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2019 Heavy subtrees of Galton-Watson trees with an application to Apollonian networks
Luc Devroye, Cecilia Holmgren, Henning Sulzbach
Electron. J. Probab. 24(none): 1-44 (2019). DOI: 10.1214/19-EJP263

Abstract

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.

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Luc Devroye. Cecilia Holmgren. Henning Sulzbach. "Heavy subtrees of Galton-Watson trees with an application to Apollonian networks." Electron. J. Probab. 24 1 - 44, 2019. https://doi.org/10.1214/19-EJP263

Information

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

zbMATH: 1406.60117
MathSciNet: MR3916322
Digital Object Identifier: 10.1214/19-EJP263

Subjects:
Primary: 60J80
Secondary: 05C80, 60J85

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