Random tree growth by vertex splitting

TL;DR

This paper introduces a vertex splitting model for growing planar trees, generalizing existing models, and provides a mean field theory to analyze degree distributions and fractal dimensions, supported by simulations.

Contribution

It develops a mean field framework for a new vertex splitting tree growth model, proving its accuracy in special cases and exploring its properties.

Findings

Mean field theory matches simulations in general cases.

Intrinsic Hausdorff dimension varies with model parameters.

Model generalizes preferential attachment and phylogenetic tree models.

Abstract

We study a model of growing planar tree graphs where in each time step we separate the tree into two components by splitting a vertex and then connect the two pieces by inserting a new link between the daughter vertices. This model generalises the preferential attachment model and Ford's $\alpha$-model for phylogenetic trees. We develop a mean field theory for the vertex degree distribution, prove that the mean field theory is exact in some special cases and check that it agrees with numerical simulations in general. We calculate various correlation functions and show that the intrinsic Hausdorff dimension can vary from one to infinity, depending on the parameters of the model.