Almost sure convergence of vertex degree densities in the vertex-splitting model

TL;DR

This paper proves that in a model of randomly growing trees, the distribution of vertex degrees converges almost surely to fixed constants, providing rigorous confirmation of earlier non-rigorous results.

Contribution

The authors establish almost sure convergence of vertex degree densities in the vertex splitting model, strengthening prior non-rigorous findings.

Findings

Vertex degree densities converge almost surely to constants.

The limiting densities satisfy a specific system of equations.

Results strengthen and rigorously confirm previous non-rigorous claims.

Abstract

We study the limiting degree distribution of the vertex splitting model introduced in \cite{DDJS:2009}. This is a model of randomly growing ordered trees, where in each time step the tree is separated into two components by splitting a vertex into two, and then inserting an edge between the two new vertices. Under some assumptions on the parameters, related to the growth of the maximal degree of the tree, we prove that the vertex degree densities converge almost surely to constants which satisfy a system of equations. Using this we are also able to strengthen and prove some previously non-rigorous results mentioned in the literature.