English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

We study the heavy path decomposition of conditional Galton-Watson trees. Ina standard Galton-Watson tree conditional on its size $n$, we order allchildren by their subtree sizes, from large (heavy) to small. A node is markedif it is among the $k$ heaviest nodes among its siblings. Unmarked nodes andtheir subtrees are removed, leaving only a tree of marked nodes, which we callthe $k$-heavy tree. We study various properties of these trees, including theirsize and the maximal distance from any original node to the $k$-heavy tree. Inparticular, under some moment condition, the $2$-heavy tree is with highprobability larger than $cn$ for some constant $c > 0$, and the maximaldistance from the $k$-heavy tree is $O(n^{1/(k+1)})$ in probability. As aconsequence, for uniformly random Apollonian networks of size $n$, the expectedsize of the longest simple path is $\Omega(n)$.

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