Advances in Applied Probability

Typical distances in ultrasmall random networks

Steffen Dereich, Christian Mönch, and Peter Mörters

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We show that in preferential attachment models with power-law exponent τ ∈ (2, 3) the distance between randomly chosen vertices in the giant component is asymptotically equal to (4 + o(1))log log N / (-log(τ - 2)), where N denotes the number of nodes. This is twice the value obtained for the configuration model with the same power-law exponent. The extra factor reveals the different structure of typical shortest paths in preferential attachment graphs.

Article information

Adv. in Appl. Probab., Volume 44, Number 2 (2012), 583-601.

First available in Project Euclid: 16 June 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 05C80: Random graphs [See also 60B20]
Secondary: 60C05: Combinatorial probability 90B15: Network models, stochastic

Scale-free network preferential attachment configuration model power-law graph conditionally Poissonian graph graph distance diameter


Dereich, Steffen; Mönch, Christian; Mörters, Peter. Typical distances in ultrasmall random networks. Adv. in Appl. Probab. 44 (2012), no. 2, 583--601. doi:10.1239/aap/1339878725.

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  • Bennett, G. (1962). Probability inequalities for the sum of independent random variables. J. Amer. Statist. Assoc. 57, 33–45.
  • Bollobás, B. and Riordan, O. (2004). The diameter of a scale-free random graph. Combinatorica 24, 5–34.
  • Cohen, R. and Havlin, S. (2003). Scale-free networks are ultrasmall. Phys. Rev. Lett. 90, 058701, 4 pp.
  • Chung, F. and Lu, L. (2003). The average distance in a random graph with given expected degrees. Internet Math. 1, 91–113.
  • Chung, F. and Lu, L. (2006). Complex Graphs and Networks (CBMS Regional Conf. Math. 107). American Mathematical Society, Providence RI.
  • Dommers, S., van der Hofstad, R. and Hooghiemstra, G. (2010). Diameters in preferential attachment models. J. Statist. Phys. 139, 72–107.
  • Dereich, S. and Mörters, P. (2009). Random networks with sublinear preferential attachment: degree evolutions. Electron. J. Prob. 14, 1222–1267.
  • Dereich, S. and Mörters, P. (2011). Random networks with concave preferential attachment rule. Jahresber. Deutschen Math. Ver. 113, 21–40.
  • Dereich, S. and Mörters, P. (2012). Random networks with sublinear preferential attachment: the giant component. To appear in Ann. Prob..
  • Dorogovtsev, S. N., Mendes, J. F. F. and Samukhin, A. N. (2003). Metric strucure of random networks. Nuclear Phys. B 653, 307–338.
  • Mönch, C. (2012). Distances in preferential attachment networks. Doctoral Thesis, University of Bath.
  • Norros, I. and Reittu, H. (2006). On a conditionally Poissonian graph process. Adv. Appl. Prob. 38, 59–75.
  • Norros, I. and Reittu, H. (2008). Network models with a `soft hierarchy': a random graph construction with loglog scalability. IEEE Network 22, 40–46.
  • Reittu, H. and Norros, I. (2002). On the effect of very large nodes in internet graphs. In GLOBECOM '02, Vol. III (Proc. Global Telecommun. Conf., Taipei, 2002), pp. 2624–2628.
  • Resnick, S. I. (2008). Extreme Values, Regular Variation and Point Processes. Springer, New York.
  • Van der Hofstad, R. (2010). Random graphs and complex networks. Lecture Notes, Eindhoven University of Technology.
  • Van der Hofstad, R., Hooghiemstra, G. and Znamenski, D. (2007). Distances in random graphs with finite mean and infinite variance degrees. Electron. J. Prob. 12, 703–766.
  • Van der Hofstad, R. and Hooghiemstra, G. (2008). Universality for distances in power-law random graphs. J. Math. Phys. 49, 125209, 14 pp.