Electronic Journal of Probability

Heavy subtrees of Galton-Watson trees with an application to Apollonian networks

Luc Devroye, Cecilia Holmgren, and Henning Sulzbach

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We study heavy subtrees of conditional Galton-Watson trees. In a standard Galton-Watson tree conditional on its size being $n$, we order all children by their subtree sizes, from large (heavy) to small. A node is marked if it is among the $k$ heaviest nodes among its siblings. Unmarked nodes and their subtrees are removed, leaving only a tree of marked nodes, which we call the $k$-heavy tree. We study various properties of these trees, including their size and the maximal distance from any original node to the $k$-heavy tree. In particular, under some moment condition, the $2$-heavy tree is with high probability larger than $cn$ for some constant $c > 0$, and the maximal distance from the $k$-heavy tree is $O(n^{1/(k+1)})$ in probability. As a consequence, for uniformly random Apollonian networks of size $n$, the expected size of the longest simple path is $\Omega (n)$. We also show that the length of the heavy path (that is, $k=1$) converges (after rescaling) to the corresponding object in Aldous’ Brownian continuum random tree.

Article information

Electron. J. Probab., Volume 24 (2019), paper no. 2, 44 pp.

Received: 8 November 2017
Accepted: 3 January 2019
First available in Project Euclid: 5 February 2019

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60J85: Applications of branching processes [See also 92Dxx] 05C80: Random graphs [See also 60B20]

branching processes fringe trees spine decomposition binary trees continuum random tree Brownian excursion exponential functionals Apollonian networks

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Devroye, Luc; Holmgren, Cecilia; Sulzbach, Henning. Heavy subtrees of Galton-Watson trees with an application to Apollonian networks. Electron. J. Probab. 24 (2019), paper no. 2, 44 pp. doi:10.1214/19-EJP263. https://projecteuclid.org/euclid.ejp/1549357219

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