Corresponding author
Additional information
B. Gerencsér is supported by NKFIH (National Research, Development and Innovation Office) Grant PD 121107. This work has been carried out during his stay at Université catholique de Louvain, Belgium. The work is supported by the DYSCO Network (Dynamical Systems, Control, and Optimization), funded by the Interuniversity Attraction Poles Programme, initiated by the Belgian Federal Science Policy Office, and by the Concerted Research Action (ARC) of the French Community of Belgium.
Rights and permissions
About this article
Cite this article
Gerencsér, B., Hendrickx, J.M. Improved Mixing Rates of Directed Cycles by Added Connection. J Theor Probab 32, 684–701 (2019). https://doi.org/10.1007/s10959-018-0861-x
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10959-018-0861-x