The heavy path approach to Galton-Watson trees with an application to Apollonian networks

TL;DR

This paper analyzes the structure of heavy path decompositions in Galton-Watson trees and applies findings to demonstrate that Apollonian networks of size n have a longest simple path of expected length proportional to n.

Contribution

It introduces the concept of the $k$-heavy tree in Galton-Watson trees and studies its properties, providing new insights into their size and structure.

Findings

The $2$-heavy tree is with high probability larger than a constant fraction of n.

The maximal distance from any node to the $k$-heavy tree is O(n^{1/(k+1)}).

Expected longest simple path in random Apollonian networks is proportional to n.

Abstract

We study the heavy path decomposition of conditional Galton-Watson trees. In a standard Galton-Watson tree conditional on its size $n$, 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 $\Omega(n)$.