## Advances in Applied Probability

### Degrees and distances in random and evolving apollonian networks

#### Abstract

In this paper we study random Apollonian networks (RANs) and evolving Apollonian networks (EANs), in d dimensions for any d≥2, i.e. dynamically evolving random d-dimensional simplices, looked at as graphs inside an initial d-dimensional simplex. We determine the limiting degree distribution in RANs and show that it follows a power-law tail with exponent τ=(2d-1)/(d-1). We further show that the degree distribution in EANs converges to the same degree distribution if the simplex-occupation parameter in the nth step of the dynamics tends to 0 but is not summable in n. This result gives a rigorous proof for the conjecture of Zhang et al. (2006) that EANs tend to exhibit similar behaviour as RANs once the occupation parameter tends to 0. We also determine the asymptotic behaviour of the shortest paths in RANs and EANs for any d≥2. For RANs we show that the shortest path between two vertices chosen u.a.r. (typical distance), the flooding time of a vertex chosen uniformly at random, and the diameter of the graph after n steps all scale as a constant multiplied by log n. We determine the constants for all three cases and prove a central limit theorem for the typical distances. We prove a similar central limit theorem for typical distances in EANs.

#### Article information

Source
Adv. in Appl. Probab., Volume 48, Number 3 (2016), 865-902.

Dates
First available in Project Euclid: 19 September 2016

Permanent link to this document
https://projecteuclid.org/euclid.aap/1474296319

Mathematical Reviews number (MathSciNet)
MR3568896

Zentralblatt MATH identifier
1348.05192

#### Citation

Kolossváry, István; Komjáthy, Júlia; Vágó, Lajos. Degrees and distances in random and evolving apollonian networks. Adv. in Appl. Probab. 48 (2016), no. 3, 865--902. https://projecteuclid.org/euclid.aap/1474296319

#### References

• Albenque, M. and Marckert, J.-F. (2008). Some families of increasing planar maps. Electron. J. Prob. 13, 1624–1671.
• Andrade, J. S., Jr., Herrmann, H. J., Andrade, R. F. S. and da Silva, L. R. (2005). Apollonian networks: simultaneously scale-free, small world, Euclidean, space filling, and with matching graphs. Phys. Rev. Lett. 94, 018702.
• Athreya, K. B. and Ney, P. E. (2004). Branching Processes. Dover, Mineola, NY.
• Barabási, A.-L. and Albert, R. (1999). Emergence of scaling in random networks. Science 286, 509–512.
• Bhamidi, S. and van der Hofstad, R. (2012). Weak disorder asymptotics in the stochastic mean-field model of distance. Ann. Appl. Prob. 22, 29–69.
• Bhamidi, S., van der Hofstad, R. and Hooghiemstra, G. (2010). First passage percolation on random graphs with finite mean degrees. Ann. Appl. Prob. 20, 1907–1965.
• Bhamidi, S., van der Hofstad, R. and Hooghiemstra, G. (2011). First passage percolation on the Erdős–Rényi random graph. Combin. Prob. Comput. 20, 683–707.
• Bhamidi, S., van der Hofstad, R. and Hooghiemstra, G. (2012). Universality for first passage percolation on sparse random graphs. Preprint. Available at http://arxiv.org/abs/1210.6839.
• Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1989). Regular Variation (Encyclopedia Math. Appl. 27). Cambridge University Press.
• Bollobás, B. and Riordan, O. (2004). The diameter of a scale-free random graph. Combinatorica 24, 5–34.
• Bollobás, B., Janson, S. and Riordan, O. (2007). The phase transition in inhomogeneous random graphs. Random Structures Algorithms 31, 3–122.
• Bollobás, B., Borgs, C., Chayes, J. and Riordan, O. (2003). Directed scale-free graphs. In Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms, ACM, New York, pp. 132–139.
• 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.
• Boyd, D. W. (1982). The sequence of radii of the Apollonian packing. Math. Comput. 39, 249–254.
• Broutin, N. and Devroye, L. (2006). Large deviations for the weighted height of an extended class of trees. Algorithmica 46, 271–297.
• Bühler, W. J. (1971). Generations and degree of relationship in supercritical Markov branching processes. Z. Wahrscheinlichkeitsth. 18, 141–152.
• Choi, V. and Golin, M. J. (2001). Lopsided trees. I. Analyses. Algorithmica 31, 240–290.
• Cooper, C., Frieze, A. and Uehara, R. (2014). The height of random $k$-trees and related branching processes. Random Structures Algorithms 45, 675–702.
• Darrasse, A. and Soria, M. (2007). Degree distribution of random Apollonian network structures and Boltzmann sampling. In 2007 Conference on Analysis of Algorithms, AofA 07, Association of Discrete Mathematics and Theoretical Computer Science, Nancy, pp. 313–324.
• Dembo, A. and Zeitouni, O. (2010). Large Deviations Techniques and Applications (Stoch. Modelling Appl. Prob. 38), 2nd edn. Springer, Berlin.
• Doye, J. P. K. and Massen, C. P. (2005). Self-similar disk packings as model spatial scale-free networks. Phys. Rev. E 71, 016128.
• Ebrahimzadeh, E. et al. (2013). On the longest paths and the diameter in random Apollonian networks. Electron. Notes Discrete Math. 43, 355–365.
• Feller, W. (1968). An Introduction to Probability Theory and Its Applications, Vol. 1, 3rd edn. John Wiley, New York.
• Frieze, A. and Tsourakakis, C. E. (2012). On certain properties of random Apollonian networks. In Algorithms and Models for the Web Graph, Springer, Berlin, pp. 93–112.
• Graham, R. L. et al. (2003). Apollonian circle packings: number theory. J. Number Theory 100, 1–45.
• Janson, S. (1999). One, two and three times log $n/n$ for paths in a complete graph with random weights. Combinatorics Prob. Comput. 8, 347–361.
• Kapoor, S. and Reingold, E. M. (1989). Optimum lopsided binary trees. J. Assoc. Comput. Mach. 36, 573–590.
• Karamata, J. (1930). Über die Hardy–Littlewoodschen Umkehrungen des Abelschen Stetigkeitssatzes. Math. Z. 32, 319–320.
• Kolossváry, I. and Komjáthy, J. (2015). First passage percolation on inhomogeneous random graphs. Adv. Appl. Prob. 47, 589–610.
• Van der Hofstad, R. (2016). Random graphs and complex networks. To appear in Cambridge Series in Statistical and probabilistic Mathematics, Vol. 1, Cambridge University Press, Cambridge.
• Zhang, Z. et al. (2008). Analytical solution of average path length for Apollonian networks. Phys. Rev. E 77, 017102.
• Zhang, Z., Rong, L. and Comellas, F. (2006). High-dimensional random Apollonian networks. Physica A 364, 610–618.
• Zhang, Z., Rong, L. and Zhou, S. (2006). Evolving Apollonian networks with small-world scale-free topologies. Phys. Rev. E 74, 046105.
• Zhang, Z., Comellas, F., Fertin, G. and Rong, L. (2006). High-dimensional Apollonian networks. J. Phys. A 39, 1811–1818.
• Zhou, T., Yan, G. and Wang, B.-H. (2005). Maximal planar networks with large clustering coefficient and power-law degree distribution. Phys. Rev. E 71, 046141.