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Longest paths in random Apollonian networks and largest r-ary subtrees of random d-ary recursive trees

  • Andrea Collevecchio (a1), Abbas Mehrabian (a2) and Nick Wormald (a1)
Abstract

Let r and d be positive integers with r<d. Consider a random d-ary tree constructed as follows. Start with a single vertex, and in each time-step choose a uniformly random leaf and give it d newly created offspring. Let 𝒯 d,t be the tree produced after t steps. We show that there exists a fixed δ<1 depending on d and r such that almost surely for all large t, every r-ary subtree of 𝒯 d,t has less than t δ vertices. The proof involves analysis that also yields a related result. Consider the following iterative construction of a random planar triangulation. Start with a triangle embedded in the plane. In each step, choose a bounded face uniformly at random, add a vertex inside that face and join it to the vertices of the face. In this way, one face is destroyed and three new faces are created. After t steps, we obtain a random triangulated plane graph with t+3 vertices, which is called a random Apollonian network. We prove that there exists a fixed δ<1, such that eventually every path in this graph has length less than t 𝛿, which verifies a conjecture of Cooper and Frieze (2015).

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Copyright
Corresponding author
* Postal address: School of Mathematical Sciences, 9 Rainforest Walk, Clayton, VIC 3800, Australia.
** Email address: andrea.collevecchio@monash.edu
*** Postal address: Department of Combinatorics and Optimization, University of Waterloo, 200 University Avenue West, Waterloo, ON N2L 3G1 Canada. Email address: amehrabi@uwaterloo.ca
**** Email address: nick.wormald@monash.edu
References
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[1] Albenque, M. and Marckert, J.-F. (2008).Some families of increasing planar maps.Electron. J. Prob. 13,16241671.
[2] Collevecchio, A.,Mehrabian, A. and Wormald, N. (2014).Longest paths in Apollonian networks and largest r-ary subtrees of random d-ary recursive trees. Available at http://arxiv.org/abs/1404.2425.
[3] Cooper, C. and Frieze, A. (2015).Long paths in random Apollonian networks.Internet Math. 11,308318.
[4] Cooper, C.,Frieze, A. and Uehara, R. (2014).The height of random $k$-trees and related branching processes.Random Structures Algorithms 45,675702.
[5] Drmota, M. (2009).Random Trees: An Interplay Between Combinatorics and Probability.Springer,Vienna.
[6] Ebrahimzadeh, E. et al. (2014).On longest paths and diameter in random Apollonian networks.Random Structures Algorithms 45,703725.
[7] Frieze, A. and Tsourakakis, C. E. (2014).Some properties of random Apollonian networks.Internet Mathematics 10,162187.
[8] Kolossváry, J. and Vág#x00F3;, L. (2013).Degrees and distances in random and evolving Apollonian networks.Preprint. Available at http://arxiv.org/abs/1310.3864v1.
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[10] Mungan, M. (2011).Comment on ‘Apollonian networks: Simultaneously scale-free, small world, Euclidean, space filling, and with matching graphs’.Phys. Rev. Lett. 106,029802.
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[12] Zhang, Z.,Comellas, F.,Fertin, G. and Rong, L. (2006).High-dimensional Apollonian networks.J. Phys. A 39,18131818.
[13] 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.
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Journal of Applied Probability
  • ISSN: 0021-9002
  • EISSN: 1475-6072
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