## The Annals of Applied Probability

### Exceptional times of the critical dynamical Erdős–Rényi graph

#### 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
Revised: August 2017
First available in Project Euclid: 9 August 2018

https://projecteuclid.org/euclid.aoap/1533780273

Digital Object Identifier
doi:10.1214/17-AAP1357

Mathematical Reviews number (MathSciNet)
MR3843829

Zentralblatt MATH identifier
06974751

#### 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

#### 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.