### The degree profile and weight in Apollonian networks and k-trees

#### Abstract

We investigate the degree profile and total weight in Apollonian networks. We study the distribution of the degrees of vertices as they age in the evolutionary process. Asymptotically, the (suitably-scaled) degree of a node with a fixed label has a Mittag-Leffler-like limit distribution. The degrees of nodes of later ages have different asymptotic distributions, influenced by the time of their appearance. The very late arrivals have a degenerate distribution. The result is obtained via triangular Pólya urns. Also, via the Bagchi-Pal urn, we show that the number of terminal nodes asymptotically follows a Gaussian law. We prove that the total weight of the network asymptotically follows a Gaussian law, obtained via martingale methods. Similar results carry over to the sister structure of the k-trees, with minor modification in the proof methods, done mutatis mutandis.

#### Article information

Source
Adv. in Appl. Probab., Volume 48, Number 1 (2016), 163-175.

Dates
First available in Project Euclid: 8 March 2016

https://projecteuclid.org/euclid.aap/1457466160

Mathematical Reviews number (MathSciNet)
MR3473572

Zentralblatt MATH identifier
1336.05124

Subjects
Primary: 05082 90B15: Network models, stochastic
Secondary: 60C05: Combinatorial probability

#### Citation

Zhang, Panpan; Mahmoud, Hosam. The degree profile and weight in Apollonian networks and k -trees. Adv. in Appl. Probab. 48 (2016), no. 1, 163--175. https://projecteuclid.org/euclid.aap/1457466160

#### References

• 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. (Erratum: 102 (2009), 079901.)
• Athreya, K. B. and Karlin, S. (1968). Embedding of urn schemes into continuous time Markov branching processes and related limit theorems. Ann. Math. Statist. 39, 1801–1817.
• Bagchi, A. and Pal, A. K. (1985). Asymptotic normality in the generalized \polya–Eggenberger urn model, with an application to computer data structures. SIAM J. Algebraic Discrete Methods 6, 394–405.
• Cooper, C. and Frieze, A. (2015). Long paths in random Apollonian networks. Internet Math. 11, 308–318.
• Cooper, C., Frieze, A. and Uehara, R. (2014). The height of random $k$-trees and related branching processes. Random Structures Algorithms 45, 675–702.
• Ebrahimzadeh, E. \et (2013). On the longest paths and the diameter in random Apollonian networks. Electron. Notes Discrete Math. 43, 355–365.
• Flajolet, P., Dumas, P. and Puyhaubert, V. (2006). Some exactly solvable models of urn process theory. in Fourth Colloquium on Mathematics and Computer Science Algorithms, Trees, Conmbinatorics and Probabilities, Association of Discrete Mathematics and Theoretical Computer Science, Nancy, pp. 59–118.
• Freedman, D. A. (1965). Bernard Friedman's urn. Ann. Math. Statist. 36, 956–970.
• Frieze, A. and Tsourakakis, C. E. (2012). On certain properties of random Apollonian networks. In Algorithms and Models for the Web Graph (Proc. WAW 2012), Springer, Berlin, pp. 93–112.
• Graham, R. L., Knuth, D. E. and Patashnik, O. (1994). Concrete Mathematics, 2nd edn. Addison-Wesley, Reading, MA.
• Hall, P. and Heyde, C. C. (1980). Martingale Limit Theory and Its Application. Academic Press, New York.
• Harary, F. and Palmer, E. M. (1968). On acyclic simplical complexes. Mathematika 15, 115–122.
• Janson, S. (2005). Asymptotic degree distribution in random recursive trees. Random Structures Algorithms 26, 69–83.
• Janson, S. (2006). Limit theorems for triangular urn schemes. Prob. Theory Relat. Fields 134, 417–452.
• Janson, S. (2010). Moments of gamma type and the Brownian supremum process area. Prob. Surveys 7, 1–52, 207–208.
• Kuba, M. and Panholzer, A. (2007). On the degree distribution of the nodes in increasing trees. J. Combin. Theory A 114, 597–618.
• Panholzer, A. and Seitz, G. (2014). Ancestors and descendants in evolving $k$-tree models. Random Structures Algorithms 44, 465–489.
• Smythe, R. T. and Mahmoud, H. M. (1996). A survey of recursive trees. Theory Prob. Math. Statist. 51, 1–27.
• Zhang, P., Chen, C. and Mahmoud, H. (2015). Explicit characterization of moments of balanced triangular \polya urns by an elementary approach. Statist. Prob. Lett. 96, 149–153.