Electronic Journal of Probability

Critical window for the configuration model: finite third moment degrees

Souvik Dhara, Remco van der Hofstad, Johan S.H. van Leeuwaarden, and Sanchayan Sen

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We investigate the component sizes of the critical configuration model, as well as the related problem of critical percolation on a supercritical configuration model. We show that, at criticality, the finite third moment assumption on the asymptotic degree distribution is enough to guarantee that the sizes of the largest connected components are of the order $n^{2/3}$ and the re-scaled component sizes (ordered in a decreasing manner) converge to the ordered excursion lengths of an inhomogeneous Brownian Motion with a parabolic drift. We use percolation to study the evolution of these component sizes while passing through the critical window and show that the vector of percolation cluster-sizes, considered as a process in the critical window, converge to the multiplicative coalescent process in the sense of finite dimensional distributions. This behavior was first observed for Erdős-Rényi random graphs by Aldous (1997) and our results provide support for the empirical evidences that the nature of the phase transition for a wide array of random-graph models are universal in nature. Further, we show that the re-scaled component sizes and surplus edges converge jointly under a strong topology, at each fixed location of the scaling window.

Article information

Electron. J. Probab., Volume 22 (2017), paper no. 16, 33 pp.

Received: 8 May 2016
Accepted: 26 January 2017
First available in Project Euclid: 15 February 2017

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Zentralblatt MATH identifier

Primary: 60C05: Combinatorial probability 05C80: Random graphs [See also 60B20]

critical configuration model finite third moment degree Brownian excursions with parabolic drift scaling window multiplicative coalescent universality

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Dhara, Souvik; van der Hofstad, Remco; van Leeuwaarden, Johan S.H.; Sen, Sanchayan. Critical window for the configuration model: finite third moment degrees. Electron. J. Probab. 22 (2017), paper no. 16, 33 pp. doi:10.1214/17-EJP29. https://projecteuclid.org/euclid.ejp/1487127644

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  • [1] Addario-Berry, L., Broutin, N., Goldschmidt, C., and Miermont, G. (2013). The scaling limit of the minimum spanning tree of the complete graph. arXiv:1301.1664.
  • [2] Aldous, D. (1997). Brownian excursions, critical random graphs and the multiplicative coalescent. Ann. Probab. 25, 2, 812–854.
  • [3] Aldous, D. and Limic, V. (1998). The entrance boundary of the multiplicative coalescent. Electron. J. Probab., 3(3):1–59.
  • [4] Aldous, D. and Pittel, B. (2000). On a random graph with immigrating vertices: emergence of the giant component. Random Structures Algorithms, 17(2):79–102.
  • [5] Bhamidi, S., Broutin, N., Sen, S., and Wang, X. (2014a). Scaling limits of random graph models at criticality: Universality and the basin of attraction of the Erdos-Rényi random graph. arXiv:1411.3417.
  • [6] Bhamidi, S., Budhiraja, A., and Wang, X. (2014). The augmented multiplicative coalescent, bounded size rules and critical dynamics of random graphs. Probab. Theory Related Fields 160, 3-4, 733–796.
  • [7] Bhamidi, S., Sen, S., and Wang, X. (2014c). Continuum limit of critical inhomogeneous random graphs arXiv:1404.4118.
  • [8] Bhamidi, S., van der Hofstad, R., and van Leeuwaarden, J. S. H. (2010). Scaling limits for critical inhomogeneous random graphs with finite third moments. Electron. J. Probab., 15(6):1682–1702.
  • [9] Bhamidi, S., van der Hofstad, R., and van Leeuwaarden, J. S. H. (2012). Novel scaling limits for critical inhomogeneous random graphs. Ann. Probab. 40, 6, 2299–2361.
  • [10] Bollobás, B. (1980). A probabilistic proof of an asymptotic formula for the number of labelled regular graphs. European J. Combin. 1, 4, 311–316.
  • [11] Clauset, A., Shalizi, C. R., and Newman, M. E. J. (2009). Power-law distributions in empirical data. SIAM Rev. 51, 4, 661–703.
  • [12] Dembo, A., Levit, A., and Vadlamani, S. (2014). Component sizes for large quantum erdos renyi graph near criticality. arXiv:1404.5705.
  • [13] Dhara, S., van der Hofstad, R., van Leeuwaarden, J. S. H., and Sen, S. Heavy-tailed configuration models at criticality. arXiv:1612.00650.
  • [14] Faloutsos, M., Faloutsos, P., and Faloutsos, C. (1999). On power-law relationships of the Internet topology. Comput. Commun. Rev., 29(4):251–262.
  • [15] Fountoulakis, N. (2007). Percolation on sparse random graphs with given degree sequence. Internet Math. 4, 4, 329–356.
  • [16] Janson, S. (2009). On percolation in random graphs with given vertex degrees. Electron. J. Probab., 14:87–118.
  • [17] Janson, S. (2010). Susceptibility of random graphs with given vertex degrees. J. Comb., 1(3-4):357–387.
  • [18] Janson, S. and Luczak, M. J. (2009). A new approach to the giant component problem. Random Structures Algorithms 34, 2, 197–216.
  • [19] Janson, S., Łuczak, T., and Rucinski, A. (2000). Random Graphs. Wiley, New York.
  • [20] Joseph, A. (2014). The component sizes of a critical random graph with given degree sequence. Ann. Appl. Probab. 24, 6, 2560–2594.
  • [21] Karatzas, I. and Shreve, S. E. (1991). Brownian Motion and Stochastic Calculus, Springer-Verlag, New York.
  • [22] Lipster, R. S. and Shiryayev, A. N. (1989). Theory of Martingales. Springer, Dordrecht.
  • [23] Massart, P. (1990). The tight constant in the Dvoretzky-Kiefer-Wolfowitz inequality. Ann. Probab. 18, 3, 1269–1283.
  • [24] Molloy, M. and Reed, B. (1995). A critical-point for random graphs with a given degree sequence. Random Structures Algorithms, 6(2-3):161–179.
  • [25] Nachmias, A. and Peres, Y. (2010a). Critical percolation on random regular graphs. Random Structures Algorithms, 36(2):111–148.
  • [26] Nachmias, A. and Peres, Y. (2010b). The critical random graph, with martingales. Israel J. Math., 176(1):29–41.
  • [27] Riordan, O. (2012). The phase transition in the configuration model. Combin. Probab. Comput., 21:265–299.
  • [28] Stegehuis, C., van der Hofstad, R., and van Leeuwaarden, J. S. H. (2016a). Epidemic spreading on complex networks with community structures. Sci. Rep., 6.
  • [29] Stegehuis, C., van der Hofstad, R., and van Leeuwaarden, J. S. H. (2016b). Power-law relations in random networks with communities. Phys. Rev. E, 94:012302.
  • [30] van der Hofstad, R. (2016). Random Graphs and Complex Networks, volume I. Cambridge University Press.
  • [31] van der Hofstad, R., Janssen, A. J. E. M., and van Leeuwaarden, J. S. H. (2010). Critical epidemics, random graphs, and Brownian motion with a parabolic drift. Adv. in Appl. Probab., 42(4):1187–1206.
  • [32] van der Hofstad, R., van Leeuwaarden, J. S. H., and Stegehuis, C. Mesoscopic scales in hierarchical configuration models. arXiv:1612.02668.
  • [33] Whitt, W. (2007). Proofs of the martingale FCLT. Probab. Surv. 4, 268–302.