The Annals of Applied Probability

Exceptional times of the critical dynamical Erdős–Rényi graph

Matthew I. Roberts and Batı Şengül

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

In this paper, we introduce a network model which evolves in time, and study its largest connected component. We consider a process of graphs $(G_{t}:t\in [0,1])$, where initially we start with a critical Erdős–Rényi graph ER$(n,1/n)$, and then evolve forward in time by resampling each edge independently at rate $1$. We show that the size of the largest connected component that appears during the time interval $[0,1]$ is of order $n^{2/3}\log^{1/3}n$ with high probability. This is in contrast to the largest component in the static critical Erdős–Rényi graph, which is of order $n^{2/3}$.

Article information

Source
Ann. Appl. Probab., Volume 28, Number 4 (2018), 2275-2308.

Dates
Received: October 2016
Revised: August 2017
First available in Project Euclid: 9 August 2018

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1533780273

Digital Object Identifier
doi:10.1214/17-AAP1357

Mathematical Reviews number (MathSciNet)
MR3843829

Zentralblatt MATH identifier
06974751

Subjects
Primary: 05C80: Random graphs [See also 60B20] 82C20: Dynamic lattice systems (kinetic Ising, etc.) and systems on graphs 60F17: Functional limit theorems; invariance principles

Keywords
Noise sensitivity Erdős–Renyi dynamical random graphs temporal networks giant component

Citation

Roberts, Matthew I.; Şengül, Batı. Exceptional times of the critical dynamical Erdős–Rényi graph. Ann. Appl. Probab. 28 (2018), no. 4, 2275--2308. doi:10.1214/17-AAP1357. https://projecteuclid.org/euclid.aoap/1533780273


Export citation

References

  • [1] Addario-Berry, L., Broutin, N. and Goldschmidt, C. (2012). The continuum limit of critical random graphs. Probab. Theory Related Fields 152 367–406.
  • [2] Aldous, D. (1997). Brownian excursions, critical random graphs and the multiplicative coalescent. Ann. Probab. 25 812–854.
  • [3] Benjamini, I., Häggström, O., Peres, Y. and Steif, J. E. (2003). Which properties of a random sequence are dynamically sensitive? Ann. Probab. 31 1–34.
  • [4] Benjamini, I., Kalai, G. and Schramm, O. (1999). Noise sensitivity of Boolean functions and applications to percolation. Inst. Hautes Études Sci. Publ. Math. 90 5–43.
  • [5] Bollobás, B. (2001). Random Graphs, 2nd ed. Cambridge Studies in Advanced Mathematics 73. Cambridge Univ. Press, Cambridge.
  • [6] Broman, E. I. and Steif, J. E. (2006). Dynamical stability of percolation for some interacting particle systems and $\epsilon$-movability. Ann. Probab. 34 539–576.
  • [7] Durrett, R. (2010). Random Graph Dynamics. Cambridge Series in Statistical and Probabilistic Mathematics 20. Cambridge Univ. Press, Cambridge.
  • [8] Erdős, P. and Rényi, A. (1960). On the evolution of random graphs. Magy. Tud. Akad. Mat. Kutató Int. Közl. 5 17–61.
  • [9] Garban, C. (2011). Oded Schramm’s contributions to noise sensitivity. In Selected Works of Oded Schramm. Volume 1, 2. Sel. Works Probab. Stat. 287–350. Springer, New York.
  • [10] Garban, C., Pete, G. and Schramm, O. (2010). The Fourier spectrum of critical percolation. Acta Math. 205 19–104.
  • [11] Garban, C. and Steif, J. E. (2015). Noise Sensitivity of Boolean Functions and Percolation. Institute of Mathematical Statistics Textbooks 5. Cambridge Univ. Press, New York.
  • [12] Häggström, O. and Pemantle, R. (1999). On near-critical and dynamical percolation in the tree case. Random Structures Algorithms 15 311–318.
  • [13] Häggström, O., Peres, Y. and Steif, J. E. (1997). Dynamical percolation. Ann. Inst. Henri Poincaré Probab. Stat. 33 497–528.
  • [14] Hammond, A., Pete, G. and Schramm, O. (2015). Local time on the exceptional set of dynamical percolation and the incipient infinite cluster. Ann. Probab. 43 2949–3005.
  • [15] Holme, P. and Saramäki, J. (2012). Temporal networks. Phys. Rep. 519 97–125.
  • [16] Janson, S., Łuczak, T. and Rucinski, A. (2000). Random Graphs. Wiley, New York.
  • [17] Jonasson, J. and Steif, J. E. (2016). Volatility of Boolean functions. Stochastic Process. Appl. 126 2956–2975.
  • [18] Lubetzky, E. and Steif, J. E. (2015). Strong noise sensitivity and random graphs. Ann. Probab. 43 3239–3278.
  • [19] Peres, Y., Schramm, O. and Steif, J. E. (2009). Dynamical sensitivity of the infinite cluster in critical percolation. Ann. Inst. Henri Poincaré Probab. Stat. 45 491–514.
  • [20] Pittel, B. (2001). On the largest component of the random graph at a nearcritical stage. J. Combin. Theory Ser. B 82 237–269.
  • [21] Roberts, M. I. (2018). The probability of unusually large components in the near-critical Erdős–Rényi graph. Adv. Appl. Probab. 50. 245–271.
  • [22] Rossignol, R. Scaling limit of dynamical percolation on the critical Erdős–Rényi random graph. Preprint. Available at arXiv:1710.09101.
  • [23] Schramm, O. and Steif, J. E. (2010). Quantitative noise sensitivity and exceptional times for percolation. Ann. of Math. (2) 171 619–672.
  • [24] Steif, J. E. (2009). A survey of dynamical percolation. In Fractal Geometry and Stochastics IV. Progress in Probability 61 145–174. Birkhäuser, Basel.
  • [25] Talagrand, M. (1994). On Russo’s approximate zero–one law. Ann. Probab. 22 1576–1587.
  • [26] van der Hofstad, R. (2017). Random Graphs and Complex Networks. Vol. 1. Cambridge Univ. Press, Cambridge.