Markov branching in the vertex splitting model

TL;DR

This paper introduces a Markov branching model for randomly growing trees with finite or infinite maximum degree, analyzing its properties including convergence, dimensions, and degree distributions, and relating it to existing models like preferential attachment.

Contribution

It develops a new Markov branching framework for vertex splitting trees, extending existing models and analyzing their geometric and probabilistic properties.

Findings

Convergence of growth measures to a measure on infinite trees with a single spine.

Hausdorff dimension is 1/α for the models studied.

Spectral dimension is 2/(1+α) in the caterpillar graph case.

Abstract

We study a special case of the vertex splitting model which is a recent model of randomly growing trees. For any finite maximum vertex degree $D$, we find a one parameter model, with parameter $\alpha \in [0,1]$ which has a so--called Markov branching property. When $D=\infty$ we find a two parameter model with an additional parameter $\gamma \in [0,1]$ which also has this feature. In the case $D = 3$, the model bears resemblance to Ford's $\alpha$--model of phylogenetic trees and when $D=\infty$ it is similar to its generalization, the $\alpha\gamma$--model. For $\alpha = 0$, the model reduces to the well known model of preferential attachment. In the case $\alpha > 0$, we prove convergence of the finite volume probability measures, generated by the growth rules, to a measure on infinite trees which is concentrated on the set of trees with a single spine. We show that the annealed Hausdorff dimension with respect to the infinite volume measure is $1/\alpha$. When $\gamma = 0$ the model reduces to a model of growing caterpillar graphs in which case we prove that the Hausdorff dimension is almost surely $1/\alpha$ and that the spectral dimension is almost surely $2/(1+\alpha)$. We comment briefly on the distribution of vertex degrees and correlations between degrees of neighbouring vertices.