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

We study the limiting degree distribution of the vertex splitting model introduced in Ref. [ 3 ] . 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 verti...

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Bibliographic Details
Published in:Stochastic models 2016-10, Vol.32 (4), p.575-592
Main Authors: Stefansson, Sigurdur O., Thörnblad, Erik
Format: Article
Language:English
Subjects:
05C05
05C80
05C80
05C05
Almost sure convergence
Convergence
degree densities
random trees
vertex splitting
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Summary:We study the limiting degree distribution of the vertex splitting model introduced in Ref. [ 3 ] . 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.
ISSN:1532-6349
1532-4214
1532-4214
DOI:10.1080/15326349.2016.1182029