Fragmentation of Random Trees

TL;DR

This paper analyzes how random recursive trees fragment into forests through node removal, revealing a power-law size distribution with a continuously increasing exponent as the process progresses.

Contribution

It provides an analytical description of the size distribution of trees during fragmentation and extends the model to include node addition, offering new insights into dynamic tree structures.

Findings

Size density follows a power-law with exponent increasing with node removal.

The tail of the size distribution becomes steeper as more nodes are removed.

The model is extended to include node addition, analyzing growing trees.

Abstract

We study fragmentation of a random recursive tree into a forest by repeated removal of nodes. The initial tree consists of N nodes and it is generated by sequential addition of nodes with each new node attaching to a randomly-selected existing node. As nodes are removed from the tree, one at a time, the tree dissolves into an ensemble of separate trees, namely, a forest. We study statistical properties of trees and nodes in this heterogeneous forest, and find that the fraction of remaining nodes m characterizes the system in the limit N --> infty. We obtain analytically the size density phi_s of trees of size s. The size density has power-law tail phi_s ~ s^(-alpha) with exponent alpha=1+1/m. Therefore, the tail becomes steeper as further nodes are removed, and the fragmentation process is unusual in that exponent alpha increases continuously with time. We also extend our analysis to the case where nodes are added as well as removed, and obtain the asymptotic size density for growing trees.