CiteExport$(function(){PrimeFaces.cw("TieredMenu","widget_formSmash_upper_j_idt871",{id:"formSmash:upper:j_idt871",widgetVar:"widget_formSmash_upper_j_idt871",autoDisplay:true,overlay:true,my:"left top",at:"left bottom",trigger:"formSmash:upper:exportLink",triggerEvent:"click"});}); $(function(){PrimeFaces.cw("OverlayPanel","widget_formSmash_upper_j_idt872_j_idt875",{id:"formSmash:upper:j_idt872:j_idt875",widgetVar:"widget_formSmash_upper_j_idt872_j_idt875",target:"formSmash:upper:j_idt872:permLink",showEffect:"blind",hideEffect:"fade",my:"right top",at:"right bottom",showCloseIcon:true});});

Degrees in Random Graphs and Tournament LimitsPrimeFaces.cw("AccordionPanel","widget_formSmash_some",{id:"formSmash:some",widgetVar:"widget_formSmash_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_all",{id:"formSmash:all",widgetVar:"widget_formSmash_all",multiple:true});
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2018 (English)Doctoral thesis, comprehensive summary (Other academic)
##### Description

##### Abstract [en]

##### Place, publisher, year, edition, pages

Uppsala: Department of Mathematics, 2018. , p. 26
##### Series

Uppsala Dissertations in Mathematics, ISSN 1401-2049 ; 105
##### Keywords [en]

Random graphs, degree distributions, degree sequences, graph limits, tournaments
##### National Category

Mathematics
##### Research subject

Mathematics
##### Identifiers

URN: urn:nbn:se:uu:diva-339025ISBN: 978-91-506-2677-3 (print)OAI: oai:DiVA.org:uu-339025DiVA, id: diva2:1175059
##### Public defence

2018-03-09, Polhemssalen, Ångströmlaboratoriet, Lägerhyddsvägen 1, Uppsala, 13:15 (English)
##### Opponent

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##### Supervisors

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#####

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt1209",{id:"formSmash:j_idt1209",widgetVar:"widget_formSmash_j_idt1209",multiple:true}); Available from: 2018-02-14 Created: 2018-01-17 Last updated: 2018-02-14
##### List of papers

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.

1. Asymptotic Degree Distribution Of A Duplication-Deletion Random Graph Model$(function(){PrimeFaces.cw("OverlayPanel","overlay1060850",{id:"formSmash:j_idt1259:0:j_idt1263",widgetVar:"overlay1060850",target:"formSmash:j_idt1259:0:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

2. The dominating colour of an infinite Pólya urn model$(function(){PrimeFaces.cw("OverlayPanel","overlay896568",{id:"formSmash:j_idt1259:1:j_idt1263",widgetVar:"overlay896568",target:"formSmash:j_idt1259:1:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

3. Almost sure convergence of vertex degree densities in the vertex splitting model$(function(){PrimeFaces.cw("OverlayPanel","overlay970984",{id:"formSmash:j_idt1259:2:j_idt1263",widgetVar:"overlay970984",target:"formSmash:j_idt1259:2:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

4. Eplett's theorem for self-converse generalised tournaments$(function(){PrimeFaces.cw("OverlayPanel","overlay1162325",{id:"formSmash:j_idt1259:3:j_idt1263",widgetVar:"overlay1162325",target:"formSmash:j_idt1259:3:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

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6. Tournament limits: Degree distributions, score functions and self-converseness$(function(){PrimeFaces.cw("OverlayPanel","overlay1162327",{id:"formSmash:j_idt1259:5:j_idt1263",widgetVar:"overlay1162327",target:"formSmash:j_idt1259:5:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

isbn
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