Annals of Applied Probability

Law of large numbers for the largest component in a hyperbolic model of complex networks

Nikolaos Fountoulakis and Tobias Müller

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We consider the component structure of a recent model of random graphs on the hyperbolic plane that was introduced by Krioukov et al. The model exhibits a power law degree sequence, small distances and clustering, features that are associated with so-called complex networks. The model is controlled by two parameters $\alpha$ and $\nu$ where, roughly speaking, $\alpha$ controls the exponent of the power law and $\nu$ controls the average degree. Refining earlier results, we are able to show a law of large numbers for the largest component. That is, we show that the fraction of points in the largest component tends in probability to a constant $c$ that depends only on $\alpha,\nu$, while all other components are sublinear. We also study how $c$ depends on $\alpha,\nu$. To deduce our results, we introduce a local approximation of the random graph by a continuum percolation model on $\mathbb{R}^{2}$ that may be of independent interest.

Article information

Ann. Appl. Probab., Volume 28, Number 1 (2018), 607-650.

Received: September 2016
Revised: March 2017
First available in Project Euclid: 3 March 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 05C80: Random graphs [See also 60B20] 05C82: Small world graphs, complex networks [See also 90Bxx, 91D30]
Secondary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65] 82B43: Percolation [See also 60K35]

Random graphs hyperbolic plane giant component law of large numbers


Fountoulakis, Nikolaos; Müller, Tobias. Law of large numbers for the largest component in a hyperbolic model of complex networks. Ann. Appl. Probab. 28 (2018), no. 1, 607--650. doi:10.1214/17-AAP1314.

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  • [1] Aizenman, M. and Newman, C. M. (1986). Discontinuity of the percolation density in one-dimensional $1/\vert x-y\vert ^{2}$ percolation models. Comm. Math. Phys. 107 611–647.
  • [2] Albert, R. and Barabási, A.-L. (2002). Statistical mechanics of complex networks. Rev. Modern Phys. 74 47–97.
  • [3] Bode, M., Fountoulakis, N. and Müller, T. (2015). On the largest component of a hyperbolic model of complex networks. Electron. J. Combin. 22 Paper 3.24, 46.
  • [4] Bode, M., Fountoulakis, N. and Müller, T. (2016). The probability of connectivity in a hyperbolic model of complex networks. Random Structures Algorithms 49 65–94.
  • [5] Bollobás, B. (2001). Random Graphs, 2nd ed. Cambridge Studies in Advanced Mathematics 73. Cambridge Univ. Press, Cambridge.
  • [6] Bollobás, B., Janson, S. and Riordan, O. (2007). The phase transition in inhomogeneous random graphs. Random Structures Algorithms 31 3–122.
  • [7] Bollobás, B. and Riordan, O. (2004). The diameter of a scale-free random graph. Combinatorica 24 5–34.
  • [8] Bollobás, B., Riordan, O., Spencer, J. and Tusnády, G. (2001). The degree sequence of a scale-free random graph process. Random Structures Algorithms 18 279–290.
  • [9] Candellero, E. and Fountoulakis, N. (2016). Clustering and the hyperbolic geometry of complex networks. Internet Math. 12 2–53.
  • [10] Chung, F. and Lu, L. (2002). The average distances in random graphs with given expected degrees. Proc. Natl. Acad. Sci. USA 99 15879–15882.
  • [11] Chung, F. and Lu, L. (2002). Connected components in random graphs with given expected degree sequences. Ann. Comb. 6 125–145.
  • [12] Chung, F. and Lu, L. (2006). Complex Graphs and Networks. CBMS Regional Conference Series in Mathematics 107. Amer. Math. Soc., Providence, RI.
  • [13] Dorogovtsev, S. N. (2010). Lectures on Complex Networks. Oxford Master Series in Physics 20. Oxford Univ. Press, Oxford.
  • [14] Erdős, P. and Rényi, A. (1959). On random graphs. I. Publ. Math. Debrecen 6 290–297.
  • [15] Fountoulakis, N. (2015). On a geometrization of the Chung–Lu model for complex networks. J. Complex Netw. 3 361–387.
  • [16] Grimmett, G. (1999). Percolation, 2nd ed. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 321. Springer, Berlin.
  • [17] Gugelmann, L., Panagiotou, K. and Peter, U. (2012). Random hyperbolic graphs: Degree sequence and clustering. In Proceedings of the 39th International Colloquium Conference on Automata, Languages, and Programming—Volume Part II, ICALP’12 573–585. Springer, Berlin.
  • [18] Janson, S., Łuczak, T. and Rucinski, A. (2000). Random Graphs. Wiley-Interscience, New York.
  • [19] Kingman, J. F. C. (1993). Poisson Processes. Oxford Studies in Probability 3. Oxford Univ. Press, New York.
  • [20] Kiwi, M. and Mitsche, D. (2015). A bound for the diameter of random hyperbolic graphs. In 2015 Proceedings of the Twelfth Workshop on Analytic Algorithmics and Combinatorics (ANALCO) 26–39. SIAM, Philadelphia, PA.
  • [21] Krioukov, D., Kitsak, M., Sinkovits, R. S., Rideout, D., Meyer, D. and Boguñá, M. (2012). Network cosmology. Nature 2 793.
  • [22] Krioukov, D., Papadopoulos, F., Kitsak, M., Vahdat, A. and Boguñá, M. (2010). Hyperbolic geometry of complex networks. Phys. Rev. E (3) 82 036106, 18.
  • [23] Meester, R. and Roy, R. (1996). Continuum Percolation. Cambridge Tracts in Mathematics 119. Cambridge Univ. Press, Cambridge.
  • [24] Mood, A. M. (1974). Introduction to the Theory of Statistics. McGraw-Hill, New York.
  • [25] Penrose, M. (2003). Random Geometric Graphs. Oxford Studies in Probability 5. Oxford Univ. Press, Oxford.
  • [26] Stillwell, J. (1992). Geometry of Surfaces. Springer, New York.
  • [27] van den Berg, J. and Keane, M. (1984). On the continuity of the percolation probability function. In Conference in Modern Analysis and Probability (New Haven, Conn., 1982). Contemp. Math. 26 61–65. Amer. Math. Soc., Providence, RI.