Entropy and Hausdorff Dimension in Random Growing Trees

TL;DR

This paper studies the growth and geometric properties of random trees formed by preferential attachment, deriving explicit formulas for the Hausdorff dimension of a measure on the leaves, revealing deep fractal structure.

Contribution

It introduces a new framework for analyzing the Hausdorff dimension of measures on limiting random trees in preferential attachment models.

Findings

Hausdorff and packing dimensions are equal and constant almost surely.

The local dimension equals the Hausdorff dimension at almost every point.

An explicit formula for the dimension based on the attachment rule is provided.

Abstract

We investigate the limiting behavior of random tree growth in preferential attachment models. The tree stems from a root, and we add vertices to the system one-by-one at random, according to a rule which depends on the degree distribution of the already existing tree. The so-called weight function, in terms of which the rule of attachment is formulated, is such that each vertex in the tree can have at most K children. We define the concept of a certain random measure mu on the leaves of the limiting tree, which captures a global property of the tree growth in a natural way. We prove that the Hausdorff and the packing dimension of this limiting measure is equal and constant with probability one. Moreover, the local dimension of mu equals the Hausdorff dimension at mu-almost every point. We give an explicit formula for the dimension, given the rule of attachment.