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Almost sure convergence of vertex degree densities in the vertex splitting model
Univ Iceland, Inst Sci, Div Math, Dunhaga 3, IS-107 Reykjavik, Iceland..
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
2016 (English)In: Communications in Statistics. Stochastic Models, ISSN 1532-6349, E-ISSN 1532-4214, Vol. 32, no 4, p. 575-592Article in journal (Refereed) Published
Abstract [en]

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.

Place, publisher, year, edition, pages
2016. Vol. 32, no 4, p. 575-592
Keyword [en]
Almost sure convergence, degree densities, random trees, vertex splitting, 05C80, 05C05
National Category
Probability Theory and Statistics
Identifiers
URN: urn:nbn:se:uu:diva-303159DOI: 10.1080/15326349.2016.1182029ISI: 000380246200003OAI: oai:DiVA.org:uu-303159DiVA, id: diva2:970984
Available from: 2016-09-15 Created: 2016-09-15 Last updated: 2018-01-17Bibliographically approved
In thesis
1. Degrees in Random Graphs and Tournament Limits
Open this publication in new window or tab >>Degrees in Random Graphs and Tournament Limits
2018 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis consists of an introduction and six papers on the topics of degree distributions in random graphs and tournaments and their limits.

The first two papers deal with a dynamic random graph, evolving in time through duplication and deletion of vertices and edges. In Paper I we study the degree densities of this model. We show that these densities converge almost surely and determine their limiting values exactly as well as asymptotically for large degrees. In Paper II we study the evolution of the maximum degree and provide a precise growth rate thereof.

Paper III deals with a dynamic random tree model known as the vertex-splitting tree model. We show that the degree densities converge almost surely and find an infinite linear system of equations which they must satisfy. Unfortunately we are not able to show that this system has a unique solution except in special cases.

Paper IV is about self-converse generalised tournaments. A self-converse generalised tournament can be seen as a matrix whose entries take values in [0,1] and whose diagonally opposite elements sum to 1. We characterise completely the marginals of such a matrix, and show that such marginals can always be realised by a self-converse generalised tournament.

In Paper V, we define and develop the theory of tournament limits and tournament kernels. We characterise transitive and irreducible tournament limits and kernels, and prove that any tournament limit and kernel has an essentially unique decomposition into irreducible tournament limits or kernels interlaced by a transitive part.

In Paper VI, we study the degree distributions of tournament limits, or equivalently, the marginals of tournament kernels. We describe precisely which distributions on [0,1] which may appear as degree distributions of tournament limits and which functions from [0,1] to [0,1] may appear as the marginals of tournament kernels. Moreover, we show that any distribution or marginal on this form may be realised by a tournament limit or tournament kernel. We also study those distributions and marginals which can be realised by a unique tournament limit or kernel, and find that only the transitive tournament limit/kernel gives rise to a degree distribution or marginal with this property.

Place, publisher, year, edition, pages
Uppsala: Department of Mathematics, 2018. p. 26
Series
Uppsala Dissertations in Mathematics, ISSN 1401-2049 ; 105
Keyword
Random graphs, degree distributions, degree sequences, graph limits, tournaments
National Category
Mathematics
Research subject
Mathematics
Identifiers
urn:nbn:se:uu:diva-339025 (URN)978-91-506-2677-3 (ISBN)
Public defence
2018-03-09, Polhemssalen, Ångströmlaboratoriet, Lägerhyddsvägen 1, Uppsala, 13:15 (English)
Opponent
Supervisors
Available from: 2018-02-14 Created: 2018-01-17 Last updated: 2018-02-14

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