The Annals of Probability

Mixing time of near-critical random graphs

Jian Ding, Eyal Lubetzky, and Yuval Peres

Full-text: Open access

Abstract

Let $\mathcal{C}_{1}$ be the largest component of the Erdős–Rényi random graph $\mathcal{G}(n,p)$. The mixing time of random walk on $\mathcal{C}_{1}$ in the strictly supercritical regime, p = c/n with fixed c > 1, was shown to have order log2n by Fountoulakis and Reed, and independently by Benjamini, Kozma and Wormald. In the critical window, p = (1 + ε)/n where λ = ε3n is bounded, Nachmias and Peres proved that the mixing time on $\mathcal{C}_{1}$ is of order n. However, it was unclear how to interpolate between these results, and estimate the mixing time as the giant component emerges from the critical window. Indeed, even the asymptotics of the diameter of $\mathcal{C}_{1}$ in this regime were only recently obtained by Riordan and Wormald, as well as the present authors and Kim.

In this paper, we show that for p = (1 + ε)/n with λ = ε3n → ∞ and λ = o(n), the mixing time on $\mathcal{C}_{1}$ is with high probability of order (n/λ)log2λ. In addition, we show that this is the order of the largest mixing time over all components, both in the slightly supercritical and in the slightly subcritical regime [i.e., p = (1 − ε)/n with λ as above].

Article information

Source
Ann. Probab., Volume 40, Number 3 (2012), 979-1008.

Dates
First available in Project Euclid: 4 May 2012

Permanent link to this document
https://projecteuclid.org/euclid.aop/1336136056

Digital Object Identifier
doi:10.1214/11-AOP647

Mathematical Reviews number (MathSciNet)
MR2962084

Zentralblatt MATH identifier
1243.05217

Subjects
Primary: 05C80: Random graphs [See also 60B20] 05C81: Random walks on graphs 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)

Keywords
Random graphs random walk mixing time

Citation

Ding, Jian; Lubetzky, Eyal; Peres, Yuval. Mixing time of near-critical random graphs. Ann. Probab. 40 (2012), no. 3, 979--1008. doi:10.1214/11-AOP647. https://projecteuclid.org/euclid.aop/1336136056


Export citation

References

  • [1] Aldous, D. and Fill, J. A. Reversible Markov Chains and random walks on graphs. Unpublished manuscript. Available at http://www.stat.berkeley.edu/~aldous/RWG/book.html.
  • [2] Aldous, D., Lovász, L. and Winkler, P. (1997). Mixing times for uniformly ergodic Markov chains. Stochastic Process. Appl. 71 165–185.
  • [3] Benjamini, I., Kozma, G. and Wormald, N. C. The mixing time of the giant component of a random graph. Preprint. Available at http://arxiv.org/abs/math/0610459.
  • [4] Benjamini, I. and Mossel, E. (2003). On the mixing time of a simple random walk on the super critical percolation cluster. Probab. Theory Related Fields 125 408–420.
  • [5] Bollobás, B. (1984). The evolution of random graphs. Trans. Amer. Math. Soc. 286 257–274.
  • [6] Bollobás, B. (2001). Random Graphs, 2nd ed. Cambridge Studies in Advanced Mathematics 73. Cambridge Univ. Press, Cambridge.
  • [7] Chandra, A. K., Raghavan, P., Ruzzo, W. L., Smolensky, R. and Tiwari, P. (1996/97). The electrical resistance of a graph captures its commute and cover times. Comput. Complexity 6 312–340.
  • [8] Ding, J., Kim, J. H., Lubetzky, E. and Peres, Y. (2010). Diameters in supercritical random graphs via first passage percolation. Combin. Probab. Comput. 19 729–751.
  • [9] Ding, J., Kim, J. H., Lubetzky, E. and Peres, Y. (2011). Anatomy of a young giant component in the random graph. Random Structures Algorithms 39 139–178.
  • [10] Erdős, P. and Rényi, A. (1959). On random graphs. I. Publ. Math. Debrecen 6 290–297.
  • [11] Fountoulakis, N. and Reed, B. A. (2007). Faster mixing and small bottlenecks. Probab. Theory Related Fields 137 475–486.
  • [12] Fountoulakis, N. and Reed, B. A. (2008). The evolution of the mixing rate of a simple random walk on the giant component of a random graph. Random Structures Algorithms 33 68–86.
  • [13] Heydenreich, M. and van der Hofstad, R. (2011). Random graph asymptotics on high-dimensional tori II: Volume, diameter and mixing time. Probab. Theory Related Fields 149 397–415.
  • [14] Kim, J. H. (2006). Poisson cloning model for random graphs. In International Congress of Mathematicians. Vol. III 873–897. Eur. Math. Soc., Zürich.
  • [15] Levin, D. A., Peres, Y. and Wilmer, E. L. (2009). Markov Chains and Mixing Times. Amer. Math. Soc., Providence, RI.
  • [16] Lindvall, T. (2002). Lectures on the Coupling Method. Dover Publications Inc., Mineola, NY.
  • [17] Lovász, L. and Kannan, R. (1999). Faster mixing via average conductance. In Annual ACM Symposium on Theory of Computing (Atlanta, GA, 1999) 282–287 (electronic). ACM, New York.
  • [18] Lovász, L. and Winkler, P. (1995). Mixing of random walks and other diffusions on a graph. In Surveys in Combinatorics, 1995 (Stirling). London Mathematical Society Lecture Note Series 218 119–154. Cambridge Univ. Press, Cambridge.
  • [19] Lovász, L. and Winkler, P. (1998). Mixing times. In Microsurveys in Discrete Probability (Princeton, NJ, 1997). DIMACS Ser. Discrete Math. Theoret. Comput. Sci. 41 85–133. Amer. Math. Soc., Providence, RI.
  • [20] Łuczak, T. (1990). Component behavior near the critical point of the random graph process. Random Structures Algorithms 1 287–310.
  • [21] Łuczak, T. (1998). Random trees and random graphs. In Proceedings of the Eighth International Conference “Random Structures and Algorithms” (Poznan, 1997) 13 485–500.
  • [22] Lyons, R. and Peres, Y. (2012). Probability on Trees and Networks. Cambridge Univ. Press. To appear. Available at http://mypage.iu.edu/~rdlyons/prbtree/book.pdf.
  • [23] Mathieu, P. and Remy, E. (2004). Isoperimetry and heat kernel decay on percolation clusters. Ann. Probab. 32 100–128.
  • [24] Nachmias, A. and Peres, Y. (2007). Component sizes of the random graph outside the scaling window. ALEA Lat. Am. J. Probab. Math. Stat. 3 133–142.
  • [25] Nachmias, A. and Peres, Y. (2008). Critical random graphs: Diameter and mixing time. Ann. Probab. 36 1267–1286.
  • [26] Pitman, J. (1998). Enumerations of trees and forests related to branching processes and random walks. In Microsurveys in Discrete Probability (Princeton, NJ, 1997). DIMACS Ser. Discrete Math. Theoret. Comput. Sci. 41 163–180. Amer. Math. Soc., Providence, RI.
  • [27] Riordan, O. and Wormald, N. (2010). The diameter of sparse random graphs. Combin. Probab. Comput. 19 835–926.
  • [28] Sinclair, A. and Jerrum, M. (1989). Approximate counting, uniform generation and rapidly mixing Markov chains. Inform. and Comput. 82 93–133.
  • [29] Tetali, P. (1991). Random walks and the effective resistance of networks. J. Theoret. Probab. 4 101–109.